Local Normal Approximations and Probability Metric Bounds for the Matrix-Variate T Distribution and Its Application to Hotelling’s T Statistic

Abstract

In this paper, we develop local expansions for the ratio of the centered matrix-variate T density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures. This work extends some previous results for the univariate Student distribution to the matrix-variate setting.

1. Introduction

For any $n\in \mathbb{N}$, define the space of (real symmetric) positive definite matrices of size $n\times n$ as follows:

$$S_{++}^n:=\left\{\mathbb{M}\in {\mathbb{R}}^{n\times n}:\mathbb{M}\mathrm{is}\mathrm{symmetric}\mathrm{and}\mathrm{positive}\mathrm{definite}\right\}.$$

For $d,m\in \mathbb{N}$, $\nu >0$, $M\in {\mathbb{R}}^{d\times m}$, $\mathrm{\Sigma }\in S_{++}^d$ and $\mathsf{\Omega }\in S_{++}^m$, the density function of the centered (and normalized) matrix-variate T distribution, hereafter denoted by $T_{d,m}(\nu ,\mathrm{\Sigma },\mathsf{\Omega })$, is defined, for all $X\in {\mathbb{R}}^{d\times m}$, by

$$\begin{array}{cc}\hfill K_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)& :=\frac{{\mathsf{\Gamma }}_d\left(\frac{1}{2}(\nu +m+d-1)\right)}{{\mathsf{\Gamma }}_d\left(\frac{1}{2}(\nu +d-1)\right)}\frac{|{\mathrm{I}}_d+{\nu }^{-1}{\mathrm{\Sigma }}^{-1}X{\mathsf{\Omega }}^{-1}{X}^{\top }{|}^{-(\nu +m+d-1)/2}}{{\left(\nu \pi \right)}^{md/2}{\left|\mathrm{\Sigma }\right|}^{m/2}{\left|\mathsf{\Omega }\right|}^{d/2}},\hfill \end{array}$$

(1)

(see, e.g., (Definition 4.2.1 in [1])) where $\nu$ is the number of degrees of freedom, and

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_d\left(z\right)& ={\int }_{\mathbb{S}\in S_{++}^d}{\left|\mathbb{S}\right|}^{z-(d+1)/2}exp(-\mathrm{tr}\left(\mathbb{S}\right))\mathrm{d}\mathbb{S}\hfill \\ \hfill & ={\pi }^{d(d-1)/4}\prod _{j=1}^d\mathsf{\Gamma }\left(z-\frac{j-1}{2}\right),\Re \left(z\right)>\frac{d-1}{2},\hfill \end{array}$$

denotes the multivariate gamma function—see, e.g., (Section 35.3 in [2]) and [3]—and

$$\mathsf{\Gamma }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}\mathrm{d}t,\Re \left(z\right)>0,$$

is the classical gamma function. The mean and covariance matrix for the vectorization of $T\sim T_{d,m}(\nu ,\mathrm{\Sigma },\mathsf{\Omega })$, namely

$$\mathrm{vec}\left(T\right):={(T_{11},T_{21},\dots ,T_{d1},T_{12},T_{22},\dots ,T_{d2},\dots ,T_{1m},T_{2m},\dots ,T_{dm})}^{\top },$$

($\mathrm{vec}(·)$ is the operator that stacks the columns of a matrix on top of each other) are known to be (see, e.g., Theorem 4.3.1 in [1], but be careful of the normalization):

$$\mathbb{E}\left[\mathrm{vec}\left(T\right)\right]={\mathbf{0}}_{dm}(\mathrm{i}.\mathrm{e}.,\mathbb{E}\left[T\right]=0_{d\times m}),$$

and

$$\mathbb{V}\mathrm{ar}\left(\mathrm{vec}\left(T^{\top }\right)\right)=\frac{\nu }{(\nu -2)}\mathrm{\Sigma }\otimes \mathsf{\Omega },\nu >2.$$

