still define $\varphi$ by continuity (canceling the corresponding factor in the numerator).

5.  The process can make infinitely many jumps in a single discrete time step (for example, when it starts from the step initial configuration $x_i(0)=-i$, $i\ge0$). However, for each $N$ the behavior of the particles $x_i$, $i\le N$, is independent from the one of with $i>N$. Therefore, the dynamics restricted to $x_1,\ldots,x_N$ is well-defined as a process with finitely many particles. These $N$-particle dynamics are compatible for various $N$, and so the dynamics of the process on infinite particle configurations having a rightmost particle is well-defined.

\[prop:nonnegativity\] Under the restrictions on parameters we have $\mathbf{P}_{\ell,g}(L)\ge0$ for all $\ell,g,L\in \mathbb{Z}_{\ge0}$, and $\sum_{L\in \mathbb{Z}_{\ge0}}\mathbf{P}_{\ell,g}(L) = 1$.

The fact that $\sum_{L\in \mathbb{Z}_{\ge0}}\mathbf{P}_{\ell,g}(L) = 1$ follows by interchanging the summations over $L$ and $p$ (the latter coming from ) and using $$\sum_{p=0}^{\ell}\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)=1,
        \qquad
        \sum_{L=p}^{\infty}\psi_{q, \nu \mu^{-1} q^{p}, \nu q^{g}, \nu^{2} q^{g+p}}(L-p)=1,$$ see .

Turning to proving the positivity, there are two cases depending on the sign of $\nu$. First, we have $\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell) \geq 0$ when $-1<\nu\le0$. Therefore, we can think that the $i$-th particle first moves $x_{i}(t) \to x_{i}(t)-p$ with 