# Concentration inequalities for the hydrodynamic limit of a two-species
stochastic particle system

Joseph Klobusicky111Department of Mathematics, The University of Scranton,
Scranton PA, 18510 (joseph.klobusicky@scranton.edu).

###### Abstract

We study a stochastic particle system which is motivated from grain boundary
coarsening in two-dimensional networks. Each particles lives on the positive
real line and is labeled as belonging to either Species 1 or Species 2.
Species 1 particles drift at unit speed toward the origin, while Species 2
particles do not move. When a particle in Species 1 hits the origin, it is
removed, and a randomly selected particle mutates from Species 2 to Species 1.
The process described is an example of a high-dimensional piecewise
deterministic Markov process (PDMP), in which deterministic flow is punctuated
with stochastic jumps. Our main result is a proof of exponential concentration
inequalities of the Kolmogorov-Smirnoff distance between empirical measures of
the particle system and solutions of limiting nonlinear kinetic equations. Our
method of proof involves a time and space discretization of the kinetic
equations, which we compare with the particle system to derive recurrence
inequalities for comparing total numbers in small intervals. To show these
recurrences occur with high probability, we appeal to a state dependent
Hoeffding type inequality at each time increment.

Keywords: piecewise deterministic Markov process, concentration inequality,
kinetic theory, grain boundary coarsening, functional law of large numbers

##  1 Introduction

An important topic in material science is the coarsening of network
microstructures such as polycrystalline metals and soap froths. Through
heating of metals or gas diffusion of foams, coarsening is induced from the
migration of network boundaries to minimize interfacial surface energy. While
tracking individual boundaries remains a multifaceted and active field of
research in numerical [3] and geometric [8] analysis, in the 1950’s von
Neumann [15] and Mullins [13] proved a simple relation between the topology
and geometry of a single cell in two-dimensional networks with isotropic
surface tension evolving through curve-shortening flow. Specifically, a cell
with area $A$ and $n$ sides has a constant growth rate

$\frac{dA}{dt}=c(n-6),$ (1)

where $c$ is a material constant. When a cell with fewer than six sides
shrinks to a point, neighboring cells may change their number of sides to
maintain the topological requirement that exactly three edges meet at at each
junction. Therefore, a grain will typically change its number of sides, and
therefore its rate of growth, several times during the coarsening process.

Several physicists [6, 5, 12, 1] used (1) as a starting point in writing
kinetic equations for densities $u_{n}(a,t)$ of cells having $n$ sides
($n$-gons) and area $a$ at time $t$. These take the form of constant
convection transport equations with intrinsic flux terms, given by

$\partial_{t}u_{n}+c(n-6)\partial_{a}u_{n}=\sum_{l=2}^{5}(l-6)u_{l}(0,t)\left(\sum_{m=2}^{M}A_{lm}(t)u_{m}(a,t)\right),\quad
n\geq 2.$ (2)

Models diverge in their choice of the matrix $A_{lm}$, which prescribes mean
field rules for how networks change topology when a grain vanishes. In [10],
Menon, Pego, and the author presented a stochastic particle system, the
$M$-species model, as an intermediate between kinetic equations and direct
simulations of grain boundary coarsening. The focus of [10] was with well-
posedness of the limiting kinetic equations and simulations. It remained,
however, to provide estimates for convergence rates of the particle systems to
their hydrodynamic limits. A study was conducted in [9] on a simplified model
of one species, in which total loss of particles is shown to be equivalent to
a diminishing urn process similar to Pittel’s model of cannibalistic behavior
[14].

In this paper, we build the groundwork for establishing concentration
inequalities for the $M$-species model by restricting our attention to a model
of two species. Specifically, each particle lives in
$\mathbb{R}_{+}=[0,\infty)$ and is tagged as belonging to either Species 1 or
Species 2. Particles in Species 2 do not drift, while particles in Species 1
drift at unit speed toward the origin. When a particle reaches the origin, it
is removed, and a particle from Species 2 immediately mutates into Species 1.
For a visual representation of the process in which mutations are represented
as vertical jumps, see Fig. 1. The process just described can be interpreted
as a minimal model of network coarsening, with the behavior of particles in
Species 1 analogous to the constant area decrease of cells with fewer than six
sides. The removal of Species 1 and mutation of Species 2 are similar to the
vanishing of faces and subsequent reassignment of neighboring cell topologies.

The hydrodynamic limits for densities $f_{j}(x,t)$ of particles in Species $j$
at position $x>0$ and time $t\geq 0$ are transport equations with nonlinear
intrinsic source terms, with

$\displaystyle\partial_{t}f_{1}(x,t)-\partial_{x}f_{1}(x,t)$
$\displaystyle=\frac{f_{1}(0,t)f_{2}(x,t)}{N_{2}(t)},$ (3)
$\displaystyle\partial_{t}f_{2}(x,t)$
$\displaystyle=-\frac{f_{1}(0,t)f_{2}(x,t)}{N_{2}(t)},$ (4) $\displaystyle
f_{1}(x,0)=\bar{f}_{1}(x),\quad$ $\displaystyle f_{2}(x,0)=\bar{f}_{2}(x),$
(5)

where $N_{j}(t)=\int_{0}^{\infty}f_{j}(x,t)dx$ is the total number of Species
$j$. To allow for nondifferentiable initial conditions, we will exclusively
work with the integral form of (3)-(5), written with Duhamel’s formula as

$\displaystyle f_{1}(x,t)$
$\displaystyle=\bar{f}_{1}(x+t)+\int_{0}^{t}f_{2}(x+t-s,s)\frac{f_{1}(0,s)}{N_{2}(s)}ds,$
(6) $\displaystyle f_{2}(x,t)$
$\displaystyle=\bar{f}_{2}(x)-\int_{0}^{t}f_{2}(x,s)\frac{f_{1}(0,s)}{N_{2}(s)}ds.$
(7)

The main result for this paper is the convergence of empirical measures of the
particle system to limiting kinetic equations (6)-(7). In Section 2, we give a
rigorous description of the particle system as a piecewise determinstic Markov
process (PDMP), lay out a deterministic discretization of the kinetic
equations, and present the main results. Section 3 gives a proof for Theorem
2, which shows that the discretization converges to the kinetic equations at
rate $\mathcal{O}(\delta+\omega(\delta,0))$, where $\delta$ is both the
spatial and temporal step size of the scheme, and $\omega(\delta,0)$ is the
modulus of continuity in the initial conditions. This is achieved through
writing recurrence relations which compare total numbers restricted to
intervals of size $\delta$. Section 4 is a proof of Theorem 3, an exponential
concentration inequality (with respect to the initial total number of
particles) between the discretization and particle system. This involves
similar recurrence inequalities seen in Section 3, but now with an added task
of showing that the inequalities occur under high probability. We will need to
apply a generalization of Hoeffding’s inequality [7], a fundamental
concentration inequality for sampling without replacement, for establishing
estimates on the total number of mutations occurring in intervals.

Theorems 2 and 3 can be combined to produce our main result, Theorem 4, which
gives a concentration inequality between empirical measures and solutions of
the kinetic equation. For sufficiently small $\varepsilon>0$ and
$n>n(\varepsilon)$, the inequality takes the form

$\displaystyle\mathbb{P}_{n}\left(\sup_{t\leq
T^{\prime}}d((\mu_{1}(t),\mu_{2}(t)),(\mu^{n}_{1}(t),\mu^{n}_{2}(t)))\geq\varepsilon\right)\leq\frac{C}{\varepsilon^{2}}\exp(-\tilde{C}\varepsilon^{5}n).$
(8)

Here, for $j=1,2$ and time $0\leq t\leq T^{\prime}$, $\mu^{n}_{j}(t)$ is the
$n$-particle empirical measure for positions of Species $j$ , and
$\mu_{j}(t,dx)=f_{j}(t,x)dx$ where $f_{j}(x,t)$ is the solution of (6)-(7).
The metric $d$ is a sum of Kolmogorov-Smirnov metrics between measures of each
species. The constants $C$, $\tilde{C}$, and $T^{\prime}$ all depend on the
initial conditions $\bar{f}$.

We conclude with Section 5, in which we derive an explicit solution of the
kinetic equations and prove the well-posedness stated in Theorem 1, relying on
several well-known facts from renewal theory. We stress that explicit
solutions are not used in either the proofs of Theorems 2 and 3, as we hope to
extend the methods used here to the $M$-species model which have no known
explicit solutions.

##  2 Particle model and statement of results

Figure 1: A schematic of the particle system. Particles in Species 1 (top
line) drift toward the origin at unit speed. When a particle (shown in black)
reaches the origin, it is removed. Another particle (shown in grey, with
dashed outline immediately before mutation and solid outline after) is
randomly selected from Species 2 (bottom line) to mutate into Species 1\.

###  2.1 A two-species particle system and its kinetic limit

We now formally define the stochastic process $\\{X^{n}(t)\\}_{t\geq 0}$ for
an initial system of $n$ particles. Each particle lives in one of two ordered
copies of $\mathbb{R}_{+}=(0,\infty)$, which we refer to as Species 1 and
Species 2. Since particles may be removed during the process, the state space
$E^{n}$ consists of states

$\mathbf{x}=\\{(x_{i},s_{i}):i=1,\dots,|\mathbf{x}|,\quad|\mathbf{x}|\leq
n\\},\quad$ (9)

with particle locations $x_{i}\in[0,\infty)$ and labels $s_{i}\in\\{1,2\\}$
denoting each particle’s species. The state space can be expressed as a
disjoint union of positive orthants

$E^{n}=\coprod_{l+m\leq n}E^{n}_{(l,m)},$ (10)

with $E^{n}_{(l,m)}=\mathbb{R}_{+}^{l}\times\mathbb{R}_{+}^{m}$ denoting
positions for $l$ particles in Species 1 and $m$ particles in Species 2.

Fix an initial state
$X^{n}(0)=\\{(x_{1}^{0},s_{1}^{0}),\dots,(x_{n}^{0},s_{n}^{0})\\}\in E^{n}$,
and denote $\alpha$ as an index for a particle in Species 1 closest to the
origin, meaning $x_{\alpha}^{0}\leq x_{i}^{0}$ for $i=1,\dots,n$ and
$s_{\alpha}^{0}=1$. Now let $\tau_{1}=x_{\alpha}^{0}$ denote the time until a
particle reaches the origin. Define $X^{n}(t)\in E^{n}$ for $t\in[0,\tau_{1})$
deterministically by advecting particles in Species 1 toward the origin at
unit speed while keeping particles in Species 2 fixed:

$s_{i}(t)=s_{i}^{0},\quad x_{i}(t)=\begin{cases}x_{i}^{0}-t,&s_{i}^{0}=1,\\\
x_{i}^{0},&s_{i}^{0}=2\\\ \end{cases},\qquad i=1,\dots,n.$ (11)

Randomness is introduced with a mutation at time $t=\tau_{1}$. At this time,
the smallest particle in Species 1 has reached the origin, and is removed from
the system. Furthermore, a particle $(x_{\iota}(\tau_{1}^{-}),2)$ selected
with uniform probability from Species 2 mutates while keeping its position,
meaning

$(x_{\iota}(\tau_{1}),s_{\iota}(\tau_{1}))=(x_{\iota}(\tau_{1}^{-}),1).$ (12)

Finally, particle indices $i>\alpha$ decrement by one so that the index set is
{1, …, $n-1\\}$. The (now stochastic) process then repeats deterministic drift
until a particle from Species 1 reaches the origin at some time $\tau_{2}$,
again triggering a random mutation, and the process continues until there are
no particles left in Species 2. For instances in which multiple particles
reach the origin simultaneously, particles in Species 2 are selected to mutate
by sampling without replacement. The process $\\{X(t)\\}_{t\geq 0}$ is an
example of the general $M$-species model, which was shown in [10] to be a
class of piecewise deterministic Markov processes (PDMPs). The stochastic
process induces a filtered probability space
$(\Omega,\mathcal{F},\mathbb{P}_{n},\\{\mathcal{F}(t)\\}_{t\geq 0})$, where
$\mathcal{F}$ is the natural filtration. Davis [2] established that PDMPs are
in fact strong Markov, which we will use in Section 4 when considering events
before and after certain mutation times.

At time $t\geq 0$, denote the number of particles in Species $j$ as
$N_{j}^{n}(t)$, and the total number as $N^{n}(t)=N_{1}^{n}(t)+N_{2}^{n}(t)$.
Empirical densities for species with $n$ initial particles are then defined as

$\mu^{n}_{j}(t)=\frac{1}{n}\sum_{i=1}^{N^{n}(t)}\delta(x_{i})\cdot\mathbf{1}(s_{i}=j),\quad\quad
j=1,2.$ (13)

The differential form of the limiting equations of the infinite particle limit
$n\rightarrow\infty$ are given by equations (3)-(5), in which for each species
$j$ the limiting measures
$\lim_{n\rightarrow\infty}\mu_{j}^{n}\rightarrow\mu_{j}$ are deterministic
with densities $f_{j}(x,t).$ We will require that that $N_{1}(0)+N_{2}(0)=1$
and $N_{2}(0)>0$. The left hand sides of (3) and (4) represent the constant
drift of Species 1 toward the origin and zero drift in Species 2. The right
hand sides give the intrinsic flux arising from mutations selected from a
normalized density of $f_{2}$, occurring at a frequency of $f_{1}(0,t)$. To
allow for nondifferentiable initial data and solutions, we will use the
integral form (6)-(7) with initial data $(\bar{f}_{1},\bar{f}_{2})\in Z^{2}$,
where $Z$ denotes the cone of positive, continuous, and locally bounded
functions under the $L^{1}(\mathbb{R}_{+})$ norm topology. Equations (6)-(7)
reach a singularity when $N_{2}=0$, corresponding to when there are no more
Species 2 particles to mutate. This occurs at time

$T(\bar{f})=\sup_{t>0}\\{N_{2}(t)>0\\}.$ (14)

The derivation of an explicit solution of (6)-(7) first relies on explicitly
solving for the removal rate, which we write as $a(t)=f_{1}(0,t)$, and
subsequently the total loss

$L(t)=\int_{0}^{t}a(s)ds,$ (15)

which may be interpreted as an “internal clock” to the system, counting
normalized total visits to the origin or, equivalently, total mutations.

######  Theorem 1.

Let $\bar{f}=(\bar{f}_{1},\bar{f}_{2})\in Z^{2}$ with $N_{2}(0)>0$,

(a) The removal rate $a(t)=f(0,t)$ and $N_{2}(t)$ may be written in terms of
the initial conditions as

$\displaystyle a(t)$
$\displaystyle=\sum_{j=0}^{\infty}\left(\frac{\bar{f}_{2}}{N_{2}(0)}\right)^{*(j)}(t)*\bar{f}_{1}(t),$
(16) $\displaystyle N_{2}(t)$ $\displaystyle=N_{2}(0)-L(t).$ (17)

Here, an exponent of $*(j)$ denotes $j$-fold self convolution (with
$f^{*(0)}=1$). For $0\leq t<T(\bar{f})$, the solution
$(f_{1}(x,t),f_{2}(x,t))\in Z^{2}$ of (6)-(7) is unique and has the explicit
form

$\displaystyle f_{1}(x,t)$
$\displaystyle=\bar{f}_{1}(x+t)+\int_{0}^{t}\frac{\bar{f}_{2}(x+t-s)}{N_{2}(0)}a(s)ds,$
(18) $\displaystyle f_{2}(x,t)$
$\displaystyle=\frac{N_{2}(t)}{N_{2}(0)}\bar{f}_{2}(x).$ (19)

(b) For $\bar{f}_{1},\bar{f}_{2}\in Z$ and $0\leq t<T(\bar{f})$, (16)-(19)
defines a continuous dynamical system in $Z^{2}$, so that the map
$(\bar{f}_{1},\bar{f}_{2},t)\mapsto(f_{1}(\cdot,t),f_{2}(\cdot,t))$ is in
$C(Z^{2}\times[0,T(\bar{f})],Z^{2})$.