The first goal of our paper (Theorem 1) is to establish an asymptotic expansion for the ratio of the centered matrix-variate T density (1) to the centered matrix-variate normal (MN) density with the same covariances. According to (Gupta and Nagar [1], Theorem 2.2.1), the density of the ${\mathrm{MN}}_{d,m}(0_{d\times m},\mathrm{\Sigma }\otimes \mathsf{\Omega })$ distribution is

$$g_{\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)=\frac{exp\left(-\frac{1}{2}\mathrm{tr}\left({\mathrm{\Sigma }}^{-1}X{\mathsf{\Omega }}^{-1}{X}^{\top }\right)\right)}{{\left(2\pi \right)}^{md/2}{\left|\mathrm{\Sigma }\right|}^{m/2}{\left|\mathsf{\Omega }\right|}^{d/2}},X\in {\mathbb{R}}^{d\times m}.$$

(2)

The second goal of our paper (Theorem 2) is to apply the log-ratio expansion from Theorem 1 to derive upper bounds on multiple probability metrics between the measures induced by the centered matrix-variate T distribution and the corresponding centered matrix-variate normal distribution. In the special case $m=1$, this gives us probability metric upper bounds between the measure induced by Hotelling’s T statistic and the associated matrix-normal measure.

To give some practical motivations for the MN distribution (2), note that noise in the estimate of individual voxels of diffusion tensor magnetic resonance imaging (DT-MRI) data has been shown to be well modeled by a symmetric form of the ${\mathrm{MN}}_{3\times 3}$ distribution in [4,5,6]. The symmetric MN voxel distributions were combined into a tensor-variate normal distribution in [7,8], which could help to predict how the whole image (not just individual voxels) changes when shearing and dilation operations are applied in image wearing and registration problems; see Alexander et al. [9]. In [10], maximum likelihood estimators and likelihood ratio tests are developed for the eigenvalues and eigenvectors of a form of the symmetric MN distribution with an orthogonally invariant covariance structure, both in one-sample problems (for example, in image interpolation) and two-sample problems (when comparing images) and under a broad variety of assumptions. This work extended significantly the previous results of Mallows [11]. In [10], it is also mentioned that the polarization pattern of cosmic microwave background (CMB) radiation measurements can be represented by $2\times 2$ positive definite matrices; see the primer by Hu and White [12]. In a very recent and interesting paper, Vafaei Sadr and Movahed [13] presented evidence for the Gaussianity of the local extrema of CMB maps. We can also mention [14], where finite mixtures of skewed MN distributions were applied to an image recognition problem.

In general, we know that the Gaussian distribution is an attractor for sums of i.i.d. random variables with finite variance, which makes many estimators in statistics asymptotically normal. Similarly, we expect the MN distribution (2) to be an attractor for sums of i.i.d. random matrices with finite variances (Hotelling’s T-squared statistic is the most natural example), thus including many estimators, such as sample covariance matrices and score statistics for matrix parameters. In particular, if a given statistic or estimator is a function of the components of a sample covariance matrix for i.i.d. observations coming from a multivariate Gaussian population, then we could study its large sample properties (such as its moments) using Theorem 1 (for example, by turning a Student-moments estimation problem into a Gaussian-moments estimation problem).

The following is a brief outline of the paper. Our main results are stated in Section 2 and proven in Section 3. Technical moment calculations are gathered in Appendix A.

Notation 1.

Throughout the paper, $a=\mathcal{O}\left(b\right)$ means that $lim\; sup|a/b|<C$ as $\nu \to \infty$, where $C>0$ is a universal constant. Whenever C might depend on some parameter, we add a subscript (for example, $a={\mathcal{O}}_d\left(b\right)$). Similarly, $a=\mathrm{o}\left(b\right)$ means that $lim|a/b|=0$, and subscripts indicate which parameters the convergence rate can depend on. If $a=(1+\mathrm{o}(1\left)\right)b$, then we write $a\sim b$. The notation $\mathrm{tr}(·)$ will denote the trace operator for matrices and $|·|$ their determinant. For a matrix $\mathbb{M}\in {\mathbb{R}}^{d\times d}$ that is diagonalizable, ${\lambda }_1\left(\mathbb{M}\right)\ge \cdots \ge {\lambda }_d\left(\mathbb{M}\right)$ will denote its eigenvalues, and we let $\mathbf{\lambda }\left(\mathbb{M}\right):={({\lambda }_1\left(\mathbb{M}\right),\dots ,{\lambda }_d\left(\mathbb{M}\right))}^{\top }$.

