# Self-Alignment of a Large-Area Dual-Atom-Interferometer Gyroscope Using
Parameter Decoupled Phase Seeking Calibrations

Zhan-Wei Yao  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Center for Cold
Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China  Hong-Hui Chen
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Innovation Academy for Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  School of Physics, University of
Chinese Academy of Sciences, Beijing 100049, China  Si-Bin Lu  State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation
Academy for Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  Center for Cold Atom Physics, Chinese Academy
of Sciences, Wuhan 430071, China  Run-Bing Li<EMAIL_ADDRESS>State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation
Academy for Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  Center for Cold Atom Physics, Chinese Academy
of Sciences, Wuhan 430071, China  Ze-Xi Lu  State Key Laboratory of Magnetic
Resonance and Atomic and Molecular Physics, Innovation Academy for Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  School of Physics, University of Chinese Academy of Sciences, Beijing
100049, China  Xiao-Li Chen  State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
School of Physics, University of Chinese Academy of Sciences, Beijing 100049,
China  Geng-Hua Yu  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Innovation Academy for Precision Measurement Science and
Technology, Chinese Academy of Sciences, Wuhan 430071, China  Center for Cold
Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China  Min Jiang
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Innovation Academy for Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  Center for Cold Atom Physics,
Chinese Academy of Sciences, Wuhan 430071, China  Chuan Sun  State Key
Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation
Academy for Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, China  School of Physics, University of Chinese
Academy of Sciences, Beijing 100049, China  Wei-Tou Ni  State Key Laboratory
of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for
Precision Measurement Science and Technology, Chinese Academy of Sciences,
Wuhan 430071, China  Jin Wang  State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China
Ming-Sheng Zhan<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China

###### Abstract

We realize a Mach-Zehnder-type dual-atom-interferometer gyroscope with an
interrogation arm of 40 cm length and the interference area up to 1.2 cm2. The
precise angular alignment of the large-scale separated Raman lasers is
demonstrated by seeking the phase intersection of Ramsey-Bord$\acute{e}$
interferometers after the gravity effect is compensated and by decoupling the
velocity dependent crosstalk phase shifts, and applied to build the Mach-
Zehnder atom interferometer. Then a compact inertial rotation sensor is
realized based on dual large-area Mach-Zehnder atom interferometers by
precisely aligning the large-scale separated Raman lasers, in which the
coherence is well preserved and the common noise is differentially suppressed.
The sensor presents a sensitivity of $1.5\times 10^{-7}$ rad/s/Hz1/2, and a
stability of $9.5\times 10^{-10}$ rad/s at 23000 s. The absolute rotation
measurement is carried out by adjusting the atomic velocity which corresponds
to modulating the scale factor.

###### pacs:

06.30.Gv, 37.25.+k, 03.75.Dg, 32.80.Qk

High-precision gyroscope plays an important role in fields of inertial
navigationLefevre2013a , fundamental physicsCiufolini2004a  and
geophysicsBrosche1998a . The studies on atom interferometers, in past two
decades, make it possible to develop an atom-interferometer gyroscope with
high rotation sensitivity and stabilityDurfee2006a ; Gauguet2009a ; Berg2015a
; Yao2018a ; Savoie2018a ; Dutta2016a ; Stockton2011a ; Xu2020a . Two types of
atom-interferometer gyroscopes with Mach-Zehnder (M-Z) and four-pulse
configurations were developed, and their features have been
discussedBarrett2014a . In principle, the M-Z interferometer is suitable for
the wider environments including the weightless condition in space, and is
much beneficial for building a spatial high-sensitivity gyroscope due to the
fact that the common-mode noise can be differentially suppressed in the dual
atom interferometersJentsch2004a ; Canuel2006a . The sensitivity of an atom-
interferometer gyroscope, based on Sagnac effect, scales with the atomic
interference-loop area. Building the large-area atom-interference loop is an
important avenue for improving the sensitivity of the atomic gyroscope. To
enlarge the interference-loop area, large-momentum-transfer techniques,
including Raman adiabatic passageKotru2015a , Bragg diffractionsChiow2011a
and Bloch oscillationMuller2009a , were developed. However, these techniques
are still in the infant hood, and their feasibilities are not well verified in
atom-interferometer gyroscopes.

As an effective and practical method for improving the sensitivity, the
interference-loop area can be enlarged by synchronously increasing the
interrogation arm and the atomic longitudinal velocity in the M-Z-type atom-
interferometer gyroscope. However, as the interrogation arm lengthened, to
build a closed interferometric loop, the separated Raman beams should be more
precisely aligned for combining two interferometric paths within the coherent
length of atomic wave packets. Using self-alignment of separated Raman lasers
with the contrast of Ramsey-Bord$\acute{e}$ (R-B) interferometers, the
interference-loop area was enlarged to 0.2 cm2 in the M-Z atom
interferometerTackmann2012a ; Yao2018a , but which is still smaller than one
in the four-pulse atom interferometerSavoie2018a ; Xu2020a . Here, to build a
larger-area M-Z-type atom interferometer, the coherent manipulation of atomic
wave packets will be limited by the angular misalignment of larger-scale
separated Raman lasers. The angular alignment of separated Raman lasers faces
a huge challenge and is still an obstacle for building the larger-area M-Z
atom interferometer. Compared with the contrast, the phase of the R-B
interferometer is more sensitive to the angular alignment of separated Raman
lasers. It was applied to analyze the systematic error in the four-pulse atom
fountain interferometer, where two Raman lasers are aligned at the 0.2
$\mu$rad levelAltorio2020a . While in the M-Z atom interferometer, to align
separated Raman lasers, the separated time ($T_{1}$) should be modulated,
which makes the precision for seeking the phase intersected point seriously
limited by the cross-coupled parameters including the gravity effect and the
velocity dependent crosstalk phase shiftTackmann2012a . Thus it is urgently
needed to develop an effective method for the precise angular alignment of
large-scale separated Raman lasers, because of which is a key step for
building the larger-area M-Z-type atom-interferometer gyroscope.

In this letter, we demonstrate an effective method for precisely aligning the
large-scale separated Raman lasers using parameter decoupled phase seeking
calibrations, and present a high-precision inertial rotation sensor with dual
large-area M-Z-type atom interferometers. First, the parameter decoupled phase
seeking calibration is performed by seeking the phase intersected point of
symmetric R-B interferometers after the gravity effect is compensated and by
decoupling the velocity dependent crosstalk phase shifts. Three large-scale
separated Raman lasers with an interrogation arm of 40 cm are precisely
aligned at the 0.4 $\mu$rad level. Then, dual M-Z atom interferometers are
realized with the interference-loop area of 1.2 cm2, which is the largest loop
area in the M-Z-type atom interferometer to our knowledge. Finally, a high-
precision inertial rotation sensor is realized based on dual large-area atom
interferometers, where the dead time is reduced and the common-mode noise is
differentially canceled. The performance is evaluated by Allan deviations and
it reaches a stability of $9.5\times 10^{-10}$ rad/s after integrating for
23000 s. The absolute rotation measurement is demonstrated by modulating the
atomic velocities.

Figure 1: (color online) Schematic diagram of the compact gyroscope with dual
large-area cold-atom interferometers (a), timing sequences of Ramsey-
Bord$\acute{e}$ (b) and Mach-Zehnder interferometers (c). The length of vacuum
chamber is less than 80 cm. The atoms trapped in two MOTs are launched and
then propagated in the vacuum chamber with opposite directions along the
symmetrical parabolic trajectories ($y$-$z$ plane). Three pairs of separated
Raman lasers, which are aligned by adjusting angles of mirrors ($x$/$y$-axes),
are counter-propagated along the gravity direction ($z$-axis), and two
symmetric large-area atom-interferometer loops are built for measuring the
horizontal rotation component. $T_{1}$ is the separated time between the
first/last two pulses, and $T_{2}$ is the delay time between the second and
third pulses in R-B interferometers. $T$ is the interrogation time between two
adjacent Raman pulses in M-Z interferometers. PBS: polarizing beam splitter,
HWP: half wave plate, and QWP: quarter wave plate.

The schematic apparatus is shown in Fig.1 (a). We build a compact inertial
sensor with dual cold atom interferometers. The length of the sensor is 80 cm,
in which the atomic propagating path is 56 cm and the interrogation arm is
extended to 40 cm. Cold atoms are counter-propagating in the x-y plane, with a
slight angle along the y-axis direction. The dead time caused by the non-
interference time during the atom propagating process, which limits the
sensitivity of the atom-interferometer gyroscopeBiedermann2013a ; Yao2016a ,
is reduced because the same vacuum window is shared between the trapping light
and the Raman lasers. The inner height of the vacuum chamber is more than 5
cm, which makes it possible to extend the interrogation time up to $T=$50 ms.
The laser system is similar to that in our previous workZhang2018a , and all
optical elements are integrated into a compact module with dimensions of 50
cm$\times$50 cm$\times$16 cm. The laser frequencies are tuned by acoustic-
optic modulators (AOMs) to match the energy levels of 87Rb atoms. The Raman
lasers are delivered to the interferometric area by a polarized-maintained
optical fiber. They are collimated by an achromatic doublet lens with a focal
length of 150 mm, and split into three pairs of Raman beams by polarized beam
splitters and half wave plates. Three tip/tilt platforms driven by the
piezoelectric ceramic (PZT), which can provide an angular resolution to less
than sub-microradians, are used to fix three mirrors.

The 87Rb atoms are loaded into two magneto-optical traps (MOTs) from the
background vapor in 0.5 s, and then are launched by the moving optical
molasses technique at a slight angle with respect to the horizontal direction.
When considering the gravity effect, the atomic trajectory has a position
displacement along the gravity direction during their propagations along the
horizontal direction. The atomic longitudinal velocities can be adjusted from
3.6 to 10 m/s by controlling the detuning of trapping lasers. When the atoms
are accelerated from the MOTs, the polarized gradient cooling is applied in a
few milliseconds. After the atoms are prepared to the magnetic insensitive
state of $|$F=1,mF=0$\rangle$ by a microwave field and the residual atoms on
the other states are blown away by the resonance lasers. The initial atoms are
manipulated by three pairs of separated Raman lasers along the gravity
direction. The geomagnetic field is compensated by three pairs of Helmholtz
coils and the ac Stark shift is canceled by adjusting the intensity ratio of
Raman lasersLi2008a ; Li2009a . The population of the state
$|$F=2,mF=0$\rangle$ is detected by the laser induced fluorescence signal, and
the atomic interference fringes are observed by scanning the differential
phase of the third pair of Raman lasers.

To construct a closed interferometric loop, the separated Raman beams should
be precisely aligned for combining two interferometric paths within the
coherent length of atomic wave packets. The alignment of Raman beams was
realized by the contrast of symmetric R-B interferometersYao2018a ;
Tackmann2012a , while it was less stringent on the alignment precision. Here,
the key avenue to realize a M-Z interferometer is how to transit from two
symmetric R-B interferometers to one M-Z interferometer. However, as the
interrogation arms elongated, the larger size disparity between the Raman
laser and the interrogation arms make it more difficult to improve alignment
precision when transiting from the two symmetric R-B interferometers with two
pairs of Raman beams to one M-Z interferometer with three pairs of Raman
beamsTackmann2012a ; Yao2018a . Compared to the contrast measurement, the
phase measurement provides a higher sensitivityXu2017a ; Altorio2020a , which
is suitable to the precise angular alignment in the four-pulse interferometer.
For the symmetric R-B interferometer, the phase imprinted by Raman lasers is
written as

$\phi_{RB}=\phi_{1}(\vec{r}_{1})-\phi_{2}(\vec{r}_{2})-\phi_{3}(\vec{r}_{3})+\phi_{4}(\vec{r}_{4})$
(1)

where, $\phi_{i}=\vec{k}_{\rm eff}\cdot\vec{r}_{i}$ ($i=1,2,3,4$) is the phase
imprinted by Raman lasers on the atoms, $k_{\rm eff}$ is the effective wave
vector of Raman lasers, and $\vec{r}_{i}$ ($x,y,z$) is the spatial position of
the atoms interacted with the Raman lasers. The tilt of Raman lasers will
change the phase shift when the Raman lasers interacted with the atoms, which
can be measured by symmetric R-B interferometers. As the Raman beams are
tilted in a tiny range, considering the first order approximation, the phase
is given by

$\phi_{RB}=k_{\rm eff}[v_{x}\theta_{x}+v_{y}\theta_{y}-(g-\frac{\alpha}{k_{\rm
eff}})(T_{1}+T_{2})]T_{1}$ (2)

where, $g$ is the gravity acceleration, and $\alpha$ is the chirped rate.
$v_{j}$ ($j=x,y$) is the velocity component, and $\theta_{j}$ is the relative
angle between two pairs of Raman lasers on the $x$\- and $y$-axis directions,
respectively. $T_{1}$ is the separated time between the first/last two pulses,
and $T_{2}$ is the delay time between the second and third pulses, as shown in
Fig.1 (b). The phase is sensitive to the separated time, and the phase
intersection of R-B interferometers with the different separated times could
give a criterion on whether two separated Raman lasers are parallel. However,
in the experiment, it is very difficult to get the phase intersected point due
to the velocity dependent crosstalk phase shifts when the separated time is
modulatedTackmann2012a .