The proof for Theorem 1 is postponed until Section 5, as neither the explicit
solution nor its derivation will be used in future results. The well-posedness
of (6)-(7) is invoked for defining constants in the convergence analysis of
Sections 3 and 4. Note, however, that the particle system in this paper is a
special case of the $M$-species model developed in [10], in which well-
posedness is derived through a Banach fixed point argument, rather than
appealing to an explicit solution.

###  2.2 Discretization scheme of kinetic equations

To enable us to write down recurrence inequalities involving total numbers
restricted to an interval, we will construct a deterministic scheme for
(6)-(7). We do so with measures
$\tilde{\mu}_{1}(t,\cdot;\delta),\tilde{\mu}_{2}(t,\cdot;\delta)\in\mathcal{M}(\mathbb{R}_{+})$
at time $t>0$ which are piecewise constant in $\delta>0$ sized time intervals
$\Delta t_{k}=[\delta(k-1),\delta k)$ for $k\geq 1$. Note that while these
measures depend on $\delta$, we will often suppress this argument in the
notation for simplicity in presentation.

Initial measures are given by

$\tilde{\mu}_{j}(t,\cdot)=\bar{\mu}_{j},\quad t\in[0,\delta),\quad j=1,2,$
(20)

with the requirement that $(\bar{\mu}_{1}+\bar{\mu}_{2})([0,\infty))=1$. For
each time step $t_{k}=k\delta$, we define the incremental loss over a time
interval as

$\Delta\tilde{L}(t_{k})=\begin{cases}0,&k=0,\\\
\tilde{\mu}_{1}(t_{k-1},[0,\delta)),&k\geq 1.\\\ \end{cases}\quad$ (21)

Total number for Species 2 then decreases by the incremental loss, with

$\tilde{N}_{2}(t_{k})=\begin{cases}\tilde{\mu}_{2}(0,[0,\infty)),&k=0,\\\
\tilde{N}_{2}(t_{k-1})-\Delta\tilde{L}(t_{k}),&k\geq 1.\\\ \end{cases}\quad$
(22)

Measures update by a shift of distance $\delta$ toward the origin in Species 1
followed by mutation in Species 2 of total number $\Delta\tilde{L}(t_{k})$.
Therefore, for $t\in[t_{k},t_{k+1})$, $k\geq 1$, we update with

$\displaystyle\tilde{\mu}_{1}(t)$
$\displaystyle=S_{\delta}(\tilde{\mu}_{2}(t_{k-1}))+\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1}),$
(23) $\displaystyle\tilde{\mu}_{2}(t)$
$\displaystyle=\tilde{\mu}_{2}(t_{k-1})\left(1-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\right).\quad$
(24)

Here $S_{h}$ is the left translation operator acting on measures, defined
through the cumulative function $F_{\mu}$ of a measure
$\mu\in\mathcal{M}(\mathbb{R}_{+})$ by

$S_{h}(F_{\mu}(x))=F_{\mu}(x+h)-F_{\mu}(h),\quad h\geq 0.$ (25)

Since Species 1 shifts before adding mutated particles from Species 2, we have
a conserved quantity $\tilde{N}_{1}(t)=\tilde{N}_{1}(0)$, and thus the total
number

$\tilde{N}(t_{k}):=\tilde{N}_{1}(t_{k})+\tilde{N}_{2}(t_{k})=1-\sum_{i=1}^{k-1}\Delta\tilde{L}(t_{i}).$
(26)

This scheme remains well-defined as long as $\tilde{N}_{2}(t)>0$. It is clear
that

$\tilde{N}_{2}(t)\geq\tilde{N}_{2}(0)-(\bar{\mu}_{1}([0,t])+\bar{\mu}_{2}[0,t]),$
(27)

so for initial measures in $Z$ we are always able to find a nonzero length
interval of existence $[0,T_{1}(\bar{f}))$, with

$T_{1}(\bar{f})=\sup\\{t:\tilde{N}_{2}(t)>\tilde{N}_{2}(0)/2\\}.$ (28)

###  2.3 Main results: convergence rates and concentration inequalities

Our first main result gives a comparison between the deterministic
discretization and solutions of (6)-(7) using the Kolmogorov-Smirnov (KS)
metric. For two measures $\nu,\eta\in\mathcal{M}(\mathbb{R}_{+}),$ with
associated cumulative functions $F_{\nu}(x)=\nu([0,x])$ and
$F_{\eta}(x)=\eta([0,x])$ the KS metric is defined as

$d_{KS}(\nu,\eta)=\sup_{x\in\mathbb{R}}|F_{\nu}(x)-F_{\eta}(x)|.$ (29)

For handling convergence of both species, we define a metric on
$\mathcal{M}(\mathbb{R}_{+})\times\mathcal{M}(\mathbb{R}_{+})$ between
$\nu=(\nu_{1},\nu_{2})$ and $\eta=(\eta_{1},\eta_{2})$ by

$d(\nu,\eta)=d_{KS}(\nu_{1},\eta_{1})+d_{KS}(\nu_{2},\eta_{2}).$ (30)

As we are often working with measures, we define measures associated to
solutions of (6)-(7) by

$\mu_{j}(t,dx)=f_{j}(x,t)dx\quad j=1,2.$ (31)

We will also need to track the modulus of continuity for solutions. For
densities $(f_{1}(x,t),f_{2}(x,t))$, we let

$\omega(\delta,t)=\sup_{x\in\mathbb{R}_{+}}\sum_{j=1}^{2}|f_{j}(x+\delta,t)-f_{j}(x,t)|.$
(32)

We will impose that initial conditions have compact support for the sake of
clarity, as calculations relating convergence and the decay of initial
conditions can become rather technical. Since initial conditions are
continuous, compact support implies that $\omega(\delta,0)\rightarrow 0$ as
$\delta\rightarrow 0$.

######  Theorem 2.

Let $\bar{f}=(\bar{f}_{1},\bar{f}_{2})\in Z^{2}$ have compact support with
$N_{2}(0)>0$. For $t<T(\bar{f})$, let $\mu(t)=(\mu_{1}(t),\mu_{2}(t))$ be
measures for the unique solution to (6)-(7). Let measures
$\tilde{\mu}(t)=(\tilde{\mu}_{1}(t),\tilde{\mu}_{2}(t))$ for the
discretization scheme with step size of $\delta>0$ also have initial
conditions $\bar{f}$. Then there exist positive constants $\delta^{d}$ and
$C^{d}$ such that for all $\delta\in(0,\delta^{d})$ and
$T^{\prime}\in[0,T(\bar{f}))$,

$\sup_{t\in[0,T^{\prime}]}d(\tilde{\mu}(t),\mu(t))\leq
C_{d}(\delta+\omega(\delta,0)).$ (33)

The constants $C^{d}$ and $\delta^{d}$ are dependent on $L^{\infty}$ and
$L^{1}$ bounds of solutions in Theorem 1, and the compact support bound
$M=\sup\\{x:\sum_{j=1,2}\bar{f}_{j}(x)>0\\}$.

To set up for the next main result, we generate initial conditions
$\bar{\mu}^{n}=(\bar{\mu}^{n}_{1},\bar{\mu}^{n}_{2})$ for the particle system
with uniform spacing through the cumulative distribution functions
$F_{j}(x)=\mu_{j}([0,x])$ for $j=1,2$. The explicit particle positions are

$\displaystyle(x_{i}(0),s_{i}(0))=\begin{cases}(F_{1}^{-1}(i/n),1)&i=1,\dots,\lfloor
nN_{1}(0)\rfloor,\\\ (F_{2}^{-1}((nN_{1}(0)-i)/n),2)&i=\lfloor
nN_{1}(0)\rfloor+1,\dots,n,\\\ \end{cases}$ (34)

where $F^{-1}$ is the quantile function of a cumulative function $F$. For
initial conditions $(\bar{f}_{1},\bar{f}_{2})\in Z^{2}$ with
$\bar{\mu}_{j}(dx)=\bar{f}_{j}dx$, it is easy to show that
$d(\bar{\mu}^{n},\bar{\mu})\rightarrow 0$ in law as $n\rightarrow\infty$, and
that there exists a positive integer $n_{0}(\bar{f})$ such that if
$n>n_{0}(\bar{f})$,

$\bar{\mu}_{1}^{n}([0,T_{1}(\bar{f})])+\bar{\mu}_{2}^{n}([0,T_{1}(\bar{f})])\leq
2N_{2}^{n}(0)/3\quad\hbox{ for }n>n_{0}(\bar{f}),$ (35)

meaning that there is always a particle available to mutate at each jumping
time in $[0,T_{1}(\bar{f})]$, and the process therefore does not reach its
cemetery state.

The next theorem, which we show in Section 4, gives an exponential
concentration inequality between the deterministic discretization and the
particle system.

######  Theorem 3.

Let $\bar{f}=(\bar{f}_{1},\bar{f}_{2})\in Z^{2}$ have compact support with
$N_{2}(0)>0$. For $t<T(\bar{f})$, let
$\mu^{n}(t)=(\mu_{1}^{n}(t),\mu_{2}^{n}(t))$ be empirical measures for
$X^{n}(t)$ generated from (34) with $\bar{f}\in Z^{2}$. Let measures
$\tilde{\mu}(t)=(\tilde{\mu}_{1}(t),\tilde{\mu}_{2}(t))$ for the
discretization scheme with step size of $\delta>0$ have initial conditions
$\bar{f}$. Then there exist positive constants
$\delta^{p},C^{p},C^{p}_{2},C_{3}^{p}$ such that for all
$\delta\in(0,\delta^{p})$ there exists $n^{p}(\delta)>0$ such that for all
integers $n>n^{p}(\delta)$ and $T^{\prime}<T(\bar{f}),$

$\mathbb{P}_{n}\left(\sup_{t\leq T^{\prime}}d(\tilde{\mu}(t),\mu^{n}(t))\geq
C^{p}(\delta+\omega(0,\delta)\right)\leq\frac{C^{p}_{2}}{\delta^{2}}\exp(-C^{p}_{3}\delta^{5}n).$
(36)

The constants $\delta^{p},C^{p},C^{p}_{2},C_{3}^{p}$ are all dependent on
$L^{\infty}$ and $L^{1}$ bounds of solutions in Theorem 1 with initial
conditions of $\bar{f}=(\bar{f}_{1},\bar{f}_{2})$, and the compact support
bound $M=\sup\\{x:\sum_{j=1,2}\bar{f}_{j}(x)>0\\}$.

From Theorems 2 and 3, it is straightforward to obtain our main concentration
inequality for initial conditions which are also locally Lipschitz.

######  Theorem 4.

Let $\bar{f}=(\bar{f}_{1},\bar{f}_{2})\in Z^{2}$ with $N_{2}(0)>0$ be both
compactly supported and locally Lipschitz, so that $\omega(0,\delta)\leq
C^{\omega}\delta$. For $C^{p}$ and $\delta^{p}$ determined from Theorem 3, let
$\varepsilon\in(0,2C^{p}C^{\omega}\delta^{p})$. Then there exist positive
constants $C,\tilde{C}$ and $N(\varepsilon)>0$ such that for all integers
$n>N(\varepsilon)$ and $T^{\prime}<T(\bar{f}),$

$\displaystyle\mathbb{P}_{n}\left(\sup_{t\leq
T^{\prime}}d(\mu(t),\mu^{n}(t))\geq\varepsilon\right)\leq\frac{C}{\varepsilon^{2}}\exp(-\tilde{C}\varepsilon^{5}n).$
(37)

###### Proof.

Since initial conditions are locally Lipschitz and compactly supported, we may
replace the $C^{d}(\delta+\omega(\delta,0))$ and
$C^{p}(\delta+\omega(\delta,0))$ terms in Theorem 2 and with 3 with
$\hat{C}^{d}\delta$ and $\hat{C}^{p}\delta$, respectively, where
$\hat{C}^{d}=C^{d}C^{\omega}$ and $\hat{C}^{p}=C^{p}C^{\omega}$.

Let $\varepsilon\in(0,2\hat{C}^{p}\delta_{p})$ and choose
$\delta=\varepsilon/(2\hat{C}^{d})$. We then have

$\displaystyle\mathbb{P}\left(\sup_{t\leq
T^{\prime}}d(\mu(t),\mu^{n}(t))\geq\varepsilon\right)\leq\mathbb{P}\left(\sup_{t\leq
T^{\prime}}d(\tilde{\mu}(t),\mu^{n}(t))+d(\mu(t),\tilde{\mu}(t))\geq\varepsilon\right)$
(38) $\displaystyle\leq\mathbb{P}\left(\sup_{t\leq
T^{\prime}}d(\tilde{\mu}(t),\mu^{n}(t))\geq\varepsilon-\hat{C}^{d}\delta\right)=\mathbb{P}\left(\sup_{t\leq
T^{\prime}}d(\tilde{\mu}(t),\mu^{n}(t))\geq\varepsilon/2\right)$
$\displaystyle\leq\frac{4(\hat{C}^{p})^{2}C_{2}^{p}}{\varepsilon^{2}}\exp\left(-\frac{C_{3}^{p}}{32(\hat{C}^{p})^{5}}\varepsilon^{5}n\right).$

The second inequality uses Theorem 2, and the third uses Theorem 3. We obtain
(37) with $C=4(\hat{C}^{p})^{2}C_{2}^{p}$ and
$\tilde{C}=C_{3}^{p}/(32(\hat{C}^{p})^{5})$. ∎

From Theorem 4, an application of the Borel-Cantelli lemma then gives us a
strong law of large numbers.

######  Corollary 1.

Under the product measure $\mathbb{Q}=\prod_{n\geq 2}\mathbb{P}_{n}$, for
$T^{\prime}<T(\bar{f})$,

$\lim_{n\rightarrow 0}\sup_{t\leq T^{\prime}}d(\mu^{n}(t),\mu(t))=0\quad\hbox{
almost surely.}$ (39)

##  3 Comparison of kinetic equations and deterministic scheme

In this section, we present a proof of Theorem 2. The discretization scheme
outlined in Section 2.2 allows us to write recursive formulas at time steps
$t_{k}$ related to measures restricted to size $\delta$ intervals, which we
denote as

$I_{l}=[(l-1)\delta,l\delta),\quad l\geq 1.$ (40)

In Section 3.1 we collect estimates related to growth of quantities for
solutions of the kinetic equations and the discretization scheme, including
the modulus of continuity $\omega(\delta,t)$ and total number contained in an
interval. Estimates related to the comparison between solutions of the kinetic
equations and the discretization are presented in Section 3.2. The main
quantity of interest is the difference of total number in intervals $I_{l}$ at
times $t_{k}$. Through constructing a closed recurrence inequality, we show
differences are of order $\delta^{2}+\delta\omega(\delta,0)$. In Section 3.3,
these differences are then summed over $[0,M]$ to establish Theorem 2.