2. Main Results

In Theorem 1 below, we prove an asymptotic expansion for the ratio of the centered matrix-variate T density to the centered matrix-variate normal (MN) density with the same covariances. The case $d=m=1$ was proven recently in [15] (see also [16] for an earlier rougher version). The result extends significantly the convergence in distribution result from Theorem 4.3.4 in [1].

Theorem 1.

Let $d,m\in \mathbb{N}$, $\mathrm{\Sigma }\in S_{++}^d$ and $\mathsf{\Omega }\in S_{++}^m$ be given. Pick any $\eta \in (0,1)$ and let

$$B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(\eta \right):=\left\{X\in {\mathbb{R}}^{d\times m}:\underset{1\le j\le d}{max}\frac{{\delta }_{{\lambda }_j}}{\sqrt{\nu -2}}\le \eta {\nu }^{-1/4}\right\}$$

denote the bulk of the centered matrix-variate T distribution, where

$${\Delta }_X={\mathrm{\Sigma }}^{-1/2}X{\mathsf{\Omega }}^{-1/2}\mathrm{and}{\delta }_{{\lambda }_j}:=\sqrt{\frac{\nu -2}{\nu }{\lambda }_j\left({\Delta }_X{\Delta }_X^{\top }\right)},1\le j\le d.$$

Then, as $\nu \to \infty$ and uniformly for $X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(\eta \right)$, we have

$$\begin{array}{cc}\hfill & log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\hfill \\ \hfill & ={\nu }^{-1}\left\{\begin{array}{c}\frac{1}{4}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)-\frac{(m+d+1)}{2}\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right)\hfill \\ +\frac{md(m+d+1)}{4}\hfill \end{array}\right\}\hfill \\ \hfill & +{\nu }^{-2}\left\{\begin{array}{c}-\frac{1}{6}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^3\right)+\frac{(m+d-1)}{4}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)\hfill \\ +\frac{md}{24}(13-2d^2-3d(-3+m)+9m-2m^2)\hfill \end{array}\right\}\hfill \\ \hfill & +{\nu }^{-3}\left\{\begin{array}{c}\frac{1}{8}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^4\right)-\frac{(m+d-1)}{6}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^3\right)\hfill \\ +\frac{md}{24}\left(\begin{array}{c}26+d^3+2d^2(-3+m)+11m\hfill \\ -6m^2+m^3+d(11-9m+2m^2)\hfill \end{array}\right)\hfill \end{array}\right\}\hfill \\ \hfill & +{\mathcal{O}}_{d,m,\eta }\left(\frac{1+\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^5\right)}{{\nu }^4}\right).\hfill \end{array}$$

(3)

Local approximations such as the one in Theorem 1 can be found for the Poisson, binomial and negative binomial distributions in [17] (based on Fourier analysis results from [18]), and [19] for the binomial distribution. Another approach, using Stein’s method, is used to study the variance-gamma distribution in [20]. Moreover, Kolmogorov and Wasserstein distance bounds are derived in [21,22] for the Laplace and variance-gamma distributions.

Below, we provide numerical evidence (displayed graphically) for the validity of the expansion in Theorem 1 when $d=m=2$. We compare three levels of approximation for various choices of $\mathbb{S}$. For any given $\mathbb{S}\in S_{++}^d$, define

$$\begin{array}{cc}\hfill E_0& :=\underset{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left({\nu }^{-1/4}\right)}{sup}\left|log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\right|,\hfill \\ \hfill E_1& :=\underset{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left({\nu }^{-1/4}\right)}{sup}\left|log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\right.\hfill \\ \hfill & \left.-{\nu }^{-1}\left\{\frac{1}{4}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)-\frac{(m+d+1)}{2}\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right)+\frac{md(m+d+1)}{4}\right\}\right|,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill E_2& :=\underset{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left({\nu }^{-1/4}\right)}{sup}\left|log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\right.\hfill \\ \hfill & -{\nu }^{-1}\left\{\frac{1}{4}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)-\frac{(m+d+1)}{2}\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right)+\frac{md(m+d+1)}{4}\right\}\hfill \\ \hfill & \left.-{\nu }^{-2}\left\{\begin{array}{c}-\frac{1}{6}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^3\right)+\frac{(m+d-1)}{4}\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)\hfill \\ +\frac{md}{24}(13-2d^2-3d(-3+m)+9m-2m^2)\hfill \end{array}\right\}\right|.\hfill \end{array}$$

In the R software [23], we use Equation (7) to evaluate the log-ratios inside $E_0$, $E_1$ and $E_2$.