Here we demonstrate an effective method for seeking the phase intersected
point of symmetric R-B interferometers by decoupling the gravity-induced and
velocity-dependent crosstalk phase shift. To improve the alignment accuracy,
we introduce a chirped rate $\alpha$ into the Raman lasers to compensate the
gravity caused phase shift. The deviation of $\alpha$ induces a systematic
error in the angular alignment. If we want to seek the phase intersection, the
uncertainty of chirped rate should be confined to a range of a few kHz/s,
which corresponds to the 10-4 level of the gravity measurement. To accurately
compensate the chirped rate, we construct a M-Z interferometer with the second
pair of Raman lasers to measure the gravity compensated chirp rateZhou2011a .
Limited by the beam size, the maximum interrogation time is about 2.6 ms,
which can give a chirped rate of 25.104 $\pm$ 0.001 MHz/s. The uncertainty of
the chirped rate induces a deviation of few microradians in the alignment,
which is sufficient to compensate the gravity-caused phase shift. The phase
dependence on the tilting angles is measured in the symmetric R-B
interferometer after the gravity is compensated by applying a chirped rate. We
first measure the contrast of symmetric R-B interferometers to get an angular
resolution of one hundred of microradians. Then an optical slit is added on
the detector to select the atomic velocity, which reduces the demand of
angular alignment by several times. With the same delay time ($T_{2}=50$ ms)
and different separated times ($T_{1}$) from 0.1 to 0.5 ms, the phase
dependence of symmetric R-B interferometer on $\theta_{y}$ is shown in Fig.2
(a). The systematic error, caused by the wavefront distortion, is ignored for
the short separated time. The phase of the R-B interferometer depends linearly
on $\theta_{y}$, and due to the much less transverse velocity component, the
phase is not sensitive to $\theta_{x}$. The velocity dependent crosstalk phase
shift between the $x$\- and $y$-axis directions is much reduced due to their
velocity components decoupled. Thus the phase seeking calibration is carried
out by compensating the gravity effect and by decoupling the velocity
dependent crosstalk phase shift. There are several phase intersected point for
different separated times, which provides a criterion for precisely seeking
the angular alignment of separated Raman lasers. The relative angular of two
Raman lasers on the $y$-axis direction is measured and given by the standard
deviation, and it is $1039.69\pm 10.60$ $\mu$rad from Fig.2 (a). For the
larger separated time, the phase is more sensitive to the angular alignment.
This is a key step to build the large-area M-Z atom interferometer.

Figure 2: (color online) (a) Phase dependence of Ramsey-Bord$\acute{e}$ atom
interferometers on the angular tilting of Raman lasers with different
separated times of $T_{1}$=0.1 ms (black squares), 0.2 ms (red dots), 0.3 ms
(green triangles), 0.4 ms (blue stars) and 0.5 ms (purple diamonds). The
experiment results are linearly fitted, and the intercross point is an
absolute zero-phase point when two pairs of Raman lasers are parallel. (b)
Contrast dependence of Mach-Zehnder interferometers on the angular tilting of
Raman lasers with the interrogation time of $T$=50 ms. The result is fitted by
the Gaussian function.

Three pairs of Raman beams are precisely aligned with the method mentioned
above, and applied to build dual large-area M-Z interferometers as shown in
Fig.1 (c). For the inertial measurement, we use the velocity sensitive pulses
to manipulate atoms. The different pulse durations select the atoms of
different velocity distributions. To reduce the contrast loss caused by the
different durations, the intensity ratio of three pairs of Raman beams are set
to $1:2:1$, which provides the ideal durations for selecting the atoms with
the same velocity group. Then, the optical slit is removed, and the
interference fringe is observed in the M-Z interferometer. To maximize the
contrast of fringes, three pairs of separated Raman beams with a consecutive
distance of 20 cm should be more precisely aligned. The alignment of three
Raman lasers is optimized by adjusting the atomic trajectories and the chirped
rate, which makes the velocity component perpendicular to the propagating
direction of Raman lasers tended to zero. In this case, the velocity-dependent
crosstalk phase shift is further reduced, and the angular alignment the
separated Raman lasers is improved by decoupling the velocity dependent
crosstalk phase shifts on $\theta_{x}$ and $\theta_{y}$. The contrast of M-Z
fringes is measured by tilting the angle on $\theta_{y}$, and is fitted by the
Gaussian function, as show in Fig.2 (b). The contrast of the fringe is up to
12$\%$ using three pairs of separated Raman beams with a consecutive time of
50 ms. The angular alignment dependent on $\theta_{y}$ is measured at the 0.4
$\mu$rad level by the contrast of M-Z interferometers. In the dual atom
interferometers, the trajectory difference makes the atoms experience the
different transition probability, which reduces the fringe contrastYao2018a .
To maximize the common noise rejection ratio, a trade-off between two
interferometers is realized by optimizing the atomic trajectories and the
Raman pulses. Finally, we realized a dual M-Z-type cold-atom-interferometer
gyroscope with contrasts of 7.6$\%$ and 8.4$\%$, which are limited by the
overlapping of velocity-sensitive Raman transitions.

Figure 3: (color online) Phases of the dual atom interferometers are measured
by using the square modulation method. Although the phases are fluctuated for
the 1st interferometer (black curve) and the 2nd interferometer (red curve),
their phase noise is canceled in the differential mode (blue curve).

The performance of the atomic gyroscope is evaluated with the same method as
used in the previous workYao2018a . The phases of dual atom interferometers
are measured by using the square modulation method as shown in Fig.3. The
signals are set to two adjacent mid-fringe points by adjusting the phase
difference of the Raman lasers. To improve rejection effect of common-mode
noise, we make the two fringes in phase by modulating the interference-loop
area. In this case, the atomic longitudinal velocity is 4.1 m/s, and the
interference area is 1.2 cm2, which is the largest area reported in M-Z-type
interferometers. The dominant noise term originates from the vibration noise,
which is estimated by measuring the environment vibration and calculating its
contribution using the weighting functionCheinet2008a . Fortunately, the
vibration noise can be rejected in dual atom interferometers. Besides of the
short-term noise, the thermal expansion of the platform changes the optical
path length among three pairs of Raman beams, which produces a phase shift. As
the interrogation arm elongated, this effect is more serious. The long-term
drift is also a common-mode phase and eliminated in dual atom interferometers.
To eliminate the long-term drift, we add a feedback loop to compensate the
temperature-caused phase drifts, which acts as a constant phase for the dual
interferometers, and can be subtracted as a common-mode phase during the
rotation measurement. The above results are evaluated by using the Allan
deviations, as shown in Fig.4. The black squares are phases for the 1st atom
interferometer, and the red triangles are those for the second one. The blue
diamonds are the differential results for the dual-atom interferometers. The
black, red and blue lines show the corresponding ideal white noise curves, as
a slope of $\tau$-1/2 in the double logarithmic coordinate system. Due to the
loop feedback, for the single interferometer, the signal deviates from
$\tau$-1/2 when the integrating time is longer than 6000 s. The phase
uncertainty after common noise rejection using two interferometers
demonstrates a 5-time noise compression compared with the single atom
interferometer, approaching to 40 mrad per shot at the integrating time of
23000 s. Considered with the cycle time of 1.44 s, the short-term rotation
sensitivity is $1.5\times 10^{-7}$ rad/s/Hz1/2. Allan standard deviations
decrease with the integration time as $\tau$-1/2 after several seconds,
reaching a long-term stability of $9.5\times 10^{-10}$ rad/s at 23000 s. To
our knowledge, this performance is the state of art of the reported M-Z-type
atom-interferometer gyroscope.

Figure 4: (color online) Performance is evaluated by the Allan deviations. The
differential result of dual atom interferometers shows a common-noise
rejection in the gyroscope.

The absolute rotation measurement is important for an inertial sensor. The
phase of an atom interferometer is usually a periodic quantity, which
introduces an extra phase shift of $2m\pi$ ($m$ is an integer number). This
becomes more obvious for the large-area atom interferometer. In our
experiment, even for the Earth’ rotation rate, the phase shift of atom
interferometer is on several tens of radians. To realize an absolute rotation
measurement, we measure the Earth’ rotation rate with different scale factors.
The scale factor is modulated by adjusting the atomic longitudinal velocity,
which is different from varying the interrogation time in the atom
gravimeterWu2019a ; Bonnin2018a ; Lautier2014a ; Avinadav2019a . The absolute
rotation rate is measured by extracting the differential phases for different
scale factors. The Earth’ rotation is measured and its rate is $(6.067\pm
0.006)\times 10^{-5}$ rad/s. The horizontal component of Earth’ rotation rate
is $6.280\times 10^{-5}$ rad/s when considering the latitude of
$30.55^{\circ}$ in Wuhan. This deviation is verified by using a commercial
compass. It is caused by the misalignment between the interference area and
the latitude direction.

In conclusion, the atom-interferometer gyroscope with the high rotation
sensitivity and long-term stability has many potential applications in
scientific and technical fields. The dual M-Z-type atom-interferometer
gyroscope is much suitable to three-axis rotation measurements. We realized a
compact M-Z-type large-area atom-interferometer gyroscope by precisely
aligning the large-scale separated Raman lasers, in which the dead time is
reduced as much as possible by elongating the interrogation arms. More
importantly, the parameter decoupled phase seeking calibration is proposed and
performed for aligning the separated Raman lasers. Three Raman lasers are
aligned at the 0.4 $\mu$rad level. For the larger separated time, the phase
measurement is more sensitive to the tilting angular alignment, which can
provide an effective method for aligning the larger-scale separated Raman
lasers by more precisely compensating the gravity effect and further
decoupling crosstalk parameters dependent phase shift. It will be useful for
realizing the larger-area interferometer, such as a large-scale atom-
interferometer gyroscopeZhan2019a , where the angular alignment is more
urgent. After precisely aligning the Raman lasers, a high-precision gyroscope
is realized using dual M-Z-type atom interferometers with each atomic
interference-loop area of 1.2 cm2, and the absolute rotation rate is measured
by modulating the atomic velocities. With the dual interferometers for
rejecting the common-mode noise, the sensor reaches a long-term stability of
$9.5\times 10^{-10}$ rad/s after integrating for 23000 s. In the future, the
rotation sensitivity could be further improved by reducing the dead time using
the continuous operationMeunier2014a  and by suppressing the amplitude noise
using the normalized detectionBiedermann2009a .

We acknowledge the financial support from the National Key Search and
Development Program of China under Grant No. 2016YFA0302002, the National
Natural Science Foundation of China under Grant Nos. 11674362, 91536221, and
91736311, the Strategic Priority Research Program of Chinese Academy of
Sciences under Grant No. XDB21010100, the Outstanding Youth Foundation of
Hubei Province of China under Grant No. 2018CFA082, the National Defense
Science and Technology Innovation Project of China, and the Youth Innovation
Promotion Association of Chinese Academy of Sciences.

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# Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive
Definite Matrices

Armine Bagyan and Donald Richards Department of Statistics, Pennsylvania State
University, University Park, PA 16802, U.S.A. E-mail address:
<EMAIL_ADDRESS>of Statistics, Pennsylvania State University,
University Park, PA 16802, U.S.A. E-mail address<EMAIL_ADDRESS>†Corresponding author.

###### Abstract

We consider random walks on the cone of $m\times m$ positive definite
matrices, where the underlying random matrices have orthogonally invariant
distributions on the cone and the Riemannian metric is the measure of distance
on the cone. By applying results of Khare and Rajaratnam (Ann. Probab., 45
(2017), 4101–4111), we obtain inequalities of Hoffmann-Jørgensen type for such
random walks on the cone. In the case of the Wishart distribution
$W_{m}(a,I_{m})$, with index parameter $a$ and matrix parameter $I_{m}$, the
identity matrix, we derive explicit and computable bounds for each term
appearing in the Hoffmann-Jørgensen inequalities.

Key words and phrases. Orthogonal invariance; Riemannian metric;
submartingale; symmetric cone; Thompson’s metric; Wishart distribution.

2020 Mathematics Subject Classification. Primary: 60E15, 62E15. Secondary:
60B20, 62E17.

Running head: Hoffmann-Jørgensen Inequalities for Positive Definite Matrices.