###  3.1 Growth estimates

Our estimates for solutions of (6)-(7) and iterations (23)-(24) will make
frequent use of the constant bounds

$\displaystyle C_{\infty}(\bar{f})$
$\displaystyle=\max_{j=1,2}\sup_{s\in[0,T_{1}(\bar{f})]}\|f_{j}(x,s)\|_{\infty},$
(41) $\displaystyle C_{b}(\bar{f})$
$\displaystyle=\max\\{1/N_{2}(T_{1}(\bar{f})),1/\tilde{N}_{2}(T_{1}(\bar{f}))\\},$
(42)

which are dependent upon the initial conditions
$\bar{f}=(\bar{f}_{1},\bar{f}_{2})$ and time of existence $T(\bar{f})$. That
$C_{\infty}(\bar{f}),C_{b}(\bar{f})$ are finite follows from well-posedness of
$(f_{1}(x,t),f_{2}(x,t))$ in Theorem 1 and the existence of $T_{1}(\bar{f})$
established from (27). For simplicity, in future estimates we will refer to
these constants simply as $C_{\infty}$ and $C_{b}$. As we will see in Lemma 2,
it will be necessary to further restrict our interval of existence to
$t\in[0,T_{2}(\bar{f})])$, with

$T_{2}(\bar{f})=T_{1}(\bar{f})\wedge 1/(8C_{\infty}C_{b}).$ (43)

We will work with solutions $(f_{1}(x,t),f_{2}(x,t))$ of (6)-(7) having
initial conditions $\bar{f}_{1},\bar{f}_{2}\in Z$ with compact support in some
interval $[0,M]\subset\mathbb{R}_{+}$. For the deterministic scheme, measures
have identical initial conditions as the kinetic equations, with
$\bar{\mu}_{j}(dx)=\bar{f}_{j}(x)dx$.

We begin with a simple estimate on the propagation of the modulus of
continuity.

######  Lemma 1.

For all $\delta>0$ and $t\leq T_{2}(\bar{f})$,

$\displaystyle\omega(\delta,t)\leq C_{1}\omega(\delta,0),$ (44)

where $C_{1}=2\exp(2C_{b}).$

###### Proof.

Since $N_{2}(t)$ is decreasing, we use (6)-(7) to show

$\displaystyle\sum_{j=1}^{2}|f_{j}(x+\delta,t)-f_{j}(x,t)|$ (45)
$\displaystyle\leq|f_{1}(x+\delta+t,0)-f_{1}(x+t,0)|+|f_{2}(x+\delta,0)-f_{2}(x,0)|$
$\displaystyle+C_{b}\int_{0}^{t}|f_{2}(x+\delta+t-s,s)-f_{2}(x+t-s,s)|+|f_{2}(x+\delta,s)-f_{2}(x,s)|dL(s),$

where $L(s)=\int_{0}^{t}f_{1}(0,s)ds$ is the total loss. By taking the
supremum of the above inequality over $x\in\mathbb{R}_{+}$, from the
definition of $\omega(\delta,t)$ given in (32),

$\displaystyle\omega(\delta,t)\leq
2\omega(\delta,0)+2C_{b}\int_{0}^{t}\omega(\delta,s)dL(s).$ (46)

From Gronwall’s inequality,

$\omega(\delta,t)\leq 2\exp(2C_{b}L(t))\omega(\delta,0).$ (47)

Since $L(t)\leq 1$, we obtain (44). ∎

Next, we turn to studying the maximum total number of a measure on length
$\delta$ intervals, denoted as

$m^{j}_{\delta}(t)=\sup_{\begin{subarray}{c}I:|I|=\delta\end{subarray}}\mu_{j}(t,I),\quad
j=1,2,\qquad m_{\delta}(t)=\sum_{j=1}^{2}m^{j}_{\delta}(t).$ (48)

We define $\tilde{m}_{\delta}(t)$ similarly.

######  Lemma 2.

For $\delta>0$ and $t<T_{2}(\bar{f})$,

$\displaystyle m_{\delta}(t)$ $\displaystyle\leq C_{2}\delta,$ (49)
$\displaystyle\tilde{m}_{\delta}(t_{k})$
$\displaystyle\leq\tilde{C}_{2}\delta,$ (50) $\displaystyle\tilde{N}(t_{k})$
$\displaystyle\geq 1-\tilde{C}_{2}\delta k.$ (51)

With constants

$C_{2}=2C_{\infty}\exp(C_{b}),\quad\tilde{C}_{2}=\frac{8C_{\infty}^{2}}{1-4C_{b}C_{\infty}T_{2}(\bar{f})}.$
(52)

###### Proof.

To show (49), for an interval $I$ with $|I|\leq\delta$, integrate (6)-(7) over
$I$ to obtain

$\displaystyle\mu_{1}(t,I)$
$\displaystyle=\mu_{1}(0,I+t)+\int_{0}^{t}\frac{\mu_{2}(s,I+t-s)}{N_{2}(s)}dL(s),$
(53) $\displaystyle\mu_{2}(t,I)$
$\displaystyle=\mu_{2}(0,I)-\int_{0}^{t}\frac{\mu_{2}(s,I)}{N_{2}(s)}dL(s)$
(54)

By taking the supremum over all length $\delta$ intervals, we find

$\displaystyle m_{\delta}^{1}(t)$ $\displaystyle\leq
m_{\delta}^{1}(0)+C_{b}\int_{0}^{t}m_{\delta}^{2}(s)dL(s),$ (55)
$\displaystyle m_{\delta}^{2}(t)$ $\displaystyle\leq m_{\delta}^{2}(0).$ (56)

We then obtain (49) by summing (55)-(56), applying Gronwall’s lemma, and
observing from initial conditions that $m_{\delta}(0)\leq 2C_{\infty}\delta$.

To show (50), we will work with

$\hat{m}^{j}_{\delta}(t)=\sup_{l\geq 1}\tilde{\mu}_{j}(t,I_{l}),\quad
j=1,2,\qquad\hat{m}_{\delta}(t)=\sum_{j=1}^{2}\hat{m}^{j}_{\delta}(t).$ (57)

Note that

$\tilde{m}^{j}_{\delta}(t)\leq 2\hat{m}^{j}_{\delta}(t).$ (58)

From evaluating measures on $I_{l}$, the recursion (23)-(24) implies that for
$l\geq 1$ and $s\in[t_{k},t_{k+1})$,

$\displaystyle\tilde{\mu}_{1}(s,I_{l})$
$\displaystyle=\tilde{\mu}_{1}(t_{k-1},I_{l+1})+\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1},I_{l}),$
(59) $\displaystyle\tilde{\mu}_{2}(s,I_{l})$
$\displaystyle=\tilde{\mu}_{2}(t_{k-1},I_{l})\left(1-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\right).$
(60)

We now take the supremum over $j\geq 1$ in (59)-(60) and sum to obtain

$\hat{m}_{\delta}(t_{k})\leq\hat{m}_{\delta}(t_{k-1})+C_{b}\left(\hat{m}_{\delta}(t_{k-1})\right)^{2}.$
(61)

For simplicity, we rescale by writing
$\hat{m}_{\delta}(t_{k})=A(t_{k})\delta$. From (61), and noting

$\hat{m}_{\delta}(0)\leq 2\tilde{m}_{\delta}(0)=2m_{\delta}(0)\leq
4C_{\infty}\delta,$ (62)

we obtain the recurrence inequalities

$\displaystyle A(t_{k})$ $\displaystyle\leq A(t_{k-1})+\delta
C_{b}(A(t_{k-1}))^{2},$ (63) $\displaystyle A(0)$ $\displaystyle\leq
4C_{\infty}.$ (64)

In the case of equality, (63)-(64) is an Euler scheme for the differential
equation $g^{\prime}(t)=C_{b}g^{2}(t)$ with $g(0)=C_{\infty}.$ We may check
directly that $A(t_{k})\leq g(t_{k})$ before blowup, or

$A(t_{k})<\frac{A(0)}{1-kA(0)C_{b}\delta}$ (65)

for $t_{k}\leq T_{2}(\bar{f})$. We then use (65), (64), and (58) to obtain
(50).

Finally, (51) follows immediately, since

$\tilde{N}_{2}(t_{k})=1-\sum_{i=1}^{k}\Delta\tilde{L}(t_{i})\geq
1-\sum_{i=1}^{k}\tilde{m}_{\delta}(t_{i}).$ (66)

∎

We finish this subsection with one more estimate related to the incremental
loss and total number in the kinetic limit.

######  Lemma 3.

For $t_{k}<T_{2}(\bar{f}),$

$\displaystyle\Delta L(t_{k}):=L(t_{k})-L(t_{k-1})\leq
C_{2}\delta+C_{b}C_{\infty}^{2}\delta^{2},$ (67) $\displaystyle N(t_{k})\geq
1-C_{2}k\delta-C_{b}C_{\infty}^{2}k\delta^{2}.$ (68)

###### Proof.

We use (6) on the removal rate $f_{1}(0,t)$ to obtain

$\displaystyle\Delta L(t_{k})$
$\displaystyle=\int_{0}^{\delta}f_{1}(0,t_{k-1}+s)ds$ (69)
$\displaystyle=\int_{0}^{\delta}\left(f_{1}(s,t_{k-1})+\int_{0}^{s}\frac{f_{1}(0,t_{k-1}+r)}{N_{2}(t_{k-1}+r)}f_{2}(s-r,t_{k-1}+r)dr\right)ds$
$\displaystyle\leq\mu_{1}(t_{k-1},[0,\delta])+C_{b}C_{\infty}^{2}\delta^{2}.$

From (49), we obtain (67). Since $N(t_{k})=1-\sum_{i=1}^{k}\Delta L(t_{i}),$
(68) also follows. ∎

###  3.2 Convergence estimates

We now use estimates from the previous subsection to establish asymptotics for
the differences of total number between the solution of (3)-(5) and its
discretization. We begin with a simple result which follows immediately from
Lemma 3 comparing incremental losses and total numbers of species.

######  Corollary 2.

For $t_{k}<T_{2}(\bar{f}),$

$\displaystyle|\Delta L(t_{k})-\Delta\tilde{L}(t_{k})|$
$\displaystyle\leq|\mu_{1}(t_{k-1},[0,\delta])-\tilde{\mu}_{1}(t_{k-1},[0,\delta])|+C_{b}C_{\infty}^{2}\delta^{2}.$
(70) $\displaystyle|N(t_{k})-\tilde{N}(t_{k})|$
$\displaystyle\leq\sum_{i=1}^{k}|\mu_{1}(t_{i},[0,\delta])-\tilde{\mu}_{1}(t_{i},[0,\delta])|+T(\bar{f})C_{b}C_{\infty}^{2}\delta.$
(71)

To compare behavior on an interval $I_{j}$, we will use a formula similar to
(59)-(60) for evolving densities over a time step $\Delta
t_{k}=[t_{k-1},t_{k})$. It follows directly from (6)-(7) that

$\displaystyle f_{1}(x,t_{k})$
$\displaystyle=f_{1}(x+\delta,t_{k-1})+\int_{\Delta
t_{k}}\frac{f_{2}(x+t_{k}-s,s)}{N_{2}(s)}f_{1}(0,s)ds,$ (72) $\displaystyle
f_{2}(x,t_{k})$ $\displaystyle=f_{2}(x,t_{k-1})-\int_{\Delta
t_{k}}\frac{f_{2}(x,s)}{N_{2}(s)}f_{1}(0,s)ds.$ (73)

To arrive at an estimate for the difference of total number in an interval, we
use (23)-(24) and (72)-(73) to express the difference of total number of an
interval in Species 1 as

$\displaystyle|\mu_{1}(t_{k},I_{l})-\tilde{\mu}_{1}(t_{k},I_{l})|\leq|\mu_{1}(t_{k-1},I_{l+1})-\tilde{\mu}_{1}(t_{k-1},I_{l+1})|$
(74) $\displaystyle+\left|\int_{I_{l}}\int_{\Delta
t_{k}}\frac{f_{2}(x+t_{k}-s,s)}{N_{2}(s)}dL(s)dx-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1},I_{l})\right|.$

For Species 2,

$\displaystyle|\mu_{2}(t_{k},I_{l})-\tilde{\mu}_{2}(t_{k},I_{l})|\leq|\mu_{2}(t_{k-1},I_{l})-\tilde{\mu}_{2}(t_{k-1},I_{l})|$
(75) $\displaystyle+\left|\int_{I_{l}}\int_{\Delta
t_{k}}\frac{f_{2}(x,s)}{N_{2}(s)}dL(s)dx-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1},I_{,})\right|.$

The following lemma allows us to estimate the integrals in (74) and (75) by
quantities at discretized times $t_{k}$.

######  Lemma 4.

For $l\geq 1$ and $t_{k}<T_{2}(\bar{f}),$

$\displaystyle\left|\int_{I_{l}}\int_{\Delta
t_{k}}\frac{f_{2}(x+t_{k}-s,s)}{N_{2}(s)}dL(s)dx-\frac{\Delta
L(t_{k})}{N_{2}(t_{k-1})}\mu_{2}(t_{k-1},I_{l})\right|$
$\displaystyle=\mathcal{O}(\delta^{3}+\delta^{2}\omega(\delta,0)).$ (76)
$\displaystyle\left|\int_{I_{l}}\int_{\Delta
t_{k}}\frac{f_{2}(x,s)}{N_{2}(s)}dL(s)dx-\frac{\Delta
L(t_{k})}{N_{2}(t_{k-1})}\mu_{2}(t_{k-1},I_{l})\right|$
$\displaystyle=\mathcal{O}(\delta^{3}).$ (77)

###### Proof.

We show (76). The proof for (77) is similar. Denote the left hand side of (76)
as $\mathcal{I}$. Through the triangle inequality, we have

$\displaystyle\mathcal{I}=\left|\int_{I_{l}}\int_{\Delta
t_{k}}\frac{f_{2}(x+t_{k}-s,s)}{N_{2}(s)}-\frac{f_{2}(x,t_{k-1})}{N_{2}(t_{k-1})}dL(s)dx\right|$
(78) $\displaystyle\leq\int_{I_{l}}\int_{\Delta
t_{k}}\left|\frac{f_{2}(x+t_{k}-s,s)-f_{2}(x,t_{k-1})}{N_{2}(s)}\right|dL(s)dx$
$\displaystyle+\int_{I_{l}}\int_{\Delta
t_{k}}\left|f_{2}(x,t_{k-1})\left(\frac{1}{N_{2}(s)}-\frac{1}{N_{2}(t_{k-1})}\right)\right|dL(s)dx$
$\displaystyle:=\mathcal{I}_{1}+\mathcal{I}_{2}.$

We first show that

$\mathcal{I}_{1}\leq
C_{b}^{2}C_{\infty}^{2}C_{2}\delta^{3}+C_{1}C_{2}C_{b}\omega(\delta,0)\delta^{2}+\mathcal{O}(\delta^{4}+\omega(\delta,0)\delta^{3}).$
(79)

This is done by another use of the triangle inequality to align arguments in
time and space,

$\displaystyle\mathcal{I}_{1}\leq C_{b}\Big{(}\int_{\Delta
t_{k}}\int_{I_{l}}|f_{2}(x+t_{k}-s,t_{k-1})-f_{2}(x,t_{k-1})|dxdL(s)$ (80)
$\displaystyle+\int_{I_{l}}\int_{\Delta
t_{k}}|f_{2}(x+t_{k}-s,s)-f_{2}(x+t_{k}-s,t_{k-1})|dL(s)dx.\Big{)}$

From (44),(49) and (67), the first integral may be bounded by

$\displaystyle\int_{\Delta
t_{k}}\int_{I_{l}}|f_{2}(x+t_{k}-s,t_{k-1})-f_{2}(x,t_{k-1})|dxdL(s)$ (81)
$\displaystyle\leq
C_{1}C_{2}\omega(\delta,0)\delta^{2}+\mathcal{O}(\omega(\delta,0)\delta^{3}).$

For the last integrand for (80), we may use (7) to obtain

$\displaystyle|f_{2}(x+t_{k}-s,s)-f_{2}(x+t_{k}-s,t_{k-1})|$ (82)
$\displaystyle=\int_{t_{k-1}}^{s}\frac{f_{2}(x+t_{k}-s,r)}{N_{2}(r)}f_{1}(0,r)dr\leq
C_{b}C_{\infty}^{2}\delta.$

Substituting (81) and (82) into (80) then gives (79).