Note that $X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left({\nu }^{-1/4}\right)$ implies $\left|\mathrm{tr}\right({\left({\Delta }_X{\Delta }_X^{\top }\right)}^k\left)\right|\le d$ for all $k\in \mathbb{N}$, so we expect from Theorem 1 that the maximum errors above ($E_0$, $E_1$ and $E_2$) will have the asymptotic behavior

$$\begin{array}{c}\hfill E_i={\mathcal{O}}_d\left({\nu }^{-(1+i)}\right),\mathrm{for}\mathrm{all}i\in \{0,1,2\},\end{array}$$

or, equivalently,

$$\begin{array}{c}\hfill \underset{\nu \to \infty }{lim\; inf}\frac{log{E}_i}{log\left({\nu }^{-1}\right)}\ge 1+i,\mathrm{for}\mathrm{all}i\in \{0,1,2\}.\end{array}$$

(5)

The property (5) is verified in Figure 1 below, for $\mathsf{\Omega }={\mathrm{I}}_2$ and various choices of ${\mathrm{\Sigma }}_{2\times 2}$. Similarly, the corresponding log-log plots of the errors as a function of $\nu$ are displayed in Figure 2. The simulations are limited to the range $5\le \nu \le 1005$. The R code that generated Figure 1 and Figure 2 can be found at Supplementary Material.

As a consequence of the previous theorem, we can derive asymptotic upper bounds on several probability metrics between the probability measures induced by the centered matrix-variate T distribution (1) and the corresponding centered matrix-variate normal distribution (2). The distance between Hotelling’s T statistic [24] and the corresponding matrix-variate normal distribution is obtained in the special case $m=1$.

Theorem 2

(Probability metric upper bounds). Let $d,m\in \mathbb{N}$, $\mathrm{\Sigma }\in S_{++}^d$ and $\mathsf{\Omega }\in S_{++}^m$ be given. Assume that $X\sim T_{d,m}(\nu ,\mathrm{\Sigma },\mathsf{\Omega })$, $Y\sim {\mathrm{MN}}_{d,m}(0_{d\times m},\mathrm{\Sigma }\otimes \mathsf{\Omega })$, and let ${\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}$ and ${\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }}$ be the laws of X and $Y\sqrt{\nu /(\nu -2)}$, respectively. Then, as $\nu \to \infty$,

$$\mathrm{dist}({\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }},{\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }})\le \frac{C{\left(md\right)}^{3/2}}{\nu }\mathrm{and}\mathcal{H}({\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }},{\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }})\le \sqrt{\frac{2C{\left(md\right)}^{3/2}}{\nu }},$$

where $C>0$ is a universal constant, $\mathcal{H}(·,·)$ denotes the Hellinger distance, and $\mathrm{dist}(·,·)$ can be replaced by any of the following probability metrics: total variation, Kolmogorov (or uniform) metric, Lévy metric, discrepancy metric, Prokhorov metric.

3. Proofs

Proof of Theorem 1. 

First, we take the expression in (1) over the one in (2):

$$\begin{array}{cc}\hfill & \frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\hfill \\ \hfill & ={\left[\frac{2}{\nu -2}\right]}^{md/2}\prod _{j=1}^d\frac{\mathsf{\Gamma }\left(\frac{1}{2}(\nu +m+d-j)\right)}{\mathsf{\Gamma }\left(\frac{1}{2}(\nu +d-j)\right)}\hfill \\ \hfill & ·exp\left(\frac{(\nu -2)}{2\nu }\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right)\right){\left|{\mathrm{I}}_d+{\nu }^{-1}{\Delta }_X{\Delta }_X^{\top }\right|}^{-(\nu +m+d-1)/2}.\hfill \end{array}$$

(6)