##  1 Introduction

In this paper, we consider inequalities of Hoffmann-Jørgensen type for random
walks on the cone of positive definite matrices. These inequalities are
obtained by adapting results of Khare and Rajaratnam [19] obtained in the
broader setting of metric semigroups. In studying these inequalities on matrix
cones, we are motivated by the appearance of random samples of positive
definite matrices in numerous fields, including: statistical inference on
Riemannian manifolds [15, 21, 22], microwave engineering [31, p. 156 ff.],
diffusion tensor imaging [11, 18], financial time series [5], wireless
communication systems [30], polarimetric radar imaging [4], factor analysis
[8], and goodness-of-fit testing [14].

The resulting generalized Hoffmann-Jørgensen inequalities on
$\mathcal{P}_{m}$, the cone of $m\times m$ positive definite (symmetric)
matrices, are derived under the assumption that the underlying random matrices
have an orthogonally invariant distribution on the cone. Specializing to the
case of the Wishart distribution $W_{m}(a,I_{m})$, with index parameter $a$
and identity matrix parameter $I_{m}$, we derive explicit and computable
bounds for each term appearing in the Hoffmann-Jørgensen inequalities.
Although we restrict our attention in this paper to $\mathcal{P}_{m}$, our
methods apply more generally to the cone of Hermitian positive definite
matrices and to abstract symmetric cones [10, 12].

In Section 2, we provide notation and preliminaries for random walks on
$\mathcal{P}_{m}$ and for the natural action of
$GL(m,\operatorname{\mathbb{R}})$, the group of nonsingular $m\times m$
matrices, on the random walks. We provide in this section some necessary
properties of the Riemannian metric, which is perhaps the most prominent
metric on $\mathcal{P}_{m}$, and we show in Proposition 2.3 that the
probability distribution of the distance between orthogonally invariant random
matrices is preserved under orthogonally invariant random walks.

In Section 3 we derive from a result of Khare and Rajaratnam [19, Theorem A] a
generalization, to the cone $\mathcal{P}_{m}$, of the classical Hoffmann-
Jørgensen inequalities. This result, which is stated in Theorem 3.1, provides
bounds on the probability distribution of the distance between $X_{n}$, the
$n$th “step” in an invariant random walk from the initial matrix $I_{m}$, the
$m\times m$ identity matrix.

In order to apply an Hoffmann-Jørgensen inequality in a practical setting, it
is necessary to calculate or obtain an upper bound for each term appearing in
the inequality; this will be seen to be a non-trivial problem for the random
walks considered here. In Section 4, we consider the special case in which the
random walk is generated by random samples from the Wishart distribution
$W_{m}(a,I_{m})$, with index parameter $a$ and matrix parameter $I_{m}$, the
identity matrix; we are particularly interested in Wishart-distributed random
walks because of their appearance in the literature on diffusion tensor
imaging, factor analysis, and financial volatility. Under the Wishart
assumption, we obtain explicit, computable bounds for each term in the
Hoffmann-Jørgensen inequalities in Theorem 3.1.

Finally, in Appendix A, we establish the submartingale properties of some
random variables appearing in the generalized Hoffmann-Jørgensen inequalities
and show that the corresponding Kolmogorov inequalities reduce to Markov’s
inequalities for those variables.

##  2 Random matrices and random walks on $\boldsymbol{\mathcal{P}_{m}}$

We denote by $G$ the general linear group $GL(m,\operatorname{\mathbb{R}})$ of
all real, $m\times m$, nonsingular matrices. The group $G$ acts on
$\mathcal{P}_{m}$ via the correspondence, $g\cdot A=gAg^{\prime}$, where $g\in
G$, $A\in\mathcal{P}_{m}$, and $g^{\prime}$ denotes the transpose of $g$. It
is elementary to verify that this correspondence is a group action, i.e.,
$g_{1}g_{2}\cdot A=g_{1}\cdot(g_{2}\cdot A)$, for all $g_{1},g_{2}\in G$.

The group action also is transitive: given $A,B\in\mathcal{P}_{m}$, there
exists $g\in G$ such that $g\cdot A=B$; simply choose $g=B^{1/2}A^{-1/2}$.
Alternatively, since $A,B\in\mathcal{P}_{m}$, there exist $g_{1},g_{2}\in G$
such that $A=g_{1}g_{1}^{\prime}$, $B=g_{2}g_{2}^{\prime}$; then
$g=g_{2}g_{1}^{-1}$ satisfies $g\cdot A=B$. Denote by $I_{m}$ the identity
matrix in $G$. Under this group action, the isotropy group of $I_{m}$ is
$O(m)$, the maximal compact subgroup of $m\times m$ orthogonal matrices. Thus,
the homogeneous space $G/O(m)$ can be identified with $\mathcal{P}_{m}$
through the correspondence $g\cdot k=(gk)(gk)^{\prime}=gg^{\prime}$ for all
$k\in O(m)$.

Whenever two random entities $X$ and $Y$ have the same probability
distribution, we will write $X\overset{\mathcal{L}}{=}Y$. Then a random matrix
$X\in\mathcal{P}_{m}$ is said to be orthogonally invariant if
$X\overset{\mathcal{L}}{=}kXk^{\prime}$ for any $k\in O(m)$. Examples of
orthogonally invariant distributions are the well-known Wishart distribution
$W_{m}(\alpha,I_{m})$ [2, Chapter 7] and the multivariate beta distribution
[2, p. 377]. Similarly, a function
$f:\mathcal{P}_{m}\to\operatorname{\mathbb{R}}$ is said to be orthogonally
invariant if $f(X)=f(kXk^{\prime})$ for all $k\in O(m)$ and
$X\in\mathcal{P}_{m}$.

If the random matrix $X=gg^{\prime}\in\mathcal{P}_{m}$ is orthogonally
invariant, where $g\in G$, then for any $k_{1},k_{2}\in O(m)$,

$(k_{1}gk_{2})(k_{1}gk_{2})^{\prime}=k_{1}gk_{2}k_{2}^{\prime}g^{\prime}k_{1}^{\prime}=k_{1}gg^{\prime}k_{1}^{\prime}=k_{1}Xk_{1}^{\prime}\overset{\mathcal{L}}{=}X.$

Therefore, for any orthogonally invariant random matrix $X\in\mathcal{P}_{m}$,
we can identify $X$ with a corresponding $g\in G$ such that $g$ is left- and
right-invariant under $O(m)$.

Suppose that the random matrix $X\in\mathcal{P}_{m}$ is orthogonally invariant
and $\operatorname{\mathbb{E}}(X)$ is finite. Since
$X\overset{\mathcal{L}}{=}kXk^{\prime}$ for all $k\in O(m)$ then
$\operatorname{\mathbb{E}}(X)=\operatorname{\mathbb{E}}(kXk^{\prime})=k\operatorname{\mathbb{E}}(X)k^{\prime}$.
It follows from Schur’s Lemma [29, p. 315] that
$\operatorname{\mathbb{E}}(X)=cI_{m}$ for some constant $c>0$.

Given orthogonally invariant random matrices $X_{1},X_{2}\in\mathcal{P}_{m}$,
the convolution product, $X_{1}\circ X_{2}$, is defined by convolving the
corresponding random matrices $g_{1},g_{2}\in G$ and then identifying the
outcome in $\mathcal{P}_{m}$ via the natural map from
$G/O(m)\to\mathcal{P}_{m}$. Concretely, if $X_{1}=g_{1}g_{1}^{\prime}$ and
$X_{2}=g_{2}g_{2}^{\prime}$ for orthogonally bi-invariant random matrices
$g_{1},g_{2}\in G$, then $X_{1}\circ
X_{2}=(g_{1}g_{2})(g_{1}g_{2})^{\prime}=g_{1}g_{2}g_{2}^{\prime}g_{1}^{\prime}$.

To express the distribution of $X_{1}\circ X_{2}$ in terms of a product of
functions of $X_{1}$ and $X_{2}$, we recall that by polar coordinates on
matrix space, $g_{1}=kX_{1}^{1/2}$ for some $k\in O(m)$. Hence, by the
orthogonal invariance of the random matrix $g$, we have $X_{1}\circ
X_{2}=g_{1}g_{2}g_{2}^{\prime}g_{1}^{\prime}=g_{1}X_{2}g_{1}^{\prime}=kX_{1}^{1/2}X_{2}(kX_{1}^{1/2})^{\prime}=kX_{1}^{1/2}X_{2}X_{1}^{1/2}k^{\prime}$.
Since $X_{1}$ and $X_{2}$ are orthogonally invariant, then replacing $X_{j}$
by $k^{\prime}X_{j}k$, $j=1,2$ and using the identity
$(kXk^{\prime})^{1/2}=kX^{1/2}k^{\prime}$ for any positive definite matrix
$X$, we obtain

$kX_{1}^{1/2}X_{2}X_{1}^{1/2}k^{\prime}\overset{\mathcal{L}}{=}kk^{\prime}X_{1}^{1/2}kk^{\prime}X_{2}kk^{\prime}X_{1}^{1/2}kk^{\prime}=X_{1}^{1/2}X_{2}X_{1}^{1/2}.$

Consequently,

$X_{1}\circ X_{2}\overset{\mathcal{L}}{=}X_{1}^{1/2}X_{2}X_{1}^{1/2}.$ (2.1)

Although the binary operation $(A,B)\mapsto A^{1/2}BA^{1/2}$,
$A,B\in\mathcal{P}_{m}$, is not associative, the following result shows that
associativity holds in distribution for mutually independent, orthogonally
invariant random matrices.

######  Lemma 2.1.

Suppose that the random matrices $X_{1},X_{2},X_{3}\in\mathcal{P}_{m}$ are
mutually independent and orthogonally invariant. Then $(X_{1}\circ X_{2})\circ
X_{3}\overset{\mathcal{L}}{=}X_{1}\circ(X_{2}\circ X_{3})$.

Proof. By (2.1),

$\displaystyle X_{1}\circ(X_{2}\circ X_{3})$
$\displaystyle\overset{\mathcal{L}}{=}X_{1}^{1/2}(X_{2}\circ
X_{3})X_{1}^{1/2}$
$\displaystyle\overset{\mathcal{L}}{=}X_{1}^{1/2}X_{2}^{1/2}X_{3}X_{2}^{1/2}X_{1}^{1/2}=\big{(}X_{2}^{1/2}X_{1}^{1/2}\big{)}^{\prime}X_{3}\big{(}X_{2}^{1/2}X_{1}^{1/2}\big{)}.$

Since
$X_{1}^{1/2}X_{2}X_{1}^{1/2}=(X_{2}^{1/2}X_{1}^{1/2})^{\prime}(X_{2}^{1/2}X_{1}^{1/2})$
then, by polar coordinates on matrix space, there exist $k\in O(m)$ such that
$X_{2}^{1/2}X_{1}^{1/2}=k(X_{1}^{1/2}X_{2}X_{1}^{1/2})^{1/2}$. Then,

$\displaystyle X_{1}\circ(X_{2}\circ X_{3})$
$\displaystyle\overset{\mathcal{L}}{=}\big{(}X_{1}^{1/2}X_{2}X_{1}^{1/2}\big{)}^{1/2}k^{\prime}X_{3}k\big{(}X_{1}^{1/2}X_{2}X_{1}^{1/2}\big{)}^{1/2}$
$\displaystyle\overset{\mathcal{L}}{=}\big{(}X_{1}^{1/2}X_{2}X_{1}^{1/2}\big{)}^{1/2}X_{3}\big{(}X_{1}^{1/2}X_{2}X_{1}^{1/2}\big{)}^{1/2}$
$\displaystyle\overset{\mathcal{L}}{=}\big{(}X_{1}\circ
X_{2}\big{)}^{1/2}X_{3}\big{(}X_{1}\circ X_{2}\big{)}^{1/2}$
$\displaystyle\overset{\mathcal{L}}{=}(X_{1}\circ X_{2})\circ X_{3}.$

The proof now is complete. ∎

######  Lemma 2.2.

Suppose that the random matrices $X_{1},X_{2}\in\mathcal{P}_{m}$ are
independent and orthogonally invariant. Then $X_{1}\circ
X_{2}\overset{\mathcal{L}}{=}X_{2}\circ X_{1}$.

More generally, if the random matrices $X_{1},\ldots,X_{n}\in\mathcal{P}_{m}$
are mutually independent and orthogonally invariant then
$X_{1}\circ\cdots\circ
X_{n}\overset{\mathcal{L}}{=}X_{\tau(1)}\circ\cdots\circ X_{\tau(n)}$ for all
permutations $\tau\in\mathfrak{S}_{n}$, the symmetric group on $n$ symbols.