By a similar calculation we may use (67) to obtain

$\displaystyle\mathcal{I}_{2}$ $\displaystyle\leq
C_{\infty}C_{b}^{2}\delta(\Delta L(t_{k}))^{2}\leq
C_{\infty}C_{b}^{2}C_{2}^{2}\delta^{3}+\mathcal{O}(\delta^{4}).$ (83)

Finally, (76) then comes from collecting estimates for
$\mathcal{I}=\mathcal{I}_{1}+\mathcal{I}_{2}$. ∎

We are now ready to compare total numbers over length $\delta$ intervals by
defining

$h_{\delta}^{j}(t_{k})=\sup_{l\geq
1}|\mu_{j}(t_{k},I_{l})-\tilde{\mu}_{j}(t_{k},I_{l})|,\quad
h_{\delta}(t_{k})=\sum_{j=1}^{2}h_{\delta}^{j}(t_{k}).$ (84)

Our next lemma gives closed recurrence inequalities for $h_{\delta}(t_{k})$.

######  Lemma 5.

There exists a $C_{3}(\bar{f})$ dependent on initial conditions such that for
$t_{k}<T_{2}(\bar{f})$, $h_{\delta}(t_{k})$ satisfies the recurrence
inequality

$\displaystyle h_{\delta}(t_{k})\leq
h_{\delta}(t_{k-1})+C_{3}\left(\delta^{2}\sum_{i=1}^{k-1}h_{\delta}(t_{i})+\delta
h(t_{k-1})+\omega(\delta,0)\delta^{2}+\delta^{3}\right).$ (85)

###### Proof.

From Lemma 4 and (74), for $l\geq 1,$

$\displaystyle|\mu_{1}(t_{k},I_{l})-\tilde{\mu}_{1}(t_{k},I_{l})|$ (86)
$\displaystyle\leq|\mu_{1}(t_{k-1},I_{l+1})-\tilde{\mu}_{1}(t_{k-1},I_{l+1})|$
$\displaystyle+\left|\frac{\Delta
L(t_{k})}{N_{2}(t_{k-1})}\mu_{2}(t_{k-1},I_{l})-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1},I_{l})\right|+\mathcal{O}(\delta^{3}+\delta^{2}\omega(\delta,0)).$

Two applications of the triangle inequality yield

$\displaystyle\left|\frac{\Delta
L(t_{k})}{N_{2}(t_{k-1})}\mu_{2}(t_{k-1},I_{l})-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1},I_{l})\right|$
(87)
$\displaystyle\leq\left|\left(\frac{1}{N_{2}(t_{k-1})}-\frac{1}{\tilde{N}_{2}(t_{k-1})}\right)\Delta
L(t_{k})\mu_{2}(t_{k-1},I_{l})\right|$
$\displaystyle+\left|\frac{1}{\tilde{N}_{2}(t_{k-1})}\mu_{2}(t_{k-1},I_{l})(\Delta
L(t_{k})-\Delta\tilde{L}(t_{k}))\right|$
$\displaystyle+\left|\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}(\mu_{2}(t_{k-1},I_{l})-\tilde{\mu}_{2}(t_{k-1},I_{l}))\right|.$

We use Lemmas 2 and 3, (86), and (87) to obtain

$\displaystyle|\mu_{1}(t_{k},I_{l})-\tilde{\mu}_{1}(t_{k},I_{l})|$ (88)
$\displaystyle\leq|\mu_{1}(t_{k-1},I_{l+1})-\tilde{\mu}_{1}(t_{k-1},I_{l+1})|$
$\displaystyle+C_{b}^{2}C_{2}^{2}\delta^{2}|N_{2}(t_{k-1})-\tilde{N}_{2}(t_{k-1})|+C_{b}C_{2}\delta|\Delta
L(t_{k})-\Delta\tilde{L}(t_{k})|$
$\displaystyle+C_{b}C_{2}\delta|\mu_{2}(t_{k-1},I_{l})-\tilde{\mu}_{2}(t_{k-1},I_{l})|+\mathcal{O}(\delta^{3}+\delta^{2}\omega(\delta,0)).$

Similar bounds hold for Species 2, with

$\displaystyle|\mu_{2}(t_{k},I_{l})-\tilde{\mu}_{2}(t_{k},I_{l})|$ (89)
$\displaystyle\leq|\mu_{2}(t_{k-1},I_{l})-\tilde{\mu}_{2}(t_{k-1},I_{l})|$
$\displaystyle+C_{b}^{2}C_{2}^{2}\delta^{2}|N_{2}(t_{k-1})-\tilde{N}_{2}(t_{k-1})|+C_{b}C_{2}\delta|\Delta
L(t_{k})-\Delta\tilde{L}(t_{k})|$
$\displaystyle+C_{b}C_{2}\delta|\mu_{2}(t_{k-1},I_{l})-\tilde{\mu}_{2}(t_{k-1},I_{l})|+\mathcal{O}(\delta^{3}+\delta^{2}\omega(\delta,0)).$

We complete the proof by taking the supremum over $l$ for (88) and (89), using
(71), and then adding to show that for some $C_{3}$ which depends on
$(\bar{f}_{1},\bar{f}_{2})$,

$\displaystyle h_{\delta}(t_{k})\leq
h_{\delta}(t_{k-1})+C_{3}\left(\delta^{2}\sum_{i=1}^{k-1}h_{\delta}(t_{i})+\delta
h_{\delta}(t_{k-1})+\omega(\delta,0)\delta^{2}+\delta^{3}\right).$ (90)

∎

We now show that the recurrence inequality (85) implies asymptotics for
$h_{\delta}(t_{k})$.

######  Lemma 6.

For $t_{k}<T_{2}(\bar{f})$,

$h_{\delta}(t_{k})=\mathcal{O}(\delta^{2}+\delta\omega(\delta,0)).$ (91)

###### Proof.

Let $b_{\delta}(t_{k})$ satisfy the recurrence equation

$\displaystyle
b_{\delta}(t_{k})=b_{\delta}(t_{k-1})+C_{3}\left(\delta^{2}\sum_{i=1}^{k-1}b_{\delta}(t_{i})+\delta
b_{\delta}(t_{k-1})+\delta\right),$ (92)

with initial condition $b_{\delta}(0)=h_{\delta}(0)=0.$ It follows immediately
from induction that
$h(t_{k})\leq(\delta^{2}+\omega(\delta,0)\delta))b_{\delta}(t_{k})$ for all
$k\geq 0$. Thus, it is sufficient to show that
$b_{\delta}(t_{k})=\mathcal{O}(1)$ for all $t_{k}<T_{2}(\bar{f})$. To see that
this holds, note that as $\delta\rightarrow 0$, (92) converges to the linear
integro-differential equation

$\displaystyle\tilde{b}^{\prime}(t)=C_{3}\left(\tilde{b}(t)+\int_{0}^{t}\tilde{b}(s)ds+1\right),\quad
t\in[0,T_{2}(\bar{f})),$ (93)

with initial condition $\tilde{b}(0)=h_{\delta}(0)$. This can be solved
through elementary second order methods to obtain the locally bounded solution

$\tilde{b}(t)=A_{1}e^{r_{1}t}+A_{2}e^{r_{2}t},$ (94)

where $A_{1}=-A_{2}=C_{3}/(r_{2}-r_{1}),$ and $r_{1},r_{2}$ are the two real
solutions of $r^{2}-C_{3}r-C_{3}=0$. ∎

We remark that this theorem also holds when
$h_{\delta}(0)=\mathcal{O}(\delta^{2}+\delta\omega(\delta,0))$. This is
important for extending the time interval of existence.

###  3.3 Proof of Theorem 2

With bounds on differences of total numbers restricted to an interval, we are
finally ready to show Theorem 2. We will require two more lemmas, which are
straightforward to show. First, we give a formula computing KS distances when
only given information about cumulative functions on grid points and bounds on
growth between grid points.

######  Lemma 7.

For $j=1,2$, suppose we have measures $\nu_{j}\in\mathcal{M}(\mathbb{R}_{+})$
with corresponding cumulative functions $F_{j}(x)=\nu_{j}([0,x])$, each with
compact support $[0,M]$. For $\delta>0$, we can bound the KS metric by

$\displaystyle d_{KS}(\nu_{1},\nu_{2})\leq\sum_{i=1}^{\lceil
M/\delta\rceil}|\nu_{1}(I_{i})-\nu_{2}(I_{i})|+\sup_{|x-y|=\delta}(|F_{1}(x)-F_{1}(y)|+|F_{2}(x)-F_{2}(y)|).$

We will also require a lemma comparing differences of solutions of kinetic
equations under small changes in time.

######  Lemma 8.

If $|t_{1}-t_{2}|\leq\delta$, then

$d((\mu_{1}(t_{1}),\mu_{2}(t_{1})),(\mu_{1}(t_{2}),\mu_{2}(t_{2})))=\mathcal{O}(\delta+\omega(\delta,0)).$
(95)

###### Proof.

Similar to the proof of Lemma 3. ∎

Proof of Theorem 2. Let $t\in[0,T_{2}(\bar{f})]$, $\delta>0$, and
$K=\lfloor\frac{t}{\delta}\rfloor$, which means $t_{K}$ is the largest
discretized time which is at most $t$. From Lemmas 2, 6, 7, and 8, and
recalling that $\tilde{\mu}_{j}(t)$ is constant in intervals
$t\in[t_{k-1},t_{k})$, we compute

$\displaystyle
d((\mu_{1}(t),\mu_{2}(t)),(\tilde{\mu}_{1}(t),\tilde{\mu}_{2}(t)))$ (96)
$\displaystyle\leq
d((\mu_{1}(t),\mu_{2}(t)),(\mu_{1}(t_{K}),\mu_{2}(t_{K})))+d((\mu_{1}(t_{K}),\mu_{2}(t_{K})),(\tilde{\mu}_{1}(t_{K}),\tilde{\mu}_{2}(t_{K})))$
$\displaystyle\leq\sum_{j=1}^{2}\sum_{l=1}^{\lceil
M/\delta\rceil}|\mu_{j}(t_{K},I_{l})-\tilde{\mu}_{j}(t_{K},I_{l})|+2\left(m_{\delta}(t_{K})+\tilde{m}_{\delta}(t_{K})\right)+\mathcal{O}(\delta+\omega(\delta,0))$
$\displaystyle\leq 2\lceil M/\delta\rceil
h_{\delta}(t_{K})+2(m_{\delta}(t_{K})+\tilde{m}_{\delta}(t_{K}))+\mathcal{O}(\delta+\omega(\delta,0))$
$\displaystyle=\mathcal{O}(\delta+\omega(\delta,0)).$

Finally, let us argue for extending the time interval of existence to any
$T^{\prime}<T(\bar{f})$. Consider initial conditions
$\mu^{(2)}(0)=\mu(T_{2}(\bar{f}))$ and
$\tilde{\mu}^{(2)}(0)=\tilde{\mu}(T_{2}(\bar{f}))$, and new time bounds

$T_{1}^{(2)}(\bar{f})=\sup\\{t:N_{2}^{(2)}(t)>2N_{2}^{(2)}(0)/3\\},\quad
T_{2}^{(2)}=T^{(2)}_{1}\wedge 1/(8\tilde{C}_{\infty}\tilde{C}_{b}),$ (97)

with

$\displaystyle\tilde{C}_{\infty}=\max_{j=1,2}\sup_{s\in[0,T^{\prime}(\bar{f})]}\|f_{j}(x,s)\|_{\infty},\quad\tilde{C}_{b}=2/\inf_{t<T^{\prime}}N_{2}(t).$
(98)

Then for sufficiently small $\delta$, it follows that Lemmas 1-6 hold, with
possibly larger constants, and therefore Theorem 2 holds for the time interval
$[0,T_{2}+T_{2}^{(2)}]$ from stitching solutions. This argument may be
repeated to produce $T^{(k)}$. Each new time interval either adds the constant
$1/(8\tilde{C}_{\infty}\tilde{C}_{b})$ (which does not depend on $k$) or
reduces $N_{2}$ by at least 4/5. After finitely many extensions we will reach
a time $T^{(K)}$ at which $N_{2}(T^{(K)})$ is arbitrarily small, so that $t$
is arbitrarily close to $T(\bar{f})$. This completes the proof of Theorem 2.

##  4 Comparison of particle system and deterministic scheme

###  4.1 Stochastic analogues of Section 3

We now compare the discretized measures described in the Section 3 with the
$n$-particle PDMP $\\{X^{n}(t)\\}_{t\geq 0}$. To do so, it will help to
express evolution of $\mu_{\sigma}^{n}$ from $t_{k}$ to $t_{k+1}$ in a form
that is similar to the iterative formulas (23)-(24). For defining analogous
notation to the the discretization scheme, we denote the number of mutations
occurring in the time interval $t\in\Delta t_{k}=[t_{k-1},t_{k})$ as $\Delta
L^{n}(t_{k})n$. At each $i\in 1,\dots,\Delta L^{n}(t_{k})n$, mutation time
$\tau_{i}^{k}$ denote when a particles at position $x_{i}\geq 0$ in Species 2
mutates into Species 1. For particles in Species 2 which mutate in interval
$I$ during $\Delta t_{k}$, the empirical measure of their positions at
mutation times is defined by

$\pi^{n}_{2}(t_{k},I)=\frac{1}{n}\sum_{i=1}^{\Delta
L^{n}(t_{k})n}\mathbf{1}(x_{i}\in I).$ (99)

We define $\pi_{1}^{n}$ as the empirical measure for the positions of mutated
particles at time $t_{k}$, with

$\displaystyle\pi^{n}_{1}(t_{k},I)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{\Delta L^{n}(t_{k})n}\mathbf{1}(x_{i}\in
I+\tau_{i}-t_{k-1})$ (100) $\displaystyle:=\frac{1}{n}\sum_{i=1}^{\Delta
L^{n}(t_{k})n}Q_{k}^{i}(I).$

Updates for measures on intervals during a time step may then be succinctly
written as

$\displaystyle\mu^{n}_{1}(t_{k},I)$
$\displaystyle=\mu^{n}_{1}(t_{k-1},I+\delta)+\pi^{n}_{1}(t_{k},I)$ (101)
$\displaystyle\mu^{n}_{2}(t_{k},I)$
$\displaystyle=\mu^{n}_{2}(t_{k-1},I)-\pi^{n}_{2}(t_{k},I).$ (102)

From (35), the time interval of existence, under sufficiently many particles
$n>n_{0}(\bar{f})$, before reaching the cemetery state is at least
$T_{1}(\bar{f})$, but since we are comparing the particle system with the
deterministic discretization, we will work with the smaller time interval
$[0,T_{2}(\bar{f})]$.