The last determinant was obtained using the fact that the eigenvalues of a product of rectangular matrices are invariant under cyclic permutations (as long as the products remain well defined). Indeed, for all $j\in \{1,2,\dots ,d\}$, we have

$$\begin{array}{cc}\hfill {\lambda }_j({\mathrm{I}}_d+{\nu }^{-1}{\mathrm{\Sigma }}^{-1}X{\mathsf{\Omega }}^{-1}{X}^{\top })& =1+{\nu }^{-1}{\lambda }_j\left({\mathrm{\Sigma }}^{-1}X{\mathsf{\Omega }}^{-1}{X}^{\top }\right)\hfill \\ \hfill & =1+{\nu }^{-1}{\lambda }_j\left({\Delta }_X{\Delta }_X^{\top }\right)={\lambda }_j({\mathrm{I}}_d+{\nu }^{-1}{\Delta }_X{\Delta }_X^{\top }).\hfill \end{array}$$

By taking the logarithm on both sides of (6), we get

$$\begin{array}{cc}\hfill & log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\hfill \\ \hfill & =-\frac{md}{2}log\left(\frac{\nu -2}{2}\right)+\sum _{j=1}^d\left[log\mathsf{\Gamma }\left(\frac{1}{2}(\nu +m+d-j)\right)-log\mathsf{\Gamma }\left(\frac{1}{2}(\nu +d-j)\right)\right]\hfill \\ \hfill & +\frac{1}{2}\sum _{j=1}^d{\delta }_{{\lambda }_j}^2-\frac{(\nu +m+d-1)}{2}\sum _{j=1}^{d}log\left(1+{\left(\frac{{\delta }_{{\lambda }_j}}{\sqrt{\nu -2}}\right)}^2\right).\hfill \end{array}$$

(7)

By applying the Taylor expansions,

$$\begin{array}{cc}\hfill & log\mathsf{\Gamma }\left(\frac{1}{2}(\nu +m+d-j)\right)-log\mathsf{\Gamma }\left(\frac{1}{2}(\nu +d-j)\right)\hfill \\ \hfill & =\frac{1}{2}(\nu +m+d-j-1)log\left(\frac{1}{2}(\nu +m+d-j)\right)-\frac{1}{2}(\nu +d-j-1)log\left(\frac{1}{2}(\nu +d-j)\right)\hfill \\ \hfill & -\frac{m}{2}+\frac{2}{12(\nu +m+d-j)}-\frac{2}{12(\nu +d-j)}\hfill \\ \hfill & -\frac{2^3}{360{(\nu +m+d-j)}^3}+\frac{2^3}{360{(\nu +d-j)}^3}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right)\hfill \\ \hfill & =\frac{m}{2}log\left(\frac{\nu }{2}\right)+\frac{m(-2+2d-2j+m)}{4\nu }-\frac{m}{12{\nu }^2}\left\{\begin{array}{c}2+3d^2+3j^2-3j(-2+m)\hfill \\ -3m+m^2+d(-6-6j+3m)\hfill \end{array}\right\}\hfill \\ \hfill & +\frac{m}{24{\nu }^3}\left\{\begin{array}{c}4d^3-4j^3-6d^2(2+2j-m)+6j^2(-2+m)+{(-2+m)}^{2}m\hfill \\ -4j(2-3m+m^2)+4d(2+3j^2-3j(-2+m)-3m+m^2)\hfill \end{array}\right\}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right).\hfill \end{array}$$

(see, e.g., (Ref. [25], p. 257)) and

$$-\frac{md}{2}log\left(\frac{\nu -2}{2}\right)+\frac{md}{2}log\left(\frac{\nu }{2}\right)=\frac{4md}{4\nu }+\frac{12md}{12{\nu }^2}+\frac{32md}{24{\nu }^3}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right),$$

and

$$log(1+y)=y-\frac{1}{2}{y}^2+\frac{1}{3}{y}^3-\frac{1}{4}{y}^4+{\mathcal{O}}_{\eta }\left(y^5\right),\left|y\right|<\eta <1,$$