Proof. Since $X_{1}^{1/2}X_{2}X_{1}^{1/2}$ and $X_{2}^{1/2}X_{1}X_{2}^{1/2}$
have the same spectrum then there exists $k\in O(m)$ such that
$X_{1}^{1/2}X_{2}X_{1}^{1/2}=kX_{2}^{1/2}X_{1}X_{2}^{1/2}k^{\prime}$;
therefore,

$X_{1}\circ
X_{2}\overset{\mathcal{L}}{=}X_{1}^{1/2}X_{2}X_{1}^{1/2}=kX_{2}^{1/2}X_{1}X_{2}^{1/2}k^{\prime}\overset{\mathcal{L}}{=}k(X_{2}\circ
X_{1})k^{\prime}.$

Since the distributions of $X_{1}$ and $X_{2}$ are orthogonally invariant then
it follows that

$\displaystyle k(X_{2}\circ X_{1})k^{\prime}$
$\displaystyle\overset{\mathcal{L}}{=}k(k^{\prime}X_{2}k\circ
k^{\prime}X_{1}k)k^{\prime}$
$\displaystyle=k(k^{\prime}X_{2}k)^{1/2}k^{\prime}X_{1}k(k^{\prime}X_{2}k)^{1/2}k^{\prime}$
$\displaystyle=kk^{\prime}X_{2}^{1/2}kk^{\prime}X_{1}kk^{\prime}X_{2}^{1/2}kk^{\prime}$
$\displaystyle=X_{2}^{1/2}X_{1}X_{2}^{1/2}$
$\displaystyle\overset{\mathcal{L}}{=}X_{2}\circ X_{1}.$

Therefore, $X_{1}\circ X_{2}\overset{\mathcal{L}}{=}X_{2}\circ X_{1}$.

Finally, the proof that the distribution of $X_{1}\circ\cdots\circ X_{n}$ is
invariant under permutation of the $X_{i}$ is obtained by using the property
that each permutation can be expressed as a product of transpositions together
with the previously-proved commutativity and associativity properties. ∎

For $A\in\mathcal{P}_{m}$, denote by $\lambda_{1}(A),\ldots,\lambda_{m}(A)$
the eigenvalues of $A$. The Riemannian metric on $\mathcal{P}_{m}$ is defined
as

$d_{R}(A,B)=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(A^{-1/2}BA^{-1/2})\big{]}^{2}\Big{)}^{1/2},$

$A,B\in\mathcal{P}_{m}$. The function $d_{R}$ is a genuine metric since, for
all $A,B,C\in\mathcal{P}_{m}$: $d_{R}(A,B)\geq 0$, $d_{R}(A,B)=d_{R}(B,A)$,
and

$d_{R}(A,C)\leq d_{R}(A,B)+d_{R}(B,C).$ (2.2)

The quantity $d_{R}(A,B)$ is the distance along the shortest geodesic path, on
the cone, starting at $A$ and ending at $B$ [17, 25, 28]. Since the matrices
$A^{-1/2}BA^{-1/2}$ and $A^{-1}B$ have the same spectrum, then we also have an
alternative formula,

$d_{R}(A,B)=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(A^{-1}B)\big{]}^{2}\Big{)}^{1/2},$
(2.3)

from which it follows that $d_{R}$ is orthogonally bi-invariant:
$d_{R}(kAk^{\prime},kBk^{\prime})=d_{R}(A,B)$ for all $k\in O(m)$ and
$A,B\in\mathcal{P}_{m}$.

######  Proposition 2.3.

Let $A,B,X\in\mathcal{P}_{m}$ be orthogonally invariant random matrices. Then

$d_{R}(X\circ A,X\circ
B)\overset{\mathcal{L}}{=}d_{R}(A,B)\overset{\mathcal{L}}{=}d_{R}(A\circ
X,B\circ X).$

Proof. By (2.1), there exist $k_{1},k_{2}\in O(m)$ such that $X\circ
A=k_{1}X^{1/2}AX^{1/2}k_{1}^{\prime}$ and $X\circ
B=k_{2}X^{1/2}BX^{1/2}k_{2}^{\prime}$. Therefore

$\displaystyle d_{R}(X\circ A,X\circ B)$
$\displaystyle=d_{R}(k_{1}X^{1/2}AX^{1/2}k_{1}^{\prime},k_{2}X^{1/2}BX^{1/2}k_{2}^{\prime})$
$\displaystyle=d_{R}(kX^{1/2}AX^{1/2}k^{\prime},X^{1/2}BX^{1/2}).$

where $k=k_{2}^{\prime}k_{1}$. By polar coordinates on matrix space, there
exists $k_{3}\in O(m)$ such that $kX^{1/2}=X^{1/2}k_{3}$; hence
$X^{1/2}k^{\prime}=k_{3}^{\prime}X^{1/2}$. Therefore,

$\displaystyle d_{R}(kX^{1/2}AX^{1/2}k^{\prime},X^{1/2}BX^{1/2})$
$\displaystyle=d_{R}(X^{1/2}k_{3}Ak_{3}^{\prime}X^{1/2},X^{1/2}BX^{1/2})$
$\displaystyle\overset{\mathcal{L}}{=}d_{R}(X^{1/2}AX^{1/2},X^{1/2}BX^{1/2})$
$\displaystyle=d_{R}(A,B),$

where the equality in distribution follows by the orthogonal invariance of $A$
and the latter equality follows from (2.3).

Again by (2.1) and (2.3), we have

$\displaystyle d_{R}(A\circ X,B\circ X)$
$\displaystyle\overset{\mathcal{L}}{=}d_{R}(A^{1/2}XA^{1/2},B^{1/2}XB^{1/2})$
$\displaystyle=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}\big{(}(A^{1/2}XA^{1/2})^{-1}B^{1/2}XB^{1/2}\big{)}\big{]}^{2}\Big{)}^{1/2}.$

Since $A^{1/2}XA^{1/2}$ and $X^{1/2}AX^{1/2}$ have the same spectrum then
there exists $k_{1}\in O(m)$ such that
$A^{1/2}XA^{1/2}=k_{1}X^{1/2}AX^{1/2}k^{\prime}_{1}$; and similarly, there
exists $k_{2}\in O(m)$ such that
$B^{1/2}XB^{1/2}=k_{2}X^{1/2}BX^{1/2}k^{\prime}_{2}$. Therefore,

$\displaystyle d_{R}(A\circ X,B\circ X)$
$\displaystyle=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}\big{(}(k_{1}X^{1/2}AX^{1/2}k^{\prime}_{1})^{-1}k_{2}X^{1/2}BX^{1/2}k^{\prime}_{2}\big{)}\big{]}^{2}\Big{)}^{1/2}$
$\displaystyle=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(k_{1}X^{-1/2}A^{-1}X^{-1/2}k^{\prime}_{1}k_{2}X^{1/2}BX^{1/2}k^{\prime}_{2})\big{]}^{2}\Big{)}^{1/2}$
$\displaystyle=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(kX^{-1/2}A^{-1}X^{-1/2}k^{\prime}X^{1/2}BX^{1/2})\big{]}^{2}\Big{)}^{1/2}.$

where $k=k^{\prime}_{2}k_{1}$. By an earlier argument, there exists $k_{3}\in
O(m)$ such that $kX^{-1/2}=X^{-1/2}k_{3}$, and hence
$X^{-1/2}k^{\prime}=k_{3}^{\prime}X^{-1/2}$. Therefore

$\displaystyle d_{R}(A\circ X,B\circ X)$
$\displaystyle=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(X^{-1/2}k_{3}A^{-1}k_{3}^{\prime}X^{-1/2}X^{1/2}BX^{1/2})\big{]}^{2}\Big{)}^{1/2}$
$\displaystyle=\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(k_{3}A^{-1}k_{3}^{\prime}B)\big{]}^{2}\Big{)}^{1/2}$
$\displaystyle\overset{\mathcal{L}}{=}\Big{(}\sum_{j=1}^{m}\big{[}\log\lambda_{j}(A^{-1}B)\big{]}^{2}\Big{)}^{1/2}$
$\displaystyle=d_{R}(A,B),$

where the equality in distribution follows from the orthogonal invariance of
$A$. ∎

######  Remark 2.4.

Many metrics on the cone $\mathcal{P}_{m}$ have been studied in the
literature, cf., [24, 26], and it is possible to extend the results in this
section to some of those metrics. We mention, in particular, the Thompson
metric,

$\displaystyle d_{T}(A,B)$
$\displaystyle:=\log\max\\{\lambda_{\max}(A^{-1/2}BA^{-1/2}),\lambda_{\max}(B^{-1/2}AB^{-1/2})\\}$
$\displaystyle=\log\max\\{\lambda_{\max}(A^{-1}B),1/\lambda_{\min}(A^{-1}B)\\},$

$A,B\in\mathcal{P}_{m}$, where $\lambda_{\max}(A)$ is the largest eigenvalue
of $A\in\mathcal{P}_{m}$ [17]. It can be shown that Proposition 2.3 holds for
$d_{T}$, also that

$d_{T}(A,B)\leq d_{R}(A,B)\leq\sqrt{m}\,d_{T}(A,B),$

so that each probabilistic inequality obtained in Section 3 for $d_{R}$ is
equivalent to a corresponding inequality for $d_{T}$.

##  3 Hoffmann-Jørgensen inequalities on $\boldsymbol{\mathcal{P}_{m}}$

Let $\\{X_{n},n\in\operatorname{\mathbb{N}}\\}$ be a sequence of mutually
independent, and orthogonally invariant, random matrices in $\mathcal{P}_{m}$.
Define the partial products, $S_{j}=X_{1}\circ\cdots\circ X_{j}$,
$j=1,2,3,\ldots$; equivalently,

$S_{1}=X_{1},\qquad S_{j+1}=S_{j}\circ X_{j+1}\equiv
S_{j}^{1/2}X_{j+1}S_{j}^{1/2},\quad j\geq 1.$ (3.1)

As we noted before, the binary relation $A\circ B$ is not commutative;
however, by Lemma 2.2, $X_{1}\circ X_{2}\overset{\mathcal{L}}{=}X_{2}\circ
X_{1}$ because the random matrices $X_{1}$ and $X_{2}$ are independent and
orthogonally invariant. Therefore, we may apply the results derived by Khare
and Rajaratnam [19] (cf., [20]) for commutative semigroups.

We now apply [19, Theorem A] with ${{\hskip 1.0pt}\mathscr{G}}$ as the
collection of orthogonally invariant random matrices on $\mathcal{P}_{m}$,
${{\hskip 1.0pt}\mathscr{G}}$ being endowed with the binary operation given in
(2.1), and $d_{{\hskip 1.0pt}\mathscr{G}}$ being the Riemannian metric
$d_{R}$. For $n=1,2,3,\ldots$ define

$M_{n}=\max\\{d_{R}(I_{m},X_{1}),\ldots,d_{R}(I_{m},X_{n})\\}$ (3.2)

and

$U_{n}=\max\\{d_{R}(I_{m},S_{1}),\ldots,d_{R}(I_{m},S_{n})\\}$ (3.3)

and let $\delta_{ij}$ denote Kronecker’s delta. We also use the notation
$\chi(\mathscr{S})$ to equal $1$ if a statement $\mathscr{S}$ is valid and $0$
if it is not.

######  Theorem 3.1.

([19, Theorem A]) Let $\\{X_{n},n\in\operatorname{\mathbb{N}}\\}$ be
independent, identically distributed (i.i.d.), orthogonally invariant random
matrices in $\mathcal{P}_{m}$. For $t_{0},t_{1},\ldots,t_{l}\in[0,\infty)$ and
integers $l,n_{1},\ldots,n_{l}\in\operatorname{\mathbb{N}}$, let
$n_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptscriptstyle\bullet$}}}}}}=n_{1}+\cdots+n_{l}$, and define

$\mathcal{I}_{0}=\Big{\\{}1\leq i\leq l:[\mathbb{P}(U_{n}\leq
t_{i})]^{n_{i}-\delta_{i1}}\leq\frac{1}{n_{i}!}\Big{\\}}.$ (3.4)

If $n_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptscriptstyle\bullet$}}}}}}\leq n+1$ then

$\mathbb{P}\Big{(}U_{n}>(n_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptscriptstyle\bullet$}}}}}}-1)t_{0}+(2n_{1}-1)t_{1}+2\sum_{j=2}^{l}n_{j}t_{j}\Big{)}\\\
\leq\mathbb{P}(M_{n}>t_{0})+[\mathbb{P}(U_{n}\leq
t_{1})]^{\chi(1\notin\mathcal{I}_{0})}\cdot\prod_{j\in\mathcal{I}_{0}}[\mathbb{P}(U_{n}>t_{j})]^{n_{j}}\cdot\prod_{j\notin\mathcal{I}_{0}}\frac{1}{n_{j}!}\left(\frac{\mathbb{P}(U_{n}>t_{j})}{\mathbb{P}(U_{n}\leq
t_{j})}\right)^{n_{j}}.$ (3.5)

Further, let $Y_{(1)}<\cdots<Y_{(n)}$ be the order statistics of
$d_{R}(I_{m},X_{1}),\ldots,d_{R}(I_{m},X_{n})$. Then the inequality (3.5) can
be strengthened by replacing $\mathbb{P}(M_{n}>t_{0})$ by