Let us present variables to be used for the particle system which are similar
to those found in Section 3. We begin with an analogue to $m_{\delta}$ given
in (48) for defining the maximum total numbers of length $\delta$ intervals as

$m^{j;n}_{\delta}(t)=\sup_{|I|\leq\delta}\mu_{j}^{n}(t,I),\quad
m_{\delta}^{n}(t)=\sum_{j=1,2}m^{j;n}_{\delta}(t).$ (103)

From (101)-(102), it follows for all realizations of $X^{n}(t)$ that

$m_{\delta}^{n}(t_{k})\leq
m_{\delta}^{n}(t_{k-1})+\sum_{j=1}^{2}\sup_{I,|I|\leq\delta}\pi^{n}_{j}(t_{k},I).$
(104)

There is also a particle system analog of $h_{\delta}$, where we now compare
total number in intervals between $X^{n}(t)$ and the discretization scheme as

$h_{\delta}^{j;n}(t_{k})=\sup_{l\leq
M/\delta}|\mu_{j}^{n}(t_{k},I_{l})-\tilde{\mu}_{j}(t_{k},I_{l})|,\quad
h_{\delta}^{n}=\sum_{j=1}^{2}h_{\delta}^{j;n}(t_{k}).$ (105)

The use of measures $\tilde{\mu}(t_{k})$ rather than $\mu(t_{k})$ comes from
ability to use the recurrence (59)-(60), along with (101)-(102) and (23)-(24),
to write the recurrence inequality

$\displaystyle h^{n}_{\delta}(t_{k})$ $\displaystyle\leq
h^{n}_{\delta}(t_{k-1})+\sum_{j=1}^{2}\max_{l\leq\tilde{M}/\delta}\left|\pi_{j}^{n}(t_{k},I_{l})-\frac{\Delta\tilde{L}(t_{k})}{\tilde{N}_{2}(t_{k-1})}\tilde{\mu}_{2}(t_{k-1},I_{l})\right|$
(106) $\displaystyle:=h^{n}_{\delta}(t_{k-1})+\Pi^{n}(t_{k}).$

From (104) and (106), the major task for controlling $h^{n}$ and $m^{n}$ will
clearly hinge on appropriate estimates for $\pi_{j}$ (and subsequently
$\Pi^{n}$). These key bounds are provided in the next two lemmas.

###  4.2 Concentration bounds for $\pi_{j}$

We begin with an easy to establish generalization of the Hoeffding inequality,
which states that for $n\geq 1$ and with $B_{i}(n,p)\sim\mathrm{Binom}(n,p)$,

$\mathbb{P}\left(\left|\frac{B_{i}(n,p)}{n}-p\right|>\varepsilon\right)\leq
2e^{-2n\varepsilon^{2}}.$ (107)

######  Corollary 3.

For $n\geq 1$, let $X^{n}=\sum_{i=1}^{n}Z_{i}$, where
$Z_{i}\sim\mathrm{Ber}(p_{i})$ are independent Bernoulli random variables with
parameters $0\leq\underline{p}\leq p_{i}\leq\bar{p}\leq 1$. Then for
$\varepsilon>0$,

$\displaystyle\mathbb{P}\left(\frac{X^{n}}{n}-\bar{p}>\varepsilon\right)$
$\displaystyle\leq
2e^{-2n\varepsilon^{2}}\quad\hbox{and}\quad\mathbb{P}\left(\frac{X^{n}}{n}-\underline{p}<-\varepsilon\right)\leq
2e^{-2n\varepsilon^{2}}.$ (108)

For the next two lemmas, we will be interested in cases where parameters
$p_{i}$ are themselves $[0,1]$-valued random variables with known lower and
upper bounds. In particular, we will use the _mutation probability_
$P_{k}^{1}(t,I)$ defined as the state-dependent probability that if a mutation
occurs at time $t\in[t_{k-1},t_{k})$, then the mutated particle would be
located in $I$ at time $t_{k}$. We also define $P_{k}^{2}(t,I)$ as the
probability that a particle would mutate in $I$ from Species 2 at time $t$.
Thus,

$\displaystyle
P_{k}^{1}(t,I)=\frac{\mu_{2}^{n}(t^{-},I+t_{k}-t)}{N_{2}^{n}(t^{-})},\quad
P_{k}^{2}(t,I)=\frac{\mu_{2}^{n}(t^{-},I)}{N_{2}^{n}(t^{-})}.$ (109)

For an initial distribution $(\mu^{n}_{1}(0),\mu_{2}^{n}(0))$ with support
$[0,M]$, tracking which intervals mutations occur in during a time interval
$[t_{k-1},t_{k})$ can be represented through a random sum of multinomials of
one draw with random selection probabilities. In particular, we perform a
total of $\Delta L^{n}(t_{k})$ draws with $\lceil M/\delta\rceil$ bins, in
which for each draw the $i$th bin has the mutation probability
$P_{k}^{2}(\tau_{i}^{-},I)$ of being selected.

The following two lemmas comparing $\pi_{j}$ with bounds for $P_{k}^{j}(t,I)$
and $\Delta L^{n}(t_{k})$ involve using Corollary 3 along with the strong
Markov property of PDMPs introduced by Davis [2]. For the first lemma, we
consider an initial state $X^{n}=X^{n}(0)$ which has pathwise bounds during
$\Delta t_{1}$ for $\Delta L^{n}(t_{1}),$ and $P^{j}_{1}(t,I_{l})$. The second
lemma assumes these bounds occur with some probability which may be less than
1. Since $X^{n}(t)$ is homogeneous, both these lemmas are readily applicable
when considering transitions during $\Delta t_{k}$ for $k>1$.

We will use common notation for stochastic ordering: for real-valued random
variables $X$ and $Y$, we write $X\leq_{ST}Y$ if
$\mathbb{P}(X>c)\leq\mathbb{P}(Y>c)$ for all real $c$. Also note in the next
two lemmas, we will suppress time arguments when no confusion will arise.
Finally, for simplicity with presentation, we will assume that $M/\delta$ is
integral.

######  Lemma 9.

Let $X^{n}(0)=\mathbf{x}\in E$ be an initial state such that for $l\in
1,\dots,M/\delta$, $j\in\\{1,2\\}$, and $t\in\Delta t_{1}$,

$\displaystyle\underline{L}\leq\Delta L^{n}\leq\bar{L}\quad\hbox{ and
}\quad\underline{p}_{l}\leq P_{1}^{j}(t,I_{l})\leq\bar{p}_{l},$ (110)

where $\underline{L},\bar{L},\underline{p}_{l},\bar{p}_{l}$ are $[0,1]$-valued
constants. Then for $\varepsilon>0$,

$\displaystyle\mathbb{P}\left(\max_{l\leq
M/\delta}\left(\pi^{n}_{j}(t_{1},I_{l})-\bar{L}\bar{p}_{l}\right)>\varepsilon\right)\leq\frac{2M}{\delta}\mathbb{\exp}(-2n\varepsilon^{2}/\bar{L})$
(111)

and

$\displaystyle\mathbb{P}\left(\min_{l\leq
M/\delta}\left(\pi^{n}_{j}(t_{1},I_{l})-\underline{L}\underline{p}_{l}\right)<-\varepsilon\right)\leq\frac{2M}{\delta}\exp(-2n\varepsilon^{2}/\underline{L}).$
(112)

###### Proof.

We will show (111) for $j=1$. The proofs for the other cases are similar. For
the parameter $q\in[0,1]$, denote $\\{B_{i}(q)\\}_{i\geq 1}$ as an iid stream
of Bernoulli random variables with $B_{1}\sim\mathrm{Ber}(q)$. We write

$\bar{Q}_{l}^{i}:=B_{i}(\bar{p}^{n}_{l}),\quad
Q_{l}^{i}:=\begin{cases}Q^{i}_{1}(I_{l})&i\leq\Delta L^{n}n,\\\
\bar{Q}_{l}^{i}&i>\Delta L^{n}n\\\ \end{cases}.$ (113)

We use iterated conditioning to show that

$\displaystyle\mathbb{P}(Q_{l}^{i}=1)=\mathbb{E}[Q_{l}^{i}]=\mathbb{E}[\mathbb{E}[Q_{l}^{i}|X(\tau_{1}^{i-})]]=\mathbb{E}[P_{1}^{1}(\tau_{1}^{i-},I_{l})]\leq\bar{p}_{l}=\mathbb{P}(\bar{Q}_{l}^{i}=1).$
(114)

This calculation implies the stochastic ordering
$Q_{l}^{i}\leq_{ST}\bar{Q}_{l}^{i}$.

Next, we show

$\pi^{n}_{1}(I_{l})\leq_{ST}\frac{1}{n}\sum_{i=1}^{\bar{L}n}Q_{l}^{i}\leq_{ST}\frac{1}{n}\sum_{i=1}^{\bar{L}n}\bar{Q}_{l}^{i}.$
(115)

The left inequality is immediate, and in fact holds for all paths in
$X^{n}(t)$. To show the right inequality, we use induction, assuming that for
$1\leq j<\bar{L}n$,

$\displaystyle\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}^{i}>c\right)$
$\displaystyle\leq\mathbb{P}\left(\sum_{i=1}^{j}\bar{Q}_{l}^{i}>c\right).$
(116)

The base case holds trivially. For the inductive step, we use the low of total
probability to show

$\displaystyle\mathbb{P}\left(\sum_{i=1}^{j+1}Q_{l}>c\right)$
$\displaystyle=\mathbb{E}\left[\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}^{i}+Q_{l}^{j+1}>c\Big{|}X((\tau_{1}^{j+1})^{-})\right)\right]$
(117)
$\displaystyle\leq\mathbb{E}\left[\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}^{i}+\bar{Q}^{j+1}_{l}>c\Big{|}X((\tau_{1}^{j+1})^{-})\right)\right]$
$\displaystyle=\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}^{i}+\bar{Q}^{j+1}_{l}>c\right)\leq\mathbb{P}\left(\sum_{i=1}^{j+1}\bar{Q}_{l}^{i}>c\right).$

The first inequality in (117) uses a well-known property for stochastic
dominance when summing random variables: if $X_{1}$ and $X_{2}$ are
independent, $Y_{1}$ and $Y_{2}$ are independent, and $X_{i}\leq_{ST}Y_{i}$
for $i=1,2$, then $X_{1}+X_{2}\leq_{ST}Y_{1}+Y_{2}$. From the strong Markov
property of PDMPs, the $\mathcal{F}(\tau_{k}^{j})$-measurable quantity
$\sum_{i=1}^{j}Q_{k;l}^{i}$ and the $\mathcal{F}(\tau_{k}^{j+1})$-measurable
quantity $Q_{k;l}^{j+1}$ are conditionally independent under
$\mathbb{P}(\cdot|X((\tau_{k}^{j+1})^{-})$. The last inequality uses the same
property of stochastic dominance along with the induction hypothesis.

With (117) we then obtain our result from Lemma 3, with

$\displaystyle\mathbb{P}\left(\max_{l\leq
M/\delta}\left(\pi^{n}_{1}(I_{l})-\bar{L}\bar{p}_{l}\right)>\varepsilon\right)\leq\sum_{l\leq
M/\delta}\mathbb{P}\left(\pi^{n}_{1}(I_{l})-\bar{L}\bar{p}_{l}>\varepsilon\right)$
(118) $\displaystyle\leq\sum_{l\leq
M/\delta}\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\bar{L}n}\bar{Q}_{l}^{i}-\bar{L}\bar{p}_{l}>\varepsilon\right)\leq\frac{2M}{\delta}\exp(-2n\varepsilon^{2}/\bar{L}).$

∎

While Lemma 9 is useful for computing total numbers in Species 1 and 2, we
find in Section 4.4 that for comparing the PDMP to our deterministic
discretization, it is necessary to consider another estimate of $\pi_{i}$ in
which the bounds on $\Delta L(t_{k})$ and $P_{k}^{j}$ may not hold with small
probability. This differs from Lemma 9, in which assume such bounds occur over
all paths given an appropriate initial condition.

For the next lemma and in many other places, we will frequently rely on an
elementary inequality derived from the law of total probability: for events
$C$ and $D$,

$\mathbb{P}(C)=\mathbb{P}(C|D)\mathbb{P}(D)+\mathbb{P}(C|D^{c})\mathbb{P}(D^{c})\leq\mathbb{P}(C|D)+\mathbb{P}(D^{c}).$
(119)

######  Lemma 10.

Let $\mathcal{T}$ be an event such that

$\displaystyle\mathcal{T}\subset\\{\underline{L}\leq\Delta
L^{n}\leq\bar{L}\\}\cap\mathcal{\\{}P_{1}^{j}(t,I_{l})\in[\underline{p}_{l},\bar{p}_{l}]:l\leq
M/\delta,j=1,2,t\in\Delta t_{1}\\},$

where $\underline{L},\bar{L},\underline{p}_{l},$ and $\bar{p}_{l}$ are
$[0,1]$-valued constants. Suppose for some $r(\delta,n)\in[0,1)$ that

$\mathbb{P}(\mathcal{T}^{c})\leq r(\delta,n).$ (120)

Then for $j=1,2$ and $\varepsilon>0$,

$\displaystyle\mathbb{P}\left(\max_{l\leq
M/\delta}\left(\pi^{n}_{j}(t_{1},I_{l})-\bar{L}\bar{p}_{l}\right)>\varepsilon\Big{|}\mathcal{T}\right)\leq\frac{2M}{\delta}\mathbb{\exp}(-2n\varepsilon^{2}/\bar{L})+\frac{M}{\delta}\left(\frac{r}{1-r}+\frac{2r\bar{L}n}{(1-r)^{2}}\right)$
(121)

and

$\displaystyle\mathbb{P}\left(\min_{l\leq
M/\delta}\left(\pi^{n}_{j}(t_{1},I_{l})-\underline{L}\underline{p}_{l}\right)<-\varepsilon\Big{|}\mathcal{T}\right)\leq\frac{2M}{\delta}\mathbb{\exp}(-2n\varepsilon^{2}/\underline{L})+\frac{M}{\delta}(2r\bar{L}n+r).$
(122)

###### Proof.

We show (121) and (122) for $j=1$. We first show by induction that

$\displaystyle\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}>c\right)\leq\mathbb{P}\left(\sum_{i=1}^{j}\bar{Q}_{l}>c\right)+\frac{2rj}{1-r},\quad
j=1,\dots,\bar{L}n.$ (123)

We will condition on the $\mathcal{F}(X(\tau_{1}^{i-}))$-measurable event

$\mathcal{T}_{i}=\\{\underline{p}_{l}\leq
P_{1}^{1}(\tau_{1}^{i-},I_{l})\leq\bar{p}_{l}\\}\supseteq\mathcal{T}.$ (124)

The base case for (123) follows from (119) and (120) since

$\displaystyle\mathbb{P}\left(Q_{1}>c\right)$
$\displaystyle\leq\mathbb{P}\left(Q_{1}>c|\mathcal{T}_{1}\right)+\mathbb{P}(\mathcal{T}_{1}^{c})\leq\frac{\mathbb{P}\left(\bar{Q}_{1}>c\right)}{1-r}+r$
(125)
$\displaystyle\leq\mathbb{P}\left(\bar{Q}_{1}>c\right)+\frac{r}{1-r}+r\leq\mathbb{P}\left(\bar{Q}_{1}>c\right)+\frac{2r}{1-r}.$