in the above equation, we obtain

$$\begin{array}{cc}\hfill & log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\hfill \\ \hfill & =\sum _{j=1}^d\frac{m(2+2d-2j+m)}{4\nu }-\sum _{j=1}^d\frac{m}{12{\nu }^2}\left\{\begin{array}{c}-10+3d^2+3j^2-3j(-2+m)\hfill \\ -3m+m^2+d(-6-6j+3m)\hfill \end{array}\right\}\hfill \\ \hfill & +\sum _{j=1}^d\frac{m}{24{\nu }^3}\left\{\begin{array}{c}32+4d^3-4j^3-6d^2(2+2j-m)+6j^2(-2+m)+{(-2+m)}^{2}m\hfill \\ -4j(2-3m+m^2)+4d(2+3j^2-3j(-2+m)-3m+m^2)\hfill \end{array}\right\}\hfill \\ \hfill & +\frac{1}{2}\sum _{j=1}^d{\delta }_{{\lambda }_j}^2-\frac{(\nu +m+d-1)}{2}\sum _{j=1}^d{\left(\frac{{\delta }_{{\lambda }_j}}{\sqrt{\nu -2}}\right)}^2\hfill \\ \hfill & +\frac{(\nu +m+d-1)}{4}\sum _{j=1}^d{\left(\frac{{\delta }_{{\lambda }_j}}{\sqrt{\nu -2}}\right)}^4-\frac{(\nu +m+d-1)}{6}\sum _{j=1}^d{\left(\frac{{\delta }_{{\lambda }_j}}{\sqrt{\nu -2}}\right)}^6\hfill \\ \hfill & +\frac{(\nu +m+d-1)}{8}\sum _{j=1}^d{\left(\frac{{\delta }_{{\lambda }_j}}{\sqrt{\nu -2}}\right)}^8+{\mathcal{O}}_{d,m,\eta }\left(\frac{1+{max}_{1\le j\le d}{\left|{\delta }_{{\lambda }_j}\right|}^{10}}{{\nu }^4}\right).\hfill \end{array}$$

(8)

Now,

$$\begin{array}{cc}\hfill \frac{1}{2}-\frac{\nu +m+d-1}{2(\nu -2)}& =-\frac{(m+d+1)}{2\nu }-\frac{(m+d+1)}{{\nu }^2}-\frac{2(m+d+1)}{{\nu }^3}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right),\hfill \\ \hfill \frac{\nu +m+d-1}{4{(\nu -2)}^2}& =\frac{1}{4\nu }+\frac{(m+d+3)}{4{\nu }^2}+\frac{(m+d+2)}{{\nu }^3}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right),\hfill \\ \hfill -\frac{\nu +m+d-1}{6{(\nu -2)}^3}& =-\frac{1}{6{\nu }^2}-\frac{(m+d+5)}{6{\nu }^3}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right),\hfill \\ \hfill \frac{\nu +m+d-1}{8{(\nu -2)}^4}& =\frac{1}{8{\nu }^3}+{\mathcal{O}}_{m,d}\left({\nu }^{-4}\right),\hfill \end{array}$$

so we can rewrite (8) as

$$\begin{array}{cc}\hfill & log\left(\frac{{[\nu /(\nu -2)]}^{md/2}{K}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}\left(X\right)}{g_{\mathrm{\Sigma },\mathsf{\Omega }}(X/\sqrt{\nu /(\nu -2)})}\right)\hfill \\ \hfill & ={\nu }^{-1}\sum _{j=1}^d\left\{\frac{1}{4}{\delta }_{{\lambda }_j}^4-\frac{(m+d+1)}{2}{\delta }_{{\lambda }_j}^2+\frac{m(2+2d-2j+m)}{4}\right\}\hfill \\ \hfill & +{\nu }^{-2}\sum _{j=1}^d\left\{\begin{array}{c}-\frac{1}{6}{\delta }_{{\lambda }_j}^6+\frac{(m+d+3)}{4}{\delta }_{{\lambda }_j}^4-(m+d+1){\delta }_{{\lambda }_j}^2\hfill \\ -\frac{m}{12}\left\{\begin{array}{c}-10+3d^2+3j^2-3j(-2+m)\hfill \\ -3m+m^2+d(-6-6j+3m)\hfill \end{array}\right\}\hfill \end{array}\right\}\hfill \\ \hfill & +{\nu }^{-3}\sum _{j=1}^d\left\{\begin{array}{c}\frac{1}{8}{\delta }_{{\lambda }_j}^8-\frac{(m+d+5)}{6}{\delta }_{{\lambda }_j}^6+(m+d+2){\delta }_{{\lambda }_j}^4-2(m+d+1){\delta }_{{\lambda }_j}^2\hfill \\ +\frac{m}{24}\left\{\begin{array}{c}32+4d^3-4j^3-6d^2(2+2j-m)+6j^2(-2+m)+{(-2+m)}^{2}m\hfill \\ -4j(2-3m+m^2)+4d(2+3j^2-3j(-2+m)-3m+m^2)\hfill \end{array}\right\}\hfill \end{array}\right\}\hfill \\ \hfill & +{\mathcal{O}}_{d,m,\eta }\left(\frac{1+{max}_{1\le j\le d}{\left|{\delta }_{{\lambda }_j}\right|}^{10}}{{\nu }^4}\right),\hfill \end{array}$$

which proves (3) after some simplifications with Mathematica.    □

Proof od Theorem 2. 