$\mathbb{P}\bigg{(}\sum_{j=n-n_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptscriptstyle\bullet$}}}}}}+2}^{n}Y_{(j)}>(n_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptscriptstyle\bullet$}}}}}}-1)t_{0}\bigg{)}.$

######  Remark 3.2.

Note that (3.5) can also be written as

$\mathbb{P}\Big{(}U_{n}>(n_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.8}{{\hskip
1.0pt}$\scriptscriptstyle\bullet$}}}}}}-1)t_{0}+(2n_{1}-1)t_{1}+2\sum_{j=2}^{l}n_{j}t_{j}\Big{)}\\\
\leq\mathbb{P}(M_{n}>t_{0})+[\mathbb{P}(U_{n}\leq
t_{1})]^{\chi(1\notin\mathcal{I}_{0})}\cdot\frac{\prod_{j=1}^{l}[\mathbb{P}(U_{n}>t_{j})]^{n_{j}}}{\prod_{j\notin\mathcal{I}_{0}}n_{j}!\,[\mathbb{P}(U_{n}\leq
t_{j})]^{n_{j}}}.$ (3.6)

######  Example 3.3.

Similar to remarks in [19, p. 4104], if in Theorem 3.1 we set $l=2$,
$n_{1}=n_{2}=1$, and $t_{1}=t_{2}=t$ then by (3.4),
$\mathcal{I}_{0}=\big{\\{}1\leq i\leq 2:[\mathbb{P}(U_{n}\leq
t)]^{1-\delta_{i1}}\leq 1\big{\\}}=\\{1,2\\}$, and (3.5) reduces to the
inequality

$\mathbb{P}(U_{n}>t_{0}+3t)\leq\mathbb{P}(M_{n}>t_{0})+[\mathbb{P}(U_{n}>t)]^{2},$

which is analogous to classical Hoffmann-Jørgensen inequalities [19, p. 4101].

We prove in Appendix A that if
$\operatorname{\mathbb{E}}(\|S_{n}\|_{\textrm{sp}})<\infty$ for all $n\geq 1$
then the sequence $\\{S_{n},n\geq 1\\}$ is a martingale and the sequences
$\\{M_{n},n\geq 1\\}$ and $\\{U_{n},n\geq 1\\}$ are submartingales. However,
these martingale properties lead only to Markov’s inequalities for
$\\{M_{n},n\geq 1\\}$ and $\\{U_{n},n\geq 1\\}$ and in any case a martingale
approach cannot lead to explicit results for specific choices of the
distribution of $\\{X_{n},n\geq 1\\}$. Therefore we develop in Section 4 an
approach that produces detailed inequalities for $\mathbb{P}(M_{n}>t)$ and
$\mathbb{P}(U_{n}>t)$ in the Wishart case.

In the following result we adopt the convention that an empty product of
matrices is the identity matrix.

######  Proposition 3.4.

Let $\\{X_{n},n\in\operatorname{\mathbb{N}}\\}$ be a sequence of i.i.d.
orthogonally invariant random matrices in $\mathcal{P}_{m}$. Then for all
$n\geq 1$,

$S_{n+1}\overset{\mathcal{L}}{=}X_{n+1}\circ
S_{n}\overset{\mathcal{L}}{=}X_{n+1}^{1/2}X_{n}^{1/2}\cdots
X_{2}^{1/2}X_{1}X_{2}^{1/2}\cdots X_{n}^{1/2}X_{n+1}^{1/2}.$ (3.7)

Proof. First, it follows from Lemma 2.2 and induction on $n$ that $S_{n}$ is
orthogonally invariant.

By Lemmas 2.1 and 2.2, the composition of independent, orthogonally invariant
random matrices is commutative and associative in distribution. Since

$\displaystyle S_{n+1}$ $\displaystyle=S_{n}\circ X_{n+1}$
$\displaystyle=(S_{n-1}\circ X_{n})\circ X_{n+1}$
$\displaystyle=((S_{n-2}\circ X_{n-1})\circ X_{n})\circ X_{n+1}$
$\displaystyle\qquad\vdots$ $\displaystyle=((((X_{1}\circ X_{2})\circ
X_{3})\circ\cdots\circ X_{n-1})\circ X_{n})\circ X_{n+1},$

then it follows that $S_{n+1}\overset{\mathcal{L}}{=}X_{1}\circ
X_{2}\circ\cdots\circ X_{n+1}\overset{\mathcal{L}}{=}X_{n+1}\circ
X_{n}\circ\cdots\circ X_{1}$. Therefore,
$S_{n+1}\overset{\mathcal{L}}{=}X_{n+1}\circ(X_{n}\circ\cdots\circ
X_{1})\overset{\mathcal{L}}{=}X_{n+1}\circ S_{n}$. This proves the first
equality (in distribution) in (3.7).

To prove the second equality in distribution in (3.7), we will use induction
on $n$. Since $X_{1}$ and $X_{2}$ are i.i.d. and $S_{1}=X_{1}$ then

$S_{2}=S_{1}\circ X_{2}=X_{1}\circ X_{2}\overset{\mathcal{L}}{=}X_{2}\circ
X_{1}=X_{2}\circ S_{1}=X_{2}^{1/2}X_{1}X_{2}^{1/2}.$

This establishes the second part of (3.7) for $n=1$.

For the induction step, suppose that (3.7) holds for $n=j-1$, where $j\geq 2$.
By (3.1) and the first equality (in distribution) in (3.7),

$\displaystyle S_{j+1}$ $\displaystyle=S_{j}\circ X_{j+1}$
$\displaystyle\overset{\mathcal{L}}{=}X_{j+1}\circ S_{j}$
$\displaystyle\overset{\mathcal{L}}{=}X_{j+1}\circ(X_{j}^{1/2}X_{j-1}^{1/2}\cdots
X_{2}^{1/2}X_{1}X_{2}^{1/2}\cdots X_{j-1}^{1/2}X_{j}^{1/2})$
$\displaystyle\overset{\mathcal{L}}{=}X_{j+1}^{1/2}X_{j}^{1/2}X_{j-1}^{1/2}\cdots
X_{2}^{1/2}X_{1}X_{2}^{1/2}\cdots X_{j-1}^{1/2}X_{j}^{1/2}X_{j+1}^{1/2}.$

Therefore (3.7) holds for $n=j$. This completes the proof by induction. ∎

##  4 The random walk with Wishart matrices

In order for the inequalities (3.5) and (3.6) to be applied in practice, it
will be necessary to obtain upper bounds for the terms $P(M_{n}>t)$,
$P(U_{n}\leq t)$, and $P(U_{n}>t)$ appearing on the right-hand sides of those
inequalities. In this section we obtain, for the case in which the random
matrices $\\{X_{n},n\in\operatorname{\mathbb{N}}\\}$ are Wishart-distributed,
upper bounds on $P(M_{n}>t)$, $P(U_{n}\leq t)$, and $P(U_{n}>t)$.

For $a>\tfrac{1}{2}(m-1)$, the multivariate gamma function is defined as

$\Gamma_{m}(a)=\pi^{m(m-1)/4}\prod_{j=1}^{m}\Gamma(a-\tfrac{1}{2}(j-1)).$

For $x=(x_{ij})\in\mathcal{P}_{m}$, let $\,\textrm{d}x=\prod_{1\leq i<j\leq
m}\,\textrm{d}x_{ij}$ denote Lebesgue measure on $\mathcal{P}_{m}$. A random
matrix $X\in\mathcal{P}_{m}$ is said to have the Wishart distribution, denoted
$X\sim W_{m}(a,I_{m})$, if the probability density function of $X$ relative to
Lebesgue measure on the cone is

$w(x)=\frac{1}{2^{ma}\,\Gamma_{m}(a)}(\det
x)^{a-\frac{1}{2}(m+1)}\exp(-\tfrac{1}{2}\operatorname{\mathop{\rm{tr}\hskip
1.0pt}}x),\qquad x\in\mathcal{P}_{m}.$ (4.1)

Throughout this section, we suppose that $X_{1},\ldots,X_{n}$ is a random
sample from the Wishart distribution $W_{m}(a,I_{m})$. Note that once we have
obtained an upper bound, $\mathbb{P}(U_{n}>t)\leq b$ with $b<1$, then it
follows that

$\frac{1}{\mathbb{P}(U_{n}\leq
t)}=\frac{1}{1-\mathbb{P}(U_{n}>t)}\leq\frac{1}{1-b};$

so we only need upper bounds for $\mathbb{P}(M_{n}>t)$, $\mathbb{P}(U_{n}>t)$,
and $\mathbb{P}(U_{n}\leq t)$.

###  4.1 A bound on the distribution function of $\boldsymbol{M_{n}}$

Define the normalizing constant,

$c_{m}=\frac{\pi^{{m^{2}}/2}}{2^{ma}\,m!\,\Gamma_{m}(a)\,\Gamma_{m}(m/2)}.$
(4.2)

For $u\leq v$ and $i,j=1,\ldots,m$, define

$\displaystyle F_{a+j}(u,v)$
$\displaystyle=\int_{u}^{v}t^{a+j-\tfrac{1}{2}(m+1)}\,\exp(-t/2)\,\textrm{d}t,$
$\displaystyle F_{a+i,a+j}(u,v)$
$\displaystyle=\int_{u}^{v}t^{a+j-\tfrac{1}{2}(m+1)}\,\exp(-t/2)\,F_{a+i}(u,t)\,\textrm{d}t,$

and

$r_{ij}(u,v)=F_{a+i-1,a+j-1}(u,v)-F_{a+j-1,a+i-1}(u,v).$

Define the $m\times m$ matrix $R=\big{(}r_{ij}(u,v)\big{)}_{i,j=1,\ldots,m}$;
also, for $l=1,\ldots,m$, define the $(m-1)\times(m-1)$ principal submatrices

$R_{l}=\big{(}r_{ij}(u,v)\big{)}_{i,j=1,\ldots,l-1,l+1,\ldots,m},$

and the corresponding Pfaffians

$H(u,v)=[\det(R)]^{1/2}\ \ \hbox{and}\ \ H_{l}(u,v)=[\det(R_{l})]^{1/2}.$

We now have the following result.

######  Proposition 4.1.

For $t>0$,

$\mathbb{P}(M_{n}>t)\leq
1-\big{[}m!\,c_{m}\,\rho_{m}\big{(}\exp(-m^{-1/2}t),\exp(m^{-1/2}t)\big{)}\big{]}^{n},$
(4.3)

where

$\rho_{m}(u,v)=\begin{cases}H(u,v),&\hbox{if }m\hbox{ is even}\\\
\operatornamewithlimits{\sum}\limits_{j=0}^{m-1}(-1)^{j}\,F_{a+j}(u,v)\,H_{j+1}(u,v),&\hbox{if
}m\hbox{ is odd}\end{cases}.$ (4.4)

Proof. Since $X_{1},\ldots,X_{n}$ are i.i.d. then

$\mathbb{P}(M_{n}\leq
t)=\mathbb{P}\big{(}\max\big{\\{}d_{R}(I_{m},X_{1}),\ldots,d_{R}(I_{m},X_{n})\big{\\}}\leq
t\big{)}=\big{[}\mathbb{P}\big{(}d_{R}(I_{m},X_{1})\leq t\big{)}\big{]}^{n}.$

By (2.3),

$\mathbb{P}\big{(}d_{R}(I_{m},X_{1})\leq
t\big{)}=\mathbb{P}\Big{(}\sum_{j=1}^{m}[\log\lambda_{j}(X_{1})]^{2}\leq
t^{2}\Big{)},$

and since

$\sum_{j=1}^{m}[\log\lambda_{j}(X_{1})]^{2}\leq
m\cdot\max\\{[\log\lambda_{1}(X_{1})]^{2},\ldots,[\log\lambda_{m}(X_{1})]^{2}\\}$

then it follows that

$\displaystyle\mathbb{P}\big{(}d_{R}(I_{m},X_{1})\leq t\big{)}$
$\displaystyle\geq\mathbb{P}\big{(}m\cdot\max\big{\\{}[\log\lambda_{1}(X_{1})]^{2},\ldots,[\log\lambda_{m}(X_{1})]^{2}\big{\\}}\leq
t^{2}\big{)}$
$\displaystyle=\mathbb{P}\bigg{(}\bigcap_{j=1}^{m}\\{[\log\lambda_{j}(X_{1})]^{2}\leq
m^{-1}t^{2}\\}\bigg{)}$
$\displaystyle=\mathbb{P}\bigg{(}\bigcap_{j=1}^{m}\\{-m^{-1/2}t\leq\log\lambda_{j}(X_{1})\leq
m^{-1/2}t\\}\bigg{)}.$ (4.5)