For the inductive step, assuming (123) holds for $1\leq j<\bar{L}n$, we use
the strong Markov property of PDMPs to show

$\displaystyle\mathbb{P}\left(\sum_{i=1}^{j+1}Q_{l}>c\right)\leq\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}+Q_{j+1}>c\Big{|}\mathcal{T}_{j+1}\right)+\mathbb{P}(\mathcal{T}_{j+1}^{c})$
(126)
$\displaystyle\leq\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}+\bar{Q}_{j+1}>c\Big{|}\mathcal{T}_{j+1}\right)+r\leq\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}+\bar{Q}_{j+1}>c\right)+\frac{2r}{1-r}$
$\displaystyle\leq\mathbb{P}\left(\sum_{i=1}^{j+1}\bar{Q}_{i}>c\right)+\frac{2rj}{1-r}+\frac{2r}{1-r}=\mathbb{P}\left(\sum_{i=1}^{j+1}\bar{Q}_{i}>c\right)+\frac{2r(j+1)}{1-r}.$

From calculations similar to (114)-(117), we use (115) to show

$\displaystyle\mathbb{P}(\pi^{n}_{1}(I_{l})>c|\mathcal{T})\leq\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\bar{L}n}Q_{l}>c\Big{|}\mathcal{T}\right)$
(127)
$\displaystyle\leq\frac{1}{1-r}\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\bar{L}n}Q_{l}>c\right)\leq\frac{1}{1-r}\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\bar{L}n}\bar{Q}_{l}>c\right)+\frac{2r\bar{L}n}{(1-r)^{2}}$
$\displaystyle\leq\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\bar{L}n}\bar{Q}_{l}>c\right)+\frac{r}{1-r}+\frac{2r\bar{L}n}{(1-r)^{2}}.$

For the lower bound, we note

$\displaystyle\mathbb{P}(Q_{l}^{i}=1)\geq\mathbb{P}(Q_{l}^{i}=1|\mathcal{T}_{i})\mathbb{P}(\mathcal{T}_{i})\geq\mathbb{P}(\underline{Q}_{l}^{i}=1)-r(\delta,n).$
(128)

We also note, for an event $A$ and $j\leq\bar{L}n$,

$\displaystyle\mathbb{P}(A)-r\leq\mathbb{P}(A)-\mathbb{P}(\mathcal{T}_{j}^{c})\leq\mathbb{P}(A)-\mathbb{P}(\mathcal{T}_{j}^{c})\mathbb{P}(A|\mathcal{T}_{j}^{c})=\mathbb{P}(\mathcal{T}_{j})\mathbb{P}(A|\mathcal{T}_{j})\leq\mathbb{P}(A|\mathcal{T}_{j}).$
(129)

We again use induction to show

$\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}>c\right)\geq\mathbb{P}\left(\sum_{i=1}^{j}\bar{Q}_{i}>c\right)-2rj,\quad
j=1,\dots,\bar{L}n.$ (130)

Showing the base case is similar to (128). For the induction step,

$\displaystyle\mathbb{P}\left(\sum_{i=1}^{j+1}Q_{l}>c\right)\geq\mathbb{P}\left(\sum_{i=1}^{j+1}Q_{l}>c\Big{|}\mathcal{T}_{j+1}\right)\mathbb{P}(\mathcal{T}_{j+1})\geq\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}+Q_{j+1}>c\Big{|}\mathcal{T}_{j+1}\right)-r$
(131)
$\displaystyle\geq\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}+\underline{Q}_{j+1}>c\Big{|}\mathcal{T}_{j+1}\right)-r\geq\mathbb{P}\left(\sum_{i=1}^{j}Q_{l}+\underline{Q}_{j+1}>c\right)-2r$
$\displaystyle\geq\mathbb{P}\left(\sum_{i=1}^{j+1}\underline{Q}_{i}>c\right)-2r(j+1).$

With (130), we then can show

$\displaystyle\mathbb{P}(\pi^{n}_{1}(I_{l})>c|\mathcal{T})\geq\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\underline{L}n}Q_{l}>c\Big{|}\mathcal{T}\right)$
(132)
$\displaystyle\geq\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\underline{L}n}Q_{l}>c\right)-r\geq\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\underline{L}n}\underline{Q}_{l}>c\right)-2r\bar{L}n-r.$

Upon taking complements, we arrive at

$\mathbb{P}(\pi^{n}_{1}(I_{l})<c|\mathcal{T})\leq\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\underline{L}n}\underline{Q}_{l}<c\right)+2r\bar{L}n+r.$
(133)

The result then follows from mirroring the calculations of (118), with

$\displaystyle\mathbb{P}\left(\max_{l\leq
M/\delta}\left(\pi^{n}_{1}(I_{l})-\bar{L}\bar{p}_{l}\right)>\varepsilon\Big{|}\mathcal{T}\right)\leq\sum_{l\leq
M/\delta}\mathbb{P}\left(\pi^{n}_{1}(I_{l})-\bar{L}\bar{p}_{l}>\varepsilon\Big{|}\mathcal{T}\right)$
$\displaystyle\leq\sum_{l\leq
M/\delta}\mathbb{P}\left(\frac{1}{n}\sum_{i=1}^{\bar{L}n}\bar{Q}_{l}^{i}-\bar{L}\bar{p}_{l}>\varepsilon\right)+\frac{M}{\delta}\left(\frac{r}{1-r}+\frac{2r\bar{L}n}{(1-r)^{2}}\right)$
$\displaystyle\leq\frac{2M}{\delta}\mathbb{\exp}(-2n\varepsilon^{2}/\bar{L})+\frac{M}{\delta}\left(\frac{r}{1-r}+\frac{2r\bar{L}n}{(1-r)^{2}}\right).$
(134)

A similar calculation using (133) yields (122). ∎

###  4.3 PDMP lemmas on growth

In this section, we derive bounds for total number in an interval. Our first
estimate is a simple pathwise bound between time intervals.

######  Lemma 11.

For all realizations of $X^{n}(t)$, if $s\in[t_{k-1},t_{k})$, then

$m_{\delta}^{n}(s)\leq 3m_{\delta}^{n}(t_{k-1}).$ (135)

###### Proof.

For Species 2, note that $m^{2;n}_{\delta}(t)$ is decreasing in $t$. As for
Species 1, a particle in an interval $I$ of size $\delta$ at time
$s\in[t_{k-1},t_{k})$ must have been located, at time $t_{k-1}$, in $I$ in
Species 2 or $I+\delta$ in either Species 1 or 2, so that

$\displaystyle\mu_{1}^{n}(s,I)$
$\displaystyle\leq\mu_{1}^{n}(t_{k-1},I)+\mu_{1}^{n}(t_{k-1},I+\delta)+\mu_{2}^{n}(t_{k-1},I+\delta),$
(136) $\displaystyle\mu_{2}^{n}(s,I)$
$\displaystyle\leq\mu_{2}^{n}(t_{k-1},I).$ (137)

Taking the supremum over length $\delta$ intervals then yields (135). ∎

From (35), there is $n_{1}(\bar{f})\geq n_{0}(\bar{f})$ such that for
$n>n_{1}(\bar{f})$, we can use the constant

$C_{4}=\inf_{n\geq n_{1}}N^{n}_{2}(T_{2}(\bar{f}))>N_{2}(0)/2.$ (138)

For our next lemma, we compare $m_{\delta}^{n}(t_{k})$ with its deterministic
analog $\bar{m}_{\delta}(\tau)$, defined through the recurrence

$\bar{m}_{\delta}(t_{k})=\bar{m}_{\delta}(t_{k-1})+24C_{4}\bar{m}_{\delta}(t_{k-1})^{2},\quad\bar{m}_{\delta}(0)=m_{\delta}^{n}(0).$
(139)

We may use the same reasoning as in Lemma 2 to show that there exists
$\delta^{p}>0$ such that for $0<\delta<\delta^{p}$, we can find
$n^{p}(\delta,\bar{f})>n_{1}(\bar{f})$ such that for $n>n^{p}(\delta,\bar{f})$
there exist positive constants $C_{5},\hat{C}_{5}>0$ such that

$\hat{C}_{5}\delta\leq\bar{m}_{\delta}(t_{k})\leq C_{5}\delta.$ (140)

For the remaining lemmas in this section, we will assume that
$0<\delta<\delta^{p}$ and $n>n^{p}(\delta)$.

Our interest is in whether interval growth in the particle system exceeds that
of $\bar{m}_{\delta}$. Whether this occurs is expressed in the sequence of
events

$\mathcal{A}_{k}=\\{m_{\delta}^{n}(t_{k})>\bar{m}_{\delta}(t_{k})\\},\quad
0\leq t_{k}\leq T_{2}.$ (141)

Our next lemma shows that conditioned under $\mathcal{A}_{k-1}^{c}$, we can
use Lemma 9 to obtain a concentration inequality for $\pi_{j}$ and
consequently $m_{\delta}^{n}(t_{k})$.

######  Lemma 12.

Let $j=1,2.$ For $0<t_{k}\leq T_{2}$,

$\mathbb{P}\left(\sup_{|I|\leq\delta}\pi^{n}_{j}(t_{k},I)>12C_{4}\bar{m}_{\delta}(t_{k-1})^{2}|\mathcal{A}_{k-1}^{c}\right)\leq\frac{2M}{\delta}\exp(-\tilde{C}_{6}\delta^{3}n),$
(142)

where $\tilde{C}_{6}=72C_{4}^{2}\hat{C}_{5}^{3}$.

###### Proof.

We give a proof for $j=1$, with the proof for $j=2$ being nearly identical.
First, observe that under $\mathcal{A}_{k-1}^{c}$ there will be at most
$\Delta L^{n}(t_{k})n\leq\bar{m}_{\delta}(t_{k-1})n$ mutations during $\Delta
t_{k}=[t_{k-1},t_{k})$, , since only particles contained in $[0,\delta)$ at
$t_{k-1}$ for Species 1 and 2 may possibly reach the origin in Species 1\.
Under the event $\mathcal{A}_{k-1}^{c}$, we use Lemma 11 to show mutation
probabilities are uniformly bounded from above by the deterministic quantity

$P_{k}^{1}(\tau,I)=\frac{\mu_{2}^{n}(t^{-},I+t_{k}-t)}{N_{2}^{n}(t^{-})}\leq
3C_{4}\bar{m}_{\delta}(t_{k-1}).$ (143)

Thus, using From (140), we can then use Lemma 9 with
$\bar{L}=\bar{m}_{\delta}(t_{k-1})$,
$\bar{p}_{l}=3C_{4}\bar{m}_{\delta}(t_{k-1})$, and
$\varepsilon=6C_{4}(\bar{m}_{\delta}(t_{k-1}))^{2}$, which gives

$\displaystyle\mathbb{P}\left(\max_{l\leq
M/\delta}\pi^{n}_{1}(t_{k},I_{l})>6C_{4}(\bar{m}_{\delta}(t_{k-1}))^{2}|\mathcal{A}_{k-1}^{c}\right)\leq\frac{2M}{\delta}\exp(-\tilde{C}_{6}\delta^{3}n).$
(144)

Note that to apply Lemma 9 we used the fact that $X^{n}(t)$ is homogeneous.
Indeed, we can write the left hand side of (144) as

$\mathbb{P}\left(\max_{l\leq
M/\delta}\pi^{n}_{1}(t_{k},I_{l})>6C_{4}(\bar{m}_{\delta}(t_{k-1}))^{2}|\mathcal{A}_{k-1}^{c}\right)=\mathbb{P}\left(\max_{l\leq
M/\delta}\pi^{n}_{1}(t_{1},I_{l})>6C_{4}(\bar{m}_{\delta}(t_{k-1}))^{2}|\tilde{\mathcal{A}}^{c}\right),$
(145)

where $\tilde{\mathcal{A}}=\\{m_{\delta}^{n}(0)>\bar{m}_{\delta}(t_{k-1})\\}$
gives requirements on the initial condition so that $\Delta
L(t_{1})\leq\bar{L}$, and $P_{k}^{1}(\tau,I_{l})\leq\bar{p}_{l}$ pathwise in
$\Delta t_{1}$.

We can extend (144) to hold over all intervals of size less than $\delta$, not
just those on a grid. This is done by noting that for any measure $\mu$, if
$\mu(I_{k})\leq a$ on a uniform grid $I_{k}$ of size $\delta$, then for any
$I$ with $|I|\leq\delta$, $\mu(I)\leq 2a$. Thus, (142) follows from (144) and
(140), since

$\displaystyle\mathbb{P}\left(\sup_{I,|I|\leq\delta}\pi^{n}_{1}(t_{k},I)>12C_{4}\bar{m}_{\delta}(t_{k-1})^{2}|\mathcal{A}_{k-1}^{c}\right)\leq\mathbb{P}\left(\max_{l\leq
M/\delta}\pi^{n}_{1}(t_{k},I_{l})>6C_{4}\bar{m}_{\delta}(t_{k-1})^{2}|\mathcal{A}_{k-1}^{c}\right).$

∎

We can now derive a concentration inequality which shows that
$m^{n}_{\delta}(t_{k})=\mathcal{O}(\delta)$ with high probability.

######  Lemma 13.

For $0\leq t_{k}\leq T_{2},$

$\displaystyle\mathbb{P}(m_{\delta}^{n}(t_{k})>C_{5}\delta)\leq\mathbb{P}\left(\mathcal{A}_{k}\right)\leq\frac{4Mk}{\delta}\exp(-\tilde{C}_{6}\delta^{3}n).$
(146)

###### Proof.

The proof follows from induction, in which we assume

$\mathbb{P}\left(\mathcal{A}_{l}\right)\leq\frac{4Ml}{\delta}\exp(-\tilde{C}_{6}\delta^{3}n)$
(147)

holds for $0\leq l<k$. The base case for when $k=0$ follows since
$\mathbb{P}(\mathcal{A}_{0})=0$. To show the inductive step, we can use Lemma
12 and the recurrence inequalities (104) and (139) to show

$\displaystyle\mathbb{P}(\mathcal{A}_{k}|\mathcal{A}_{k-1}^{c})$ (148)
$\displaystyle\leq\mathbb{P}\left(m_{\delta}^{n}(t_{k-1})+\sum_{j=1}^{2}\sup_{I,|I|\leq\delta}\pi^{n}_{j}(t_{k},I)>\bar{m}_{\delta}(t_{k-1})+24C_{4}\bar{m}_{\delta}(t_{k-1})^{2}\Big{|}\mathcal{A}_{k-1}^{c}\right)$
$\displaystyle\leq\sum_{j=1}^{2}\mathbb{P}\left(\sup_{I,|I|\leq\delta}\pi^{n}_{j}(t_{k},I)>12C_{4}\bar{m}_{\delta}(t_{k-1})^{2}|\mathcal{A}_{k-1}^{c}\right)\leq\frac{4M}{\delta}\exp(-\tilde{C}_{6}\delta^{3}n).$

We may then apply (119) with $C=\mathcal{A}_{k}$ and
$D=\mathcal{A}_{k-1}^{c}$, and then apply (147) for $l=k-1$, along with (148)
and (140) to obtain the right inequality of (146) (the left inequality follows
immediately from (140)). ∎

We finish this subsection with an estimate for total mutations during $\Delta
t_{k}$ analogous to Lemma 3.