Let $\lambda_{\min}(X_{1})$ and $\lambda_{\max}(X_{1})$ denote the smallest
and largest eigenvalues, respectively, of $X_{1}$. Solving the inequalities in
(4.1) for $\lambda_{j}(X_{1})$, we obtain

$\displaystyle\mathbb{P}(d_{R}(I_{m},X_{1})\leq t)$
$\displaystyle\geq\mathbb{P}\bigg{(}\bigcap_{j=1}^{m}\\{\exp(-m^{-1/2}t)\leq\lambda_{j}(X_{1})\leq\exp(m^{-1/2}t)\\}\bigg{)}$
$\displaystyle=\mathbb{P}\big{(}\exp(-m^{-1/2}t)\leq\lambda_{\min}(X_{1})<\lambda_{\max}(X_{1})\leq\exp(m^{-1/2}t)\big{)},$

Therefore,

$\displaystyle\mathbb{P}(M_{n}>t)$ $\displaystyle=1-\mathbb{P}(M_{n}\leq t)$
$\displaystyle=1-\big{[}\mathbb{P}\big{(}d_{R}(I_{m},X_{1})\leq
t\big{)}\big{]}^{n}$ $\displaystyle\leq
1-\big{[}\mathbb{P}\big{(}\exp(-m^{-1/2}t)\leq\lambda_{\min}(X_{1})<\lambda_{\max}(X_{1})\leq\exp(m^{-1/2}t)\big{)}\big{]}^{n}.$
(4.6)

From a result of Krishnaiah and Chang [23, Theorem 3.2] for the joint
distribution function of $\\{\lambda_{\min}(X_{1}),\lambda_{\max}(X_{1})\\}$,
we obtain the explicit, computable expression,

$\mathbb{P}\big{(}u\leq\lambda_{\min}(X_{1})<\lambda_{\max}(X_{1})\leq
v\big{)}=m!\,c_{m}\,\rho_{m}(u,v),$

where $c_{m}$ and $\rho_{m}(u,v)$ are defined in (4.2) and (4.4),
respectively. Applying this expression to (4.1), we obtain (4.3). ∎

###  4.2 Bounds on the distribution function of $\boldsymbol{U_{n}}$

It is more difficult to derive bounds for the cumulative distribution function
of $U_{n}$ than for $M_{n}$, the reason being that $M_{n}$ is defined in terms
of the i.i.d. random matrices $X_{1},\ldots,X_{n}$ whereas $U_{n}$ is defined
in terms of the dependent random matrices $S_{1},\ldots,S_{n}$.

For $0<\theta<1$, define

$I_{m}(\theta)=(1-\theta)^{-m[(2a-m-1)(1-\theta)+m]}\prod_{j=1}^{m}j!\,\Gamma\big{(}(2a-m-1)(1-\theta)+j\big{)}.$
(4.7)

Also let

$\mathcal{D}=\\{(v,\theta):0\leq
v<\tfrac{1}{2}((2a-m-1)\theta+1),\,0<\theta<1\\},$ (4.8)

and define

$K(v,\theta)=\int_{0}^{\infty}\lambda^{(2a-m-1)\theta}\exp(2v|\log\lambda|-\theta\lambda)\,\textrm{d}\lambda,$
(4.9)

and

$G(v,\theta)=2\log c_{m}+\log I_{m}(\theta)+m\log K(v,\theta),$ (4.10)

$(v,\theta)\in\mathcal{D}$, where $c_{m}$ is given in (4.2). We now establish
the following result.

######  Proposition 4.2.

For $t>0$,

$\mathbb{P}(U_{n}>t)\leq\
n\exp\Big{[}\inf_{(v,\theta)\in\mathcal{D}}\big{(}-vt+\tfrac{1}{2}nG(v,\theta)\big{)}\Big{]}.$
(4.11)

Proof. By the exponential Markov (or Chernoff) inequality,

$\mathbb{P}(U_{n}>t)=\mathbb{P}\big{(}\exp(vU_{n})>\exp(vt)\big{)}\leq\exp(-vt)\operatorname{\mathbb{E}}\big{(}\exp(vU_{n})\big{)}$
(4.12)

for $t,v>0$. To obtain a bound for
$\operatorname{\mathbb{E}}\big{(}\exp(vU_{n})\big{)}$ we apply an approach
from the area of concentration inequalities, [7, p. 375], [9]. For
$j=1,\ldots,n$, consider
$\xi_{j}(v)=\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},S_{j})\big{)}$,
$v\geq 0$, the moment-generating function of $d_{R}(I_{m},S_{j})$. For $j\geq
2$, we have

$\displaystyle\xi_{j}(v)$
$\displaystyle=\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},S_{j})\big{)}$
$\displaystyle=\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},X_{j}^{1/2}S_{j-1}X_{j}^{1/2})\big{)}=\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(X_{j}^{-1},S_{j-1})\big{)}.$

By the triangle inequality (2.2), we have

$\displaystyle d_{R}(X_{j}^{-1},S_{j-1})$ $\displaystyle\leq
d_{R}(X_{j}^{-1},I_{m})+d_{R}(I_{m},S_{j-1})$
$\displaystyle=d_{R}(I_{m},X_{j})+d_{R}(I_{m},S_{j-1}).$

Exponentiating the latter inequality, taking expectations, applying the
independence of $X_{j}$ and $S_{j-1}$, and the i.i.d. property of
$X_{1},\ldots,X_{n}$, we obtain

$\displaystyle\xi_{j}(v)$
$\displaystyle\leq\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},X_{j})\big{)}\cdot\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},S_{j-1})\big{)}$
$\displaystyle=\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},X_{1})\big{)}\cdot\operatorname{\mathbb{E}}\exp\big{(}v\>d_{R}(I_{m},S_{j-1})\big{)}$
$\displaystyle=[\xi_{1}(v)]\xi_{j-1}(v).$

Iterating this inequality for $j=2,\ldots,n$ we obtain

$\xi_{j}(v)\leq[\xi_{1}(v)]^{j}$ (4.13)

for all $j\geq 1$. Therefore

$\displaystyle\operatorname{\mathbb{E}}\big{(}\exp(vU_{n})\big{)}$
$\displaystyle=\operatorname{\mathbb{E}}\big{[}\exp\big{(}v\cdot\max\\{d_{R}(I_{m},S_{1}),\ldots,d_{R}(I_{m},S_{n})\\}\big{)}\big{]}$
$\displaystyle=\operatorname{\mathbb{E}}\big{[}\max\big{\\{}\exp\big{(}vd_{R}(I_{m},S_{1})\big{)},\ldots,\exp\big{(}vd_{R}(I_{m},S_{n})\big{)}\big{\\}}\big{]}$
$\displaystyle\leq\sum_{j=1}^{n}\operatorname{\mathbb{E}}\big{[}\exp\big{(}vd_{R}(I_{m},S_{j})\big{)}\big{]}$
$\displaystyle=\sum_{j=1}^{n}\xi_{j}(v).$ (4.14)

Since $U_{j}\geq 0$ then $\xi_{j}(v)\geq 1$ for all $v\geq 0$; hence
$\xi_{j}(v)\leq[\xi_{1}(v)]^{j}\leq[\xi_{1}(v)]^{n}$ for all $v\geq 0$ and
$j=1,\ldots,n$. By (4.2),
$\operatorname{\mathbb{E}}\big{(}\exp(vU_{n})\big{)}\leq n[\xi_{1}(v)]^{n}$;
also, by (4.12),

$\mathbb{P}(U_{n}>t)\leq n\exp(-vt)[\xi_{1}(v)]^{n},$ (4.15)

a bound which is valid for all $v>0$ such that $\xi_{1}(v)$ exists. Let
$\operatorname{\mathop{\rm{Dom}\hskip 0.2pt}}(\xi_{1})=\\{v\geq
0:\xi_{1}(v)<\infty\\}$; then we obtain, by minimization with respect to $v$,
the inequality

$\displaystyle\mathbb{P}(U_{n}>t)$ $\displaystyle\leq
n\inf_{v\in\operatorname{\mathop{\rm{Dom}\hskip
0.2pt}}(\xi_{1})}\exp(-vt)[\xi_{1}(v)]^{n}$ (4.16) $\displaystyle\equiv
n\inf_{v\in\operatorname{\mathop{\rm{Dom}\hskip
0.2pt}}(\xi_{1})}\exp\big{(}-vt+n\log\xi_{1}(v)\big{)}.$

We shall show later that
$[0,a-\tfrac{1}{2}(m-1))\subseteq\operatorname{\mathop{\rm{Dom}\hskip
0.2pt}}(\xi_{1})$; since we have assumed that $a>\tfrac{1}{2}(m-1)$ then it
follows that $\operatorname{\mathop{\rm{Dom}\hskip 0.2pt}}(\xi_{1})$ contains
a non-empty open set.

Since each $X\in\mathcal{P}_{m}$ is symmetric then, by diagonalization,
$X=kDk^{\prime}$ where $k\in O(m)$ and $D$ is a diagonal matrix whose diagonal
entries are the eigenvalues of $X$. It follows that $f(X)=f(D)$ for any
orthogonally invariant function
$f:\mathcal{P}_{m}\to\operatorname{\mathbb{C}}$. Denote by
$\lambda_{1},\ldots,\lambda_{m}$ the eigenvalues of $x\in\mathcal{P}_{m}$ and
denote by $\operatorname{\mathbb{R}}_{+}$ the positive real line; then by
Anderson [2, p. 538, Theorem 13.3.1], for any integrable, orthogonally
invariant function
$f:\mathcal{P}_{m}(\operatorname{\mathbb{R}})\to\operatorname{\mathbb{C}}$,

$\int_{\mathcal{P}_{m}}f(x)\,\textrm{d}x=\frac{\pi^{m^{2}/2}}{m!\,\Gamma_{m}(m/2)}\int_{\operatorname{\mathbb{R}}_{+}^{m}}f\big{(}{\rm{diag}}(\lambda_{1},\ldots,\lambda_{m})\big{)}\prod_{1\leq
i<j\leq m}|\lambda_{i}-\lambda_{j}|\prod_{j=1}^{m}\,\textrm{d}\lambda_{j}.$
(4.17)

Recalling that

$\xi_{1}(v)=\operatorname{\mathbb{E}}d_{R}(I_{m},X)=\operatorname{\mathbb{E}}\exp\bigg{(}v\Big{(}\sum_{j=1}^{m}[\log\lambda_{j}(X)]^{2}\Big{)}^{1/2}\bigg{)},$
(4.18)

then, since $X\sim W_{m}(a,I_{m})$, it follows from (4.1), (4.2), (4.17), and
(4.18) that

$\displaystyle\xi_{1}(v)$
$\displaystyle=\frac{1}{2^{ma}\Gamma_{m}(a)}\int_{\mathcal{P}_{m}}\exp\bigg{[}v\Big{(}\sum_{j=1}^{m}[\log\lambda_{j}(x)]^{2}\Big{)}^{1/2}\bigg{]}(\det
x)^{a-\tfrac{1}{2}(m+1)}\exp(-\tfrac{1}{2}\operatorname{\mathop{\rm{tr}\hskip
1.0pt}}x)\,\textrm{d}x$
$\displaystyle=c_{m}\int_{\operatorname{\mathbb{R}}_{+}^{m}}\exp\bigg{[}v\Big{(}\sum_{j=1}^{m}(\log\lambda_{j})^{2}\Big{)}^{1/2}\bigg{]}\prod_{1\leq
i<j\leq
m}|\lambda_{i}-\lambda_{j}|\cdot\prod_{j=1}^{m}\lambda_{j}^{a-\tfrac{1}{2}(m+1)}e^{-\lambda_{j}/2}\,\textrm{d}\lambda_{j}.$
(4.19)

Also, it is straightforward to verify that

$\Big{(}\sum_{j=1}^{m}(\log\lambda_{j})^{2}\Big{)}^{1/2}\leq\sum_{j=1}^{m}|\log\lambda_{j}|,$
(4.20)

and by substituting this inequality into (4.2) we obtain

$\xi_{1}(v)\leq
c_{m}\int_{\operatorname{\mathbb{R}}_{+}^{m}}\exp\Big{[}\sum_{j=1}^{m}(v|\log\lambda_{j}|-\tfrac{1}{2}\lambda_{j})\Big{]}\prod_{1\leq
i<j\leq
m}|\lambda_{i}-\lambda_{j}|\prod_{j=1}^{m}\lambda_{j}^{a-\tfrac{1}{2}(m+1)}\,\textrm{d}\lambda_{j}.$
(4.21)

To verify that $\operatorname{\mathop{\rm{Dom}\hskip
0.2pt}}(\xi_{1})\neq\emptyset$, we now obtain via (4.21) an open interval that
is contained in $\operatorname{\mathop{\rm{Dom}\hskip 0.2pt}}(\xi_{1})$.

For each $\tau\in\mathfrak{S}_{m}$, the group of permutations on $m$ symbols,
denote by $\mathop{\rm{sgn}}(\tau)$ the signature of $\tau$. Recall the well-
known Vandermonde determinant formula,

$(-1)^{m(m-1)/2}\prod_{1\leq i<j\leq
m}(\lambda_{i}-\lambda_{j})=\det(\lambda_{i}^{j-1})=\sum_{\tau\in\mathfrak{S}_{m}}\mathop{\rm{sgn}}(\tau)\prod_{j=1}^{m}\lambda_{\tau(j)}^{j-1};$

consequently, for $\lambda_{1},\ldots,\lambda_{m}>0$,

$\prod_{1\leq i<j\leq
m}|\lambda_{i}-\lambda_{j}|\leq\sum_{\tau\in\mathfrak{S}_{m}}\prod_{j=1}^{m}\lambda_{\tau(j)}^{j-1}.$
(4.22)

Substituting the bound (4.22) into (4.21), and using the symmetry of the
resulting integrand, we find that the integral of each term on the right-hand
side of (4.22) is the same. Therefore $\xi_{1}(v)<\infty$ if

$\displaystyle\int_{\operatorname{\mathbb{R}}_{+}^{m}}\exp\Big{(}v\sum_{j=1}^{m}$
$\displaystyle|\log\lambda_{j}|\Big{)}\prod_{j=1}^{m}\lambda_{j}^{a-\tfrac{1}{2}(m+1)+j-1}e^{-\frac{1}{2}\lambda_{j}}\,\textrm{d}\lambda_{j}$
(4.23)
$\displaystyle\equiv\prod_{j=1}^{m}\int_{0}^{\infty}\exp(v|\log\lambda_{j}|-\tfrac{1}{2}\lambda_{j})\lambda_{j}^{(2a-m+2j-3)/2}\,\textrm{d}\lambda_{j}<\infty.$

The $j$th integral in this product can be written as a sum of an integral over
the interval $(0,1]$ and a second integral over $(1,\infty)$, each of which is
an incomplete gamma function. Applying well-known convergence properties of
the incomplete gamma function we find that the $j$th integral converges if and
only if $0\leq v<(2a-m+2j-1)/2$; and from the case in which $j=1$ we deduce
that $[0,a-\tfrac{1}{2}(m-1))\subseteq\operatorname{\mathop{\rm{Dom}\hskip
0.2pt}}(\xi_{1})$.

For $0<\theta<1$, write

$\prod_{j=1}^{m}\lambda_{j}^{a-\tfrac{1}{2}(m+1)}e^{-\tfrac{1}{2}\lambda_{j}}=\prod_{j=1}^{m}\lambda_{j}^{\big{(}a-\tfrac{1}{2}(m+1)\big{)}\theta}e^{-\tfrac{1}{2}\theta\lambda_{j}}\cdot\prod_{j=1}^{m}\lambda_{j}^{\big{(}a-\tfrac{1}{2}(m+1)\big{)}(1-\theta)}e^{-\tfrac{1}{2}(1-\theta)\lambda_{j}},$

and substitute the right-hand side into the integrand in (4.2). By the Cauchy-
Schwarz inequality,

$\displaystyle[\xi_{1}(v)]^{2}$ $\displaystyle\leq
c_{m}^{2}\int_{\operatorname{\mathbb{R}}_{+}^{m}}\exp\bigg{[}2v\Big{(}\sum_{j=1}^{m}(\log\lambda_{j})^{2}\Big{)}^{1/2}\bigg{]}\prod_{j=1}^{m}\lambda_{j}^{(2a-m-1)\theta}e^{-\theta\lambda_{j}}\,\textrm{d}\lambda_{j}$
$\displaystyle\qquad\cdot\int_{\operatorname{\mathbb{R}}_{+}^{m}}\prod_{1\leq
i<j\leq
m}|\lambda_{i}-\lambda_{j}|^{2}\cdot\prod_{j=1}^{m}\lambda_{j}^{(2a-m-1)(1-\theta)}e^{-(1-\theta)\lambda_{j}}\,\textrm{d}\lambda_{j}.$

By Selberg’s integral [3, p. 406], the latter integral can be evaluated
explicitly:

$\displaystyle I_{m}(\theta)$
$\displaystyle:=\int_{\operatorname{\mathbb{R}}_{+}^{m}}\prod_{1\leq i<j\leq
m}|\lambda_{i}-\lambda_{j}|^{2}\cdot\prod_{j=1}^{m}\lambda_{j}^{(2a-m-1)(1-\theta)}e^{-(1-\theta)\lambda_{j}}\,\textrm{d}\lambda_{j}$
(4.24)
$\displaystyle=(1-\theta)^{-m[(2a-m-1)(1-\theta)+m]}\int_{\operatorname{\mathbb{R}}_{+}^{m}}\prod_{1\leq
i<j\leq
m}|\lambda_{i}-\lambda_{j}|^{2}\cdot\prod_{j=1}^{m}\lambda_{j}^{(2a-m-1)(1-\theta)}e^{-\lambda_{j}}\,\textrm{d}\lambda_{j}$
$\displaystyle=(1-\theta)^{-m[(2a-m-1)(1-\theta)+m]}\prod_{j=1}^{m}j!\,\Gamma\big{(}(2a-m-1)(1-\theta)+j\big{)},$

as stated in (4.7). Applying the log-inequality (4.20), we obtain

$\displaystyle[\xi_{1}(v)]^{2}$ $\displaystyle\leq
c_{m}^{2}I_{m}(\theta)\int_{\operatorname{\mathbb{R}}_{+}^{m}}\exp\bigg{[}2v\sum_{j=1}^{m}|\log\lambda_{j}|\bigg{]}\prod_{j=1}^{m}\lambda_{j}^{(2a-m-1)\theta}\exp(-\theta\lambda_{j})\,\textrm{d}\lambda_{j}$
$\displaystyle=c_{m}^{2}I_{m}(\theta)\bigg{(}\int_{0}^{\infty}\exp(2v|\log\lambda|-\theta\lambda)\lambda^{(2a-m-1)\theta}\,\textrm{d}\lambda\bigg{)}^{m}.$
(4.25)

By the convergence properties of the incomplete gamma function, we deduce that
the integral in (4.2) converges if and only if $(v,\theta)\in\mathcal{D}$, the
set defined in (4.8). Therefore, by (4.9) and (4.2),

$\xi_{1}(v)\leq c_{m}[I_{m}(\theta)]^{1/2}\big{(}K(v,\theta)\big{)}^{m/2},$
(4.26)

$(v,\theta)\in\mathcal{D}$, and by substituting (4.26) into (4.16) we obtain

$\mathbb{P}(U_{n}>t)\leq n\inf_{(v,\theta)\in\mathcal{D}}\exp\big{[}-vt+n\log
c_{m}+\tfrac{1}{2}n\log I_{m}(\theta)+\tfrac{1}{2}mn\log K(v,\theta)\big{]}.$

Interchanging the infimum and the exponential function, and using the
definition of $G(v,\theta)$ from (4.10), we obtain (4.11). ∎

######  Remark 4.3.

(i) Proposition 4.2 implies that, for large $t$, $\mathbb{P}(U_{n}>t)$
converges to zero at a rate that is at least $O(e^{-ct})$, where $c>0$.

(ii) A bound on $\mathbb{P}(U_{n}>t)$ can also be obtained by using (4.13) and
the resulting 3inequality,

$\displaystyle\operatorname{\mathbb{E}}\big{(}\exp(vU_{n})\big{)}\leq\sum_{j=1}^{n}\xi_{j}(v)$
$\displaystyle\leq\sum_{j=1}^{n}[\xi_{1}(v)]^{j}$
$\displaystyle\leq\sum_{j=1}^{n}[c_{m}[I_{m}(\theta)]^{1/2}\big{(}K(v,\theta)\big{)}^{m/2}]^{j}$
$\displaystyle=\frac{\big{(}c_{m}[I_{m}(\theta)]^{1/2}(K(v,\theta))^{m/2}\big{)}^{n+1}-1}{c_{m}[I_{m}(\theta)]^{1/2}(K(v,\theta))^{m/2}-1}-1.$

This bound is sharper than the one given in (4.16), so it can be expected to
lead to sharper inequalities for $\mathbb{P}(U_{n}>t)$. However, the resulting
numerical calculations will be more involved.

We now consider the issue of calculating the infimum in (4.11) for given
values of $m$, $n$, and $a$. We will see that the objective function,
$(v,\theta)\mapsto-vt+\tfrac{1}{2}mG(v,\theta)$, $(v,\theta)\in\mathcal{D}$,
is particularly well-behaved as it is strictly convex in $\mathcal{D}$;
consequently, the infimum occurs at a unique point in $\mathcal{D}$.

######  Proposition 4.4.

(i) For $(v,\theta)\in\mathcal{D}$, the function $K(v,\theta)$, is strictly
log-convex in $(v,\theta)$ jointly, strictly log-convex in $v$ for fixed
$\theta$, and strictly log-convex in $\theta$ for fixed $v$.

(ii) The function $I_{m}(\theta)$, $\theta\in(0,1)$, is strictly log-convex.

(iii) For $t\geq 0$, the objective function $(v,\theta)\mapsto-
vt+\tfrac{1}{2}mG(v,\theta)$, $(v,\theta)\in\mathcal{D}$, is strictly convex.

Proof. These properties follow readily from the fact that $K(v,\theta)$ and
$I_{m}(\theta)$ can be viewed as modified Laplace transforms of positive
measures. For instance, let us write $K(v,\theta)$ in the form

$K(v,\theta)=\int_{0}^{\infty}\exp\big{(}2v\phi_{1}(\lambda)-\theta\lambda+\phi_{2}(\lambda)\big{)}\,\textrm{d}\lambda,$

where $\phi_{1}(\lambda)=|\log\lambda|$ and
$\phi_{2}(\lambda)=(2a-m-1)\theta\log\lambda$. By proceeding as in the theory
of exponential families of distributions (Barndorff-Nielsen [6, p. 103 ff.]),
the standard method of applying Hölder’s inequality leads to the conclusion
that $K(v,\theta)$ is log-convex in $(v,\theta)$ jointly. Moreover, the strict
log-convexity of $K(v,\theta)$ follows from the fact that the function

$\bigg{|}\begin{matrix}\phi_{1}(\lambda_{1})&\phi_{1}(\lambda_{2})\\\
\phi_{2}(\lambda_{1})&\phi_{2}(\lambda_{2})\\\ \end{matrix}\bigg{|}^{2},$

$(\lambda_{1},\lambda_{2})\in\operatorname{\mathbb{R}}_{+}^{2}$, is strictly
positive on an open subset of $\operatorname{\mathbb{R}}_{+}^{2}$.

The remaining statements in (i) can be proved similarly, (ii) can be proved
analogously using the integral representation given in (4.24), and (iii)
follows from (i) and (ii). ∎

Once the strict convexity property of the objective function has been
established, it is numerically straightforward to compute in (4.11) both the
value of the infimum and its point of location. We remark that it is
especially simple to compute numerical examples using, e.g., the SageMath
software system [32].

Next, we obtain an upper bound for the distribution function,
$\mathbb{P}(U_{n}\leq t)$. Denoting by $\chi^{2}_{\alpha}$ a random variable
having the chi-squared distribution with degrees-of-freedom $\alpha$, we have
the following result.

######  Proposition 4.5.

Let $R_{1},\ldots,R_{n}$ be i.i.d., $\chi^{2}_{(2a-m-1)m}$-distributed random
variables. Then,

$\mathbb{P}(U_{n}\leq t)\leq\min_{1\leq j\leq
n}\mathbb{P}\Big{(}\prod_{i=1}^{j}R_{i}\leq m^{j}\exp(m^{-1/2}t)\Big{)},$
(4.27)

$t>0$. In particular,

$\mathbb{P}(U_{n}\leq t)\leq\mathbb{P}\big{(}\chi^{2}_{(2a-m-1)m}\leq
m\exp(m^{-1/2}t)\big{)}.$ (4.28)

Proof. By Bonferroni’s inequality,

$\displaystyle\mathbb{P}(U_{n}\leq t)$
$\displaystyle=\mathbb{P}\big{(}\max\\{d_{R}(I,S_{1}),\ldots,d_{R}(I,S_{n})\\}\leq
t\big{)}$ $\displaystyle=\mathbb{P}\big{(}d_{R}(I,S_{1})\leq
t,\ldots,d_{R}(I,S_{n})\leq t\big{)}$ $\displaystyle\leq\min_{1\leq j\leq
n}\mathbb{P}\big{(}d_{R}(I,S_{j})\leq t\big{)}.$ (4.29)

For $j=1,\ldots,n$,

$\displaystyle\mathbb{P}\big{(}d_{R}(I,S_{j})\leq t\big{)}$
$\displaystyle=\mathbb{P}\big{(}[d_{R}(I,S_{j})]^{2}\leq
t^{2}\big{)}=\mathbb{P}\Big{(}\sum_{i=1}^{m}[\log\lambda_{i}(S_{j})]^{2}\leq
t^{2}\Big{)}.$

By the Cauchy-Schwarz inequality,

$\displaystyle\sum_{i=1}^{m}[\log\lambda_{i}(S_{j})]^{2}$
$\displaystyle\geq\frac{1}{m}\Big{(}\sum_{i=1}^{m}\log\lambda_{i}(S_{j})\Big{)}^{2}$
$\displaystyle=\frac{1}{m}\Big{(}\log\prod_{i=1}^{m}\lambda_{i}(S_{j})\Big{)}^{2}=\frac{1}{m}\big{[}\log(\det
S_{j})\big{]}^{2}.$

Therefore,

$\displaystyle\mathbb{P}\big{(}d_{R}(I,S_{j})\leq t\big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\frac{1}{m}\big{[}\log(\det
S_{j})\big{]}^{2}\leq t^{2}\Big{)}=\mathbb{P}\big{(}(\det
S_{j})\leq\exp(m^{1/2}t)\big{)}.$

By (3.7), $\det S_{j}\overset{\mathcal{L}}{=}\prod_{i=1}^{j}(\det X_{i})$, and
we proceed to obtain a lower bound on the distribution of this product.