######  Lemma 14.

For $0<t_{k}\leq T_{2}$,

$\mathbb{P}(|\Delta
L^{n}(t_{k})-\mu_{1}^{n}(t_{k-1},[0,\delta])|>C_{6}\delta^{2})\leq\frac{4Mk}{\delta}\exp(-\tilde{C}_{7}\delta^{3}n),$
(149)

with $C_{6}=4C_{2}C_{5}^{2}$ and
$\tilde{C}_{7}=\tilde{C}_{6}\wedge(18C_{4}^{2}C_{5}^{3})$.

###### Proof.

Denote $\mathcal{M}_{\delta}^{n}(t_{k})$ as the normalized total number of
mutations that affect Species 2 particles in the interval $[0,\delta)$ during
time $[t_{k-1},t_{k})$. This may be written as

$\mathcal{M}_{\delta}^{n}(t_{k})=\frac{1}{n}\sum_{i=1}^{\Delta
L(t_{k})}M_{i}^{k},$ (150)

where $M_{i}^{k}$ is an indicator random variable for the event that the $i$th
mutation during $\Delta t_{k}$ occurs within $[0,\delta)$.

All particles of Species 1 located in $[0,\delta)$ at $t_{k-1}$ with reach the
origin and trigger a mutation by time $t_{k}$. The only other particles in the
system which potentially hit the origin are those initially located in
$[0,\delta)$ in Species 2 which have mutated during $\Delta t_{k}$. It follows
that under all paths in $X^{n}(t)$,

$|\Delta
L^{n}(t_{k})-\mu_{1}^{n}(t_{k-1},[0,\delta])|\leq\mathcal{M}^{n}_{\delta}(t_{k}).$
(151)

Thus proving (149) follows from showing an equivalent estimate on
$\mathcal{M}_{\delta}^{n}(t_{k})$. Toward that end, note that under
$\mathcal{A}_{k-1}^{c}$ the number of mutations during time $\Delta t_{k}$ is
less than $\bar{L}n=C_{5}\delta n$. Also, the probability of selecting a
Species 2 particle to mutate in $[0,\delta]$ at each mutation time during
$\Delta t_{k}$ is bounded by $\bar{p}=3C_{4}C_{5}\delta$. Let $B_{i}(q)$
denote an iid stream of Bernoulli random variables with parameter $q\in[0,1]$.
It follows that under $\mathbb{P}(\cdot|\mathcal{A}_{k-1}^{c})$, from
arguments similar to Lemma 9, that

$\mathcal{M}_{\delta}^{n}(t_{k})\leq_{ST}\frac{1}{n}\sum_{i=1}^{\bar{L}n}B_{i}(\bar{p}).\quad$
(152)

For $C_{6}=6C_{2}C_{5}^{2}$, from the Hoeffding inequality,

$\displaystyle\mathbb{P}(\mathcal{M}^{n}_{\delta}(t_{k})>C_{6}\delta^{2}|\mathcal{A}_{k-1}^{c})\leq\mathbb{P}\left(\sum_{i=1}^{\bar{L}n}B_{i}(\bar{p})/(\bar{L}n)-\bar{p}>\bar{p}\right)$
$\displaystyle\leq 2\exp(-2\bar{L}\bar{p}^{2}n)$ (153)
$\displaystyle=2\exp(-18C_{4}^{2}C_{5}^{3}\delta^{3}n).$

The lemma then follows from applying Lemma 13, (119), (147),and (153). ∎

###  4.4 Difference of total number on an interval

We now begin our estimates comparing the deterministic discretization
$\tilde{\mu}_{j}$ and empirical measures $\mu^{n}_{j}$. As in Section 3.2, our
focus is on differences of measures restricted to length $\delta$ intervals.

For inequalities related to bounding $P^{j}_{k}$ with $h_{\delta}^{j;n}$, we
will need to consider a modulus of continuity for the deterministic
discretization, defined by

$\tilde{\omega}(s,t_{k})=\sup_{I:|I|=\delta}\sum_{j=1}^{2}|\tilde{\mu}_{j}(t_{k},I+s)-\tilde{\mu}_{j}(t_{k},I)|,$
(154)

Fortunately, we can compare $\tilde{\omega}(\delta,t_{k})$ with
$\omega(\delta,t_{k})$, the modulus of continuity for the solutions to the
kinetic equations, through the following lemma.

######  Lemma 15.

There exists $C_{8}>0$ such that for $0<s\leq\delta$,

$\tilde{\omega}(s,t_{k})\leq C_{8}(\omega(\delta,0)\delta+\delta^{2}).$ (155)

###### Proof.

For $j=1,2$,

$\displaystyle|\tilde{\mu}_{j}(t_{k},I+s)-\tilde{\mu}_{j}(t_{k},I)|$ (156)
$\displaystyle\leq|\tilde{\mu}_{j}(t_{k},I+s)-\mu_{j}(t_{k},I+s)|+|\mu_{j}(t_{k},I+s)-\mu_{j}(t_{k},I)|+|\mu_{j}(t_{k},I)-\tilde{\mu}_{j}(t_{k},I)|.$

Summing over $j$ and taking suprema gives

$\displaystyle\tilde{\omega}(s,t_{k})\leq\omega(s,t_{k})\delta+2h_{\delta}(t_{k}).$
(157)

The result then follows from Lemmas 1 and 6. ∎

We now give a pathwise inequality over $\Delta t_{k}$ for comparing mutation
probabilities.

######  Lemma 16.

For $\tau\in\Delta t_{k}$, there exists $C_{9}>0$ such that for all paths in
$X^{n}(t),$

$\displaystyle|P^{j}_{k}(\tau,I)-P^{j}_{k}(t_{k-1},I)|\leq
C_{9}(h^{n}(t_{k-1})+\pi_{2}^{n}(t_{k})+(m^{n}(t_{k-1}))^{2}+\omega(\delta,0)\delta+\delta^{2}).$
(158)

###### Proof.

For $j=1$ (the proof for $j=2$ is similar), we write

$\displaystyle|P^{1}_{k}(\tau,I)-P^{1}_{k}(t_{k-1},I)|=\left|\frac{\mu_{2}^{n}(\tau,I+t_{k}-\tau)}{N^{n}(\tau)}+\frac{\mu_{2}^{n}(t_{k-1},I+\delta)}{N^{n}(t_{k-1})}\right|$
(159) $\displaystyle\leq
C_{4}(|\mu_{2}^{n}(\tau,I+t_{k}-\tau)-\mu_{2}^{n}(t_{k-1},I+\delta)|+\mu_{2}^{n}(t_{k-1},I+\delta)(N^{n}(\tau)-N^{n}(t_{k-1})))$
$\displaystyle\leq
C_{4}(|\mu_{2}^{n}(\tau,I+t_{k}-\tau)-\mu_{2}^{n}(t_{k-1},I+\delta)|+(m^{n}(t_{k-1}))^{2}.$

We may then break up terms further, with

$\displaystyle|\mu_{2}^{n}(\tau,I+t_{k}-\tau)-\mu_{2}^{n}(t_{k-1},I+\delta)|$
(160)
$\displaystyle\leq|\mu_{2}^{n}(\tau,I+t_{k}-\tau)-\mu_{2}^{n}(t_{k-1},I+t_{k}-\tau)|$
$\displaystyle+|\mu_{2}^{n}(t_{k-1},I+t_{k}-\tau)-\tilde{\mu}_{2}(t_{k-1},I+t_{k}-\tau)|$
$\displaystyle+|\tilde{\mu}_{2}(t_{k-1},I+t_{k}-\tau)-\tilde{\mu}_{2}(t_{k-1},I+\delta)|+|\tilde{\mu}_{2}(t_{k-1},I+\delta)-\mu_{2}^{n}(t_{k-1},I+\delta)|$
$\displaystyle\leq\pi_{2}^{n}(t_{k})+2h^{n}(t_{k-1})+\tilde{\omega}(\delta-
t_{k}-\tau,t_{k-1}).$

The result follows from applying Lemma (15). ∎

Here we collect previous results to form a high-probability event under which
we can bound $h^{n}_{\delta}(t_{k})$.

######  Lemma 17.

There exist positive constants $C_{10},C_{11},C_{12}$ such that the events

$\displaystyle\mathcal{C}(t_{k})$
$\displaystyle=\left\\{\sup_{l}\pi_{2}^{n}(t_{k},I_{l})>C_{10}\delta^{2}\right\\},$
(161) $\displaystyle\quad\mathcal{L}(t_{k})$ $\displaystyle=\cup_{t_{k}\leq
T_{2}}\\{|\Delta
L^{n}(t_{k})-\mu_{1}^{n}(t_{k-1},[0,\delta])|>C_{10}\delta^{2}\\},$ (162)
$\displaystyle\mathcal{D}(t_{k+1})$
$\displaystyle=\mathcal{C}(t_{k+1})^{c}\cap\mathcal{A}(t_{k})^{c}\cap\mathcal{L}(t_{k+1})^{c}.$

satisfy

$\displaystyle\mathbb{P}(\mathcal{D}(t_{k})^{c}|\mathcal{A}(t_{k-1})^{c})\leq\frac{C_{12}}{\delta}\exp(-C_{11}\delta^{3}n).$
(163)

###### Proof.

This follows immediately from applying (142) with (151) and (153) to the event

$\mathbb{P}(\mathcal{D}(t_{k})^{c}|\mathcal{A}(t_{k-1})^{c})\leq\mathbb{P}(\mathcal{C}(t_{k})|\mathcal{A}(t_{k-1})^{c})+\sum_{k\leq\lceil
T_{2}/\delta\rceil}\mathbb{P}(\mathcal{L}(t_{k})|\mathcal{A}(t_{k-1})^{c}).\qed$

The upshot of using $\mathcal{D}(t_{k})$ is that we may bound selection
probability and total losses by quantities which are $\mathcal{F}(t_{k-1})$
measurable. Furthermore, we may we use $\mathcal{D}(t_{k})$ for the event
$\mathcal{T}$ in Lemma 10 to produce concentration bounds for $h(t_{k})$.

######  Lemma 18.

Under $\mathcal{D}(t_{k})$, there exists $C_{13}$ such that for $t\in\Delta
t_{k}$,

$P_{k}^{j}(t,I_{l})\in[\underline{p}_{l}(t_{k-1}),\bar{p}_{l}(t_{k-1})],\qquad\Delta
L(t_{k})\in[\underline{L}^{n}(t_{k-1}),\bar{L}^{n}(t_{k-1})],$ (164)

where

$\displaystyle\bar{p}_{l}(t_{k-1})$
$\displaystyle=P^{2}_{k}(t_{k-1},I_{l})+C_{13}(h^{n}(t_{k-1})+\omega(\delta,0)\delta+\delta^{2}),$
(165) $\displaystyle\underline{p}_{l}(t_{k-1})$
$\displaystyle=P^{2}_{k}(t_{k-1},I_{l})-C_{13}(h^{n}(t_{k-1})+\omega(\delta,0)\delta+\delta^{2}),$
(166) $\displaystyle\bar{L}^{n}(t_{k-1})$
$\displaystyle=\mu_{1}^{n}(t_{k-1},[0,\delta])+C_{10}\delta^{2},$ (167)
$\displaystyle\underline{L}^{n}(t_{k-1})$
$\displaystyle=\mu_{1}^{n}(t_{k-1},[0,\delta]).$ (168)

###### Proof.

We obtain (165)-(166) from Lemma 16, and (167)-(168) follows from Lemma 14. ∎

######  Lemma 19.

Let

$H^{n}(t_{k}):=\delta^{3}+\omega(\delta,0)\delta^{2}+\delta
h^{n}(t_{k})+\delta^{2}\sum_{i<k}h^{n}(t_{i}),$ (169)

and let $\Pi$ be defined as in (106). Then there are positive constants
$C_{16},\tilde{C}_{16},C_{17}$ such that

$\mathbb{P}(\Pi^{n}(t_{k})>C_{17}H^{n}(t_{k-1})|\mathcal{D}(t_{k}))\leq\frac{C_{16}}{\delta}\exp(-\tilde{C}_{16}\delta^{5}n).$
(170)

###### Proof.

We begin with breaking up $\Pi^{n}$ as

$\displaystyle\Pi^{n}(t_{k})$ $\displaystyle\leq 2\max_{l\leq
M/\delta}\left|\frac{\tilde{\mu}_{2}(t_{k-1},I_{l})}{\tilde{N}_{2}(t_{k-1})}\Delta\tilde{L}(t_{k})-\mu^{n}_{1}(t_{k-1},[0,\delta])P^{2}_{k}(t_{k-1},I)\right|$
(171) $\displaystyle+\sum_{j=1}^{2}\max_{l\leq
M/\delta}\left|\pi_{j}^{n}(t_{k},I_{l})-\mu^{n}_{1}(t_{k-1},[0,\delta])P^{2}_{k}(t_{k-1},I_{l})\right|$
$\displaystyle:=G^{n}(t_{k})+\tilde{\Pi}^{n}(t_{k}).$

From Lemma 18, we can rewrite $\mu_{1}^{n}(t_{k-1},[0,\delta])$ in terms of
$\underline{L}^{n}(t_{k-1})$ and $\bar{L}^{n}(t_{k-1})$, and also
$P^{2}_{k}(t_{k-1},I_{l})$ in terms of $\underline{p}_{l}(t_{k-1})$ and
$\bar{p}_{l}(t_{k-1})$, from which we can then bound $\tilde{\Pi}^{n}(t_{k})$,
for some $C_{14}>0$, by

$\displaystyle\mathbb{P}(\tilde{\Pi}^{n}(t_{k})>\varepsilon|\mathcal{D}(t_{k}))$
(172)
$\displaystyle\leq\sum_{j=1}^{2}\mathbb{P}(\max_{l}\left|\pi_{j}^{n}(t_{k},I_{l})-\mu^{n}_{1}(t_{k-1},[0,\delta])P^{2}_{k}(t_{k-1},I_{l})\right|>\varepsilon/2|\mathcal{D}(t_{k}))$
$\displaystyle\leq\sum_{j=1}^{2}\Big{[}\mathbb{P}(\max_{l}(\pi_{j}^{n}(t_{k},I_{l})-\mu^{n}_{1}(t_{k-1},[0,\delta])P^{2}_{k}(t_{k-1},I_{l}))>\varepsilon/2|\mathcal{D}(t_{k}))$
$\displaystyle+\mathbb{P}(\min_{l}(\pi_{j}^{n}(t_{k},I_{l})-\mu^{n}_{1}(t_{k-1},[0,\delta])P^{2}_{k}(t_{k-1},I_{l}))<-\varepsilon/2|\mathcal{D}(t_{k}))\Big{]}$
$\displaystyle\leq\sum_{j=1}^{2}\Big{[}\mathbb{P}(\max_{l}(\pi_{j}^{n}(t_{k},I_{l})-\bar{L}^{n}(t_{k})\bar{p}^{n}_{l}(t_{k-1}))+C_{14}H(t_{k-1})>\varepsilon/2|\mathcal{D}(t_{k}))$
$\displaystyle+\mathbb{P}(\min_{l}(\pi_{j}^{n}(t_{k},I_{l})-\underline{L}^{n}(t_{k})\underline{p}^{n}_{l}(t_{k-1}))-C_{14}H(t_{k-1})<-\varepsilon/2|\mathcal{D}(t_{k}))\Big{]}.$

For all paths in $\mathcal{D}(t_{k}),$ by calculations similar to (87)-(90),
there exists a positive constant $C_{15}>C_{14}$ such that