For random variables $Q_{1}$ and $Q_{2}$, recall that $Q_{1}$ is said to be
stochastically greater than $Q_{2}$, written
$Q_{1}\overset{\mathcal{L}}{\geq}Q_{2}$, if $\mathbb{P}(Q_{1}\leq
t)\leq\mathbb{P}(Q_{2}\leq t)$ for all $t\in\operatorname{\mathbb{R}}$. It is
well-known that if $Q_{ij}$, $i,j=1,2$ are positive random variables such that
$Q_{11}\overset{\mathcal{L}}{\geq}Q_{21}$ and
$Q_{12}\overset{\mathcal{L}}{\geq}Q_{22}$ then
$Q_{11}Q_{12}\overset{\mathcal{L}}{\geq}Q_{21}Q_{22}$.

Since $X_{j}\sim W_{m}(a,I_{m})$ then, by [13, Theorem 4],

$m(\det X_{j})^{1/m}\overset{\mathcal{L}}{\geq}\chi^{2}_{(2a-m-1)m}.$

Therefore, since $R_{1},\ldots,R_{n}$ are i.i.d. and
$\chi^{2}_{(2a-m-1)m}$-distributed then it follows that

$m^{j}(\det S_{j})^{1/m}\overset{\mathcal{L}}{=}\prod_{i=1}^{j}m(\det
X_{i})^{1/m}\overset{\mathcal{L}}{\geq}\prod_{i=1}^{j}R_{i},$

and therefore

$\displaystyle\mathbb{P}\big{(}d_{R}(I,S_{j})\leq t\big{)}$
$\displaystyle\leq\mathbb{P}\big{(}\det S_{j}\leq\exp(m^{1/2}t)\big{)}$
$\displaystyle=\mathbb{P}\big{(}m^{j}(\det S_{j})^{1/m}\leq
m^{j}\exp(m^{-1/2}t)\big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\prod_{i=1}^{j}R_{i}\leq
m^{j}\exp(m^{-1/2}t)\Big{)}.$

Applying (4.2), we now obtain (4.27). Also, by choosing $j=1$ we obtain
(4.28). ∎

We remark that much is known about the exact distribution of a product of
independent chi-square random variables; cf. [2, 13]. Such results can be used
to obtain sharper, but more complicated, upper bounds on the distribution
function of $U_{n}$.

Acknowledgments: We are very grateful to Apoorva Khare for providing us with
comments that led to improvements in an earlier draft of this manuscript. We
also thank an anonymous referee for reading our manuscript with great
thoroughness.

Declaration: No financial support was received to underwrite the preparation
of this manuscript, and the authors have no relevant financial or non-
financial interests to disclose.

Data Availability Statement: Data sharing is not applicable to this article as
no datasets were generated or analysed during the current study.

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##  Appendix A Martingale properties of $\boldsymbol{S_{n}}$,
$\boldsymbol{M_{n}}$, and $\boldsymbol{U_{n}}$

Let $\\{Y_{n},n\in\operatorname{\mathbb{N}}\\}$ be a sequence of random
entities taking values in a finite-dimensional Euclidean space
$(\operatorname{\mathbb{R}}^{d},\|\cdot\|)$, where
$\operatorname{\mathbb{E}}(\|Y_{n}\|)<\infty$. The sequence
$\\{Y_{n},n\in\operatorname{\mathbb{N}}\\}$ is called a martingale if
$Y_{n}=\operatorname{\mathbb{E}}(Y_{n+1}|Y_{1},\ldots,Y_{n})$ for all $n\geq
1$. If each $Y_{n}$ is scalar-valued then the sequence
$\\{Y_{n},n\in\operatorname{\mathbb{N}}\\}$ is called a submartingale if
$Y_{n}\leq\operatorname{\mathbb{E}}(Y_{n+1}|Y_{1},\ldots,Y_{n})$ for all
$n\geq 1$.

######  Proposition A.1.

Let $\\{X_{n},n\in\operatorname{\mathbb{N}}\\}$ be a sequence of orthogonally
invariant, i.i.d. random matrices in
$\mathcal{P}_{m}(\operatorname{\mathbb{R}})$ such that
$\operatorname{\mathbb{E}}(X_{n})=I_{n}$ for all $n$. Then the sequence
$\\{S_{n},n\in\operatorname{\mathbb{N}}\\}$ in (3.1) is a martingale, and the
sequences $\\{M_{n},n\in\operatorname{\mathbb{N}}\\}$ in (3.2) and
$\\{U_{n},n\in\operatorname{\mathbb{N}}\\}$ in (3.3) are submartingales.

Proof. Let $\mathcal{S}_{m}(\operatorname{\mathbb{R}})$ denote the vector
space of $m\times m$ symmetric matrices, and endow
$\mathcal{S}_{m}(\operatorname{\mathbb{R}})$ with the spectral norm,
$\|X\|_{\textrm{sp}}:=\max\\{|\lambda_{1}(X)|,\ldots,|\lambda_{m}(X)|\\}$, the
maximum absolute value of the eigenvalues of $X$. For
$X\in\mathcal{P}_{m}(\operatorname{\mathbb{R}})$, it is evident that
$\|X^{1/2}\|_{\textrm{sp}}=\|X\|_{\textrm{sp}}^{1/2}$. Also, by [16, p. 343,
Theorem 5.6.2], this norm is submultiplicative:
$\|XY\|_{\textrm{sp}}\leq\|X\|_{\textrm{sp}}\cdot\|Y\|_{\textrm{sp}}$.
Therefore,

$\displaystyle\|S_{n}\|_{\textrm{sp}}\overset{\mathcal{L}}{=}\|X_{n}\circ
S_{n-1}\|_{\textrm{sp}}$
$\displaystyle=\|X_{n}^{1/2}S_{n-1}X_{n}^{1/2}\|_{\textrm{sp}}$
$\displaystyle\leq(\|X_{n}\|_{\textrm{sp}}^{1/2})^{2}\cdot\|S_{n-1}\|_{\textrm{sp}}=\|X_{n}\|_{\textrm{sp}}\|S_{n-1}\|_{\textrm{sp}}.$

By iterating this inequality, or by applying (3.7), we obtain
$\|S_{n}\|_{\textrm{sp}}\overset{\mathcal{L}}{\leq}\|X_{1}\|_{\textrm{sp}}\cdots\|X_{n}\|_{\textrm{sp}}$.

Since $X_{1},\ldots,X_{n}$ are i.i.d. then it follows that

$\operatorname{\mathbb{E}}(\|S_{n}\|_{\textrm{sp}})\leq\operatorname{\mathbb{E}}(\|X_{1}\|_{\textrm{sp}}\cdots\|X_{n}\|_{\textrm{sp}})=(E\|X_{1}\|_{\textrm{sp}})^{n}.$

Also, since $\|X_{1}\|_{\textrm{sp}}<\operatorname{\mathop{\rm{tr}\hskip
1.0pt}}X_{1}$ then

$\operatorname{\mathbb{E}}(\|X_{1}\|_{\textrm{sp}})\leq\operatorname{\mathbb{E}}(\operatorname{\mathop{\rm{tr}\hskip
1.0pt}}X_{1})=\operatorname{\mathop{\rm{tr}\hskip
1.0pt}}\operatorname{\mathbb{E}}(X_{1})=m<\infty;$

therefore, $\operatorname{\mathbb{E}}(\|S_{n}\|_{\textrm{sp}})\leq
m^{n}<\infty$.

By (3.7) and the independence of $X_{n+1}$ and $\\{X_{1},\ldots,X_{n}\\}$,

$\displaystyle\operatorname{\mathbb{E}}(S_{n+1}|S_{1},\ldots,S_{n})$
$\displaystyle=\operatorname{\mathbb{E}}(S_{n}^{1/2}X_{n+1}S_{n}^{1/2}|S_{1},\ldots,S_{n})$
$\displaystyle=S_{n}^{1/2}\operatorname{\mathbb{E}}(X_{n+1}|S_{1},\ldots,S_{n})S_{n}^{1/2}=S_{n}^{1/2}\operatorname{\mathbb{E}}(X_{n+1})S_{n}^{1/2}=S_{n}.$

Therefore, the sequence $\\{S_{n},n\in\operatorname{\mathbb{N}}\\}$ is a
martingale.

We now show that $\\{M_{n},n\in\operatorname{\mathbb{N}}\\}$ is a
submartingale. First,

$\operatorname{\mathbb{E}}(M_{n})=\operatorname{\mathbb{E}}\max_{1\leq j\leq
n}d_{R}(I_{m},X_{j})\leq\sum_{j=1}^{n}\operatorname{\mathbb{E}}d_{R}(I_{m},X_{j})=n\operatorname{\mathbb{E}}d_{R}(I_{m},X_{1}).$

By (4.20),

$\operatorname{\mathbb{E}}d_{R}(I_{m},X_{1})=\operatorname{\mathbb{E}}\Big{(}\sum_{j=1}^{n}[\log\lambda_{j}(X_{1})]^{2}\Big{)}^{1/2}\leq\operatorname{\mathbb{E}}\sum_{j=1}^{n}|\log\lambda_{j}(X_{1})|.$

Applying the same analysis as at (4.23), we find that the latter expected
value is finite. See also the functions $\xi_{1}(v)$ and $k(v,\theta)$ in
(4.2) and (4.9). Therefore $\operatorname{\mathbb{E}}(M_{n})<\infty$.

Also, it is obvious that $M_{n+1}\geq M_{n}$ for all $n\geq 1$. Therefore

$\operatorname{\mathbb{E}}(M_{n+1}|M_{1},\ldots,M_{n})\geq\operatorname{\mathbb{E}}(M_{n}|M_{1},\ldots,M_{n})=M_{n},$

so $\\{M_{n},n\in\operatorname{\mathbb{N}}\\}$ is a submartingale.

As for $\\{U_{n},n\in\operatorname{\mathbb{N}}\\}$, it follows from (4.15)
that

$\operatorname{\mathbb{E}}(U_{n})=\int_{0}^{\infty}\mathbb{P}(U_{n}>t)\,\textrm{d}t\leq
nv^{-1}[\xi_{1}(v)]^{n},$

and therefore

$\operatorname{\mathbb{E}}(U_{n})\leq
n\inf_{v\in\operatorname{\mathop{\rm{Dom}\hskip
0.2pt}}(\xi_{1})}v^{-1}[\xi_{1}(v)]^{n}<\infty.$

Further, $U_{n+1}\geq U_{n}$ for all $n\geq 1$ and therefore

$\operatorname{\mathbb{E}}(U_{n+1}|U_{1},\ldots,U_{n})\geq\operatorname{\mathbb{E}}(U_{n}|U_{1},\ldots,U_{n})=U_{n},$

so $\\{U_{n},n\in\operatorname{\mathbb{N}}\\}$ also is a submartingale. ∎

By Kolmogorov’s inequalities for submartingales [27, p. 314],

$\mathbb{P}(M_{n}\geq t)\leq t^{-1}\operatorname{\mathbb{E}}(M_{n})$

and

$\mathbb{P}(U_{n}\geq t)\leq t^{-1}\operatorname{\mathbb{E}}(U_{n}),$

$t>0$. Since the sequences $\\{M_{n},n\in\operatorname{\mathbb{N}}\\}$ and
$\\{U_{n},n\in\operatorname{\mathbb{N}}\\}$ are nonnegative and increasing
then it follows that Kolmogorov’s inequalities for $M_{n}$ and $U_{n}$ are the
same as Markov’s inequalities.