$\displaystyle G^{n}(t_{k})\leq C_{15}H^{n}(t_{k-1}).$ (173)

and thus

$\displaystyle\mathbb{P}(\Pi^{n}(t_{k})>8C_{15}H^{n}(t_{k-1})|\mathcal{D}(t_{k}))\leq\mathbb{P}(\tilde{\Pi}^{n}(t_{k})>4C_{15}H^{n}(t_{k-1})|\mathcal{D}(t_{k})).$
(174)

Consider a path $\omega:[0,t_{k-1}]\rightarrow E$ such that
$\omega\in\mathcal{A}(t_{k-1})^{c}$. We now invoke (174), (172), and Lemma 10
with $\varepsilon(t_{k-1})=C_{15}H^{n}(t_{k-1})$, $C_{17}=8C_{15}$, and
$r(\delta,n)=\frac{C_{12}}{\delta}\exp(-C_{11}\delta^{3}n)$ from (163) to
obtain

$\displaystyle\mathbb{P}(\Pi^{n}(t_{k})>C_{17}H^{n}(t_{k-1};\omega)|\mathcal{D}(t_{k}))$
(175) $\displaystyle\leq\sum_{j=1}^{2}\mathbb{P}\left(\max_{l\leq
M/\delta}\left(\pi^{n}_{j}(t_{k},I_{l})-\bar{L}^{n}(t_{k-1};\omega)\bar{p}^{n}_{l}(t_{k-1};\omega)\right)>\varepsilon^{n}(t_{k-1};\mathbf{x})|\mathcal{D}(t_{k})\right)$
$\displaystyle+\sum_{j=1}^{2}\mathbb{P}\left(\min_{l\leq
M/\delta}\left(\pi^{n}_{j}(t_{k},I_{l})-\underline{L}^{n}(t_{k-1};\omega)\underline{p}^{n}_{l}(t_{k-1};\omega)\right)<-\varepsilon^{n}(t_{k-1};\mathbf{x})|\mathcal{D}(t_{k})\right)$
$\displaystyle\leq\frac{8M}{\delta}\exp(-2C_{15}^{2}H^{n}(t_{k-1};\omega)^{2}n/\bar{L}^{n}(t_{k-1};\omega))$
(176)
$\displaystyle+\frac{2M}{\delta}\left(\frac{r}{1-r}+\frac{2rn\bar{L}^{n}(t_{k-1};\omega)}{1-r}+2rn\bar{L}^{n}(t_{k-1};\omega)+r\right)$
(177) $\displaystyle:=J_{1}(t_{k-1},n;\omega)+J_{2}(t_{k-1},n;\omega).$ (178)

By increasing $n^{p}$ used in obtain (140), if necessary, elementary
calculations show that for $n>n^{p}$,

$J_{2}(t_{k-1},n;\omega)\leq\frac{18M}{\delta^{2}}\exp(-C_{11}\delta^{3}n/2)\cdot\bar{L}^{n}(t_{k-1};\omega).$
(179)

On the other hand, since $\mathcal{D}(t_{k})\subset\mathcal{A}(t_{k-1})^{c}$,
for $X(t_{k-1})=\omega^{\prime}\in\mathcal{A}(t_{k-1})$ we have

$\mathbb{P}(\Pi^{n}(t_{k})>C_{17}H^{n}(t_{k-1};\omega^{\prime})|\mathcal{D}(t_{k}))=0.$
(180)

From the law of total probability,

$\displaystyle\mathbb{P}(\Pi^{n}(t_{k})>C_{17}H^{n}(t_{k-1})|\mathcal{D}(t_{k}))$
(181)
$\displaystyle\leq\mathbb{E}[J_{1}(t_{k-1},n)|\mathcal{A}(t_{k-1})^{c}]+\mathbb{E}[J_{2}(t_{k-1},n)|\mathcal{A}(t_{k-1})^{c}]$
$\displaystyle\leq\frac{C_{16}}{\delta}\exp(-\tilde{C}_{16}\delta^{5}n)$

for a sufficiently small $\tilde{C}_{16}>0$ and sufficiently large $C_{16}>0$.
In the last inequality, we used the simple pathwise bound of
$H^{n}(t_{k})\geq\delta^{3}$ and that $\bar{L}=\mathcal{O}(\delta)$ under
$\mathcal{A}(t_{k-1})^{c}$. ∎

####  4.4.1 Proof of Theorem 3

We consider the events

$\displaystyle\mathcal{H}(t_{k})=\cup_{l\leq
k}\\{h^{n}(t_{k})>h^{n}(t_{k-1})+C_{17}H^{n}(t_{k-1})\\},$ (182)
$\displaystyle\mathcal{B}(t_{k};C)=\\{d(\tilde{\mu}(t_{k}),\mu^{n}(t_{k}))>C(\delta+\omega(\delta,0))\\}.$
(183)

Using the same argument given in Lemma 6, under $\cap_{t_{k}\leq
T_{2}}\mathcal{H}(t_{k})^{c}$ and a sufficiently large $C_{18}$,

$h^{n}(t_{k})\leq C_{18}\mathcal{(}\delta^{2}+\delta\omega(\delta,0))\quad
t_{k}\leq T_{2}.$ (184)

We may then use (184) with Lemma 7 to show that for a sufficiently large
$C_{19}$,

$\cap_{t_{k}\leq T_{2}}\mathcal{H}(t_{k})^{c}\subseteq\cap_{l\leq
k}\mathcal{B}(t_{k};C_{19})^{c}$ (185)

and that for sufficiently large $C_{20}$ and small $\tilde{C}_{20}$,

$\displaystyle\mathbb{P}\left(\max_{t_{k}\leq
T_{2}}d(\tilde{\mu}(t_{k}),\mu^{n}(t_{k}))>C_{19}(\delta+\omega(\delta))\right)=\mathbb{P}(\cup_{t_{k}\leq
T_{2}}\mathcal{B}(t_{k};C_{19}))$ (186) $\displaystyle\leq\sum_{t_{k}\leq
T_{2}}\mathbb{P}(\mathcal{H}(t_{k})|\mathcal{D}(t_{k}))+\mathbb{P}(\mathcal{D}(t_{k})^{c})$
$\displaystyle\leq\sum_{t_{k}\leq
T_{2}}\mathbb{P}(\Pi^{n}(t_{k})>C_{17}H^{n}(t_{k-1})|\mathcal{D}(t_{k}))+\mathbb{P}(\mathcal{D}(t_{k})^{c})$
$\displaystyle\leq\frac{C_{20}}{\delta^{2}}\exp(-\tilde{C}_{20}\delta^{5}n).$

To conclude, we may replace the maximum in (186) with a supremum. Indeed,
since $\tilde{\mu}(t)$ is constant during time intervals $\Delta t_{k}$, for
$t\leq T_{2}$ and $K=\lfloor\frac{t}{\delta}\rfloor$,

$d(\tilde{\mu}(t),\mu^{n}(t))\leq
d(\tilde{\mu}(t_{K}),\mu^{n}(t_{K}))+d(\mu^{n}(t),\mu^{n}(t_{K})).$ (187)

During $\Delta t_{k}$, an interval can change its total number by at most
$\sum_{j=1,2}\pi_{j}(t_{k},I)$, and thus

$d(\mu^{n}(t),\mu^{n}(t_{K}))\leq\sum_{\begin{subarray}{c}l\leq M/\delta\\\
j=1,2\end{subarray}}\pi_{j}(t_{k},I_{l})\leq\frac{M}{\delta}\sup_{\begin{subarray}{c}l\leq
M/\delta\\\ j=1,2\end{subarray}}\pi_{j}(t_{k},I_{l}).$ (188)

Then for sufficiently large $C_{p}>2C_{19}$ and $C_{2}^{p}$, and sufficiently
small $C_{3}^{p}$.

$\displaystyle\mathbb{P}\left(\sup_{t\leq
T_{2}}d(\tilde{\mu}(t),\mu^{n}(t))>C^{p}(\delta+\omega(\delta))\right)$ (189)
$\displaystyle\leq\mathbb{P}\left(\max_{t_{k}\leq
T_{2}}d(\tilde{\mu}(t_{k}),\mu^{n}(t_{k}))>C_{19}(\delta+\omega(\delta))\right)+\sum_{j=1}^{2}\mathbb{P}\left(\sup_{\begin{subarray}{c}l\leq
M/\delta\\\ t_{k}\leq
T_{2}\end{subarray}}\pi_{j}(t_{k},I_{l})>\frac{C_{p}\delta^{2}}{2M}\right)$
$\displaystyle\leq\frac{C_{2}^{p}}{\delta^{2}}\exp(-C_{3}^{p}\delta^{5}n).$

Finally, through a similar stitching argument argument appealed to in the
proof of Theorem 2 at the end of Section 3, we may extend (189) to any
$T^{\prime}<T(\bar{f})$, which yields Theorem 3.

##  5 Proof of Theorem 1

For deriving an explicit solution for (6)-(7), we assume
$(f_{1}(x,t),f_{2}(x,t))\in Z^{2}$ for $t\in T(\bar{f})$, and show that such a
solution must be given by the explicit expressions (16)-(19), and later verify
that this solution is in fact in $Z^{2}$.

We begin by integrating (7) over space, giving

$N_{2}(t)=N_{2}(0)-L(t).$ (190)

This implies that $N_{2}(t)$ is differentiable, with

$\dot{N}_{2}(t)=-a(t).$ (191)

Substituting (191) into (7) yields the simple form

$f_{2}(x,t)=\frac{N_{2}(t)}{N_{2}(0)}\bar{f}_{2}(x).$ (192)

Another substitution of (192) into (6) then gives

$\displaystyle
f_{1}(x,t)=\bar{f}_{1}(x+t)+\int_{0}^{t}\frac{\bar{f}_{2}(x+t-s)}{N_{2}(0)}a(s)ds.$
(193)

It remains to express $a(t)$ and $N_{2}(t)$ in terms of initial conditions.
Setting $x=0$, we arrive at the closed equation

$a(t)=\bar{f}_{1}(t)+\int_{0}^{t}\frac{\bar{f}_{2}(x+t-s)}{N_{2}(0)}a(s)ds.$
(194)

Denoting the probability density
$\hat{f}_{2}(s)=\frac{\bar{f}_{2}(s)}{N_{2}(0)}$, we may rewrite (194) as the
integral equation

$a(t)=\bar{f}_{1}(t)+\int_{0}^{t}a(t-s)\hat{f}_{2}(s)ds.$ (195)

Equation (195) is a renewal equation, which has been studied extensively in
probability theory (see [4, Chapter XI] for an introduction). It is well-known
that there exists a unique solution for (195) given by

$\displaystyle a(t)$
$\displaystyle=\sum_{j=0}^{\infty}\hat{f}_{2}^{*(j)}(t)*\bar{f}_{1}(t):=Q_{\hat{f}_{2}}(t)*\bar{f}_{1}(t),$
(196)

where the exponent $*(k)$ denotes $k$-fold self-convolution. Then (17) follows
directly from (190). For a locally bounded density $p$, the operator
$Q_{p}(t)$ is also locally bounded, (see Thm. 3.18 of [11]) . Thus, it is
clear that $a(t)$ and $N(t)$ are both positive, locally bounded, and
continuous for $0\leq t<T(\bar{f})$, and subsequently that
$(f_{1}(x,t),f_{2}(x,t))\in Z^{2}$. This completes the derivation for part (a)
for Theorem 1.

For showing part (b), the only ambiguity is in establishing for a fixed
$t\in[0,T(\bar{f}))$, the map $p\mapsto Q_{p}(t)$ is in
$C(L^{1}([0,T(\bar{f})),\mathbb{R}_{+})$. We will use a probabilistic
argument. Let $X_{i}^{(p)}$ $i=1,2,\dots$ denote a sequence of $[0,\infty)$
valued, iid random variables, each with a probability density $p\in
L^{1}(\mathbb{R}_{+})$. The number of renewals up to time $t<\infty$ is given
by

$N_{p}(t)=\sup\left\\{k:\sum_{i=1}^{k}X_{i}^{(p)}\leq t\right\\}.$ (197)

In renewal theory, $Q_{p}(t)$ is the well-known renewal density, satisfying

$\int_{0}^{t}Q_{p}(t)=\mathbb{E}[N_{p}(t)]-1.$ (198)

Each term in the sum of $Q_{p}(t)$ also has a probabilistic interpretation,
with

$c_{k}^{(p)}(t):=\int_{0}^{t}f^{*(k)}(t)=\mathbb{P}\left(\sum_{i=1}^{k}X_{i}^{(p)}\leq
t\right),\quad k\geq 1.$ (199)

Estimates for the decay of $c_{k}^{(p)}(t)$ as $k\rightarrow\infty$ can be
obtained from Markov’s inequality, with

$\displaystyle c_{k}^{(p)}(t)$
$\displaystyle=\mathbb{P}\left(\exp\left(-\sum_{i=1}^{k}X_{i}^{(p)}\right)\geq
e^{-t}\right)$ (200) $\displaystyle\leq
e^{t}\mathbb{E}[\exp(-X_{1}^{(p)})]^{k}.$ (201)

As $X^{(p)}$ is a non-deficient random variable, $\mathbb{P}(X=0)\neq 1.$ Then
$\mathbb{E}[\exp(-X_{1}^{(p)})]<1$, and thus $c_{k}^{(p)}(t)$ decays
exponentially as $k\rightarrow\infty$.

To show continuity, fix $p\in L^{1}(\mathbb{R}_{+})$, $t\in[0,T(\bar{f}))$,
and let $\varepsilon>0$. From (200)-(201), $c_{k}^{p}(t)$ is summable in $k$,
so we may choose a $K>0$ such that

$\sum_{i=K}^{\infty}c_{k}^{(p)}(t)<\varepsilon/6.$ (202)

Since $\mathbb{E}[\exp(-X_{1}^{(p)})]$ varies continuously with respect with
$p$ in $L^{1}(\mathbb{R}_{+})$, a similar calculation to (200)-(201) implies
that the map $p\mapsto c_{k}^{(p)}(t)$ is also in
$C(L^{1}(\mathbb{R}_{+}),\mathbb{R}_{+})$ for all $k\geq 1$. Furthermore, tail
sums of $c_{k}^{(p)}(t)$ also vary continuously in the $p$ variable. Thus,
there exists $\delta>0$ such that if $\tilde{p}\in L^{1}(\mathbb{R}_{+})$
satisfies $\|p-\tilde{p}\|_{L^{1}(\mathbb{R}_{+})}<\delta$, then both

$\sum_{i=K}^{\infty}c_{k}^{(\tilde{p})}(t)<\varepsilon/3\quad\hbox{and}\quad\sum_{i=1}^{K-1}|c^{(p)}_{k}(t)-c^{(\tilde{p})}_{k}(t)|<\varepsilon/2$
(203)

hold. It then follows that

$\displaystyle|Q_{p}(t)-Q_{\tilde{p}}(t)|$
$\displaystyle\leq\sum_{i=1}^{K-1}|c^{(p)}_{k}(t)-c^{(\tilde{p})}_{k}(t)|+\sum_{i=K}^{\infty}c_{k}^{(p)}(t)+\sum_{i=K}^{\infty}c_{k}^{(\tilde{p})}(t)<\varepsilon.$
(204)

Acknowledgments: The author wishes to thank Anthony Kearsley and Paul Patrone
for providing guidance during his time as a National Research Council
Postdoctoral Fellow at the National Institute of Standards and Technology, and
also Govind Menon for several discussions regarding the manuscript.

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