By the comparison of the total variation norm $\parallel ·\parallel$ with the Hellinger distance on page 726 of Carter [26], we already know that

$$\begin{array}{cc}\hfill & \parallel {\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}-{\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }}\parallel \hfill \\ \hfill & \le \sqrt{2\mathbb{P}\left(X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}^c(1/2)\right)+\mathbb{E}\left[log\left(\frac{\mathrm{d}{\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}}{\mathrm{d}{\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }}}\left(X\right)\right){\mathbb{1}}_{\{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}(1/2)\}}\right]}.\hfill \end{array}$$

(9)

Given that ${\Delta }_X={\mathrm{\Sigma }}^{-1/2}X{\mathsf{\Omega }}^{-1/2}\sim T_{d,m}(\nu ,{\mathrm{I}}_d,{\mathrm{I}}_m)$ by Theorem 4.3.5 in [1], we know, by Theorem 4.2.1 in [1], that

$${\Delta }_X\stackrel{\mathrm{law}}{=}{\left({\nu }^{-1}\mathbb{S}\right)}^{-1/2}Z,$$

for $\mathbb{S}\sim {\mathrm{Wishart}}_{d\times d}(\nu +d-1,{\mathrm{I}}_d)$ and $Z\sim {\mathrm{MN}}_{d\times m}(0_{d\times m},{\mathrm{I}}_d\otimes {\mathrm{I}}_m)$ that are independent, so that, by Theorems 3.3.1 and 3.3.3 in [1], we have

$${\Delta }_X{\Delta }_X^{\top }|\mathbb{S}\sim {\mathrm{Wishart}}_{d\times d}(m,\nu {\mathbb{S}}^{-1}).$$

(10)

Therefore, by conditioning on $\mathbb{S}$, and then by applying the sub-multiplicativity of the largest eigenvalue for nonnegative definite matrices, and a large deviation bound on the maximum eigenvalue of a Wishart matrix (which is sub-exponential), we get, for $\nu$ large enough,

$$\begin{array}{cc}\hfill \mathbb{P}\left(X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}^c(1/2)\right)& \le \mathbb{E}\left[\left.\mathbb{P}\left({\lambda }_1\left({\Delta }_X{\Delta }_X^{\top }\right)>\frac{{\nu }^{1/2}}{4}\right|\mathbb{S}\right)\right]\hfill \\ \hfill & \le \mathbb{E}\left[\left.\mathbb{P}\left({\lambda }_1\left({\left({\nu }^{-1}\mathbb{S}\right)}^{-1/2}\right){\lambda }_1\left(Z{Z}^{\top }\right){\lambda }_1\left({\left({\nu }^{-1}\mathbb{S}\right)}^{-1/2}\right)>\frac{{\nu }^{1/2}}{4}\right|\mathbb{S}\right)\right]\hfill \\ \hfill & =\mathbb{E}\left[\left.\mathbb{P}\left({\lambda }_1\left(Z{Z}^{\top }\right)>\frac{{\lambda }_d\left(\mathbb{S}\right)}{4{\nu }^{1/2}}\right|\mathbb{S}\right)\right]\hfill \\ \hfill & \le C_{m,d}exp\left(-\frac{{\nu }^{1/2}}{{10}^{4}md}\right),\hfill \end{array}$$

(11)

for some positive constant $C_{m,d}$ that depends only on m and d. By Theorem 1, we also have

$$\begin{array}{cc}\hfill & \mathbb{E}\left[log\left(\frac{\mathrm{d}{\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}}{\mathrm{d}{\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }}}\left(X\right)\right){\mathbb{1}}_{\{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}(1/2)\}}\right]\hfill \\ \hfill & ={\nu }^{-1}\left\{\begin{array}{c}\frac{1}{4}·\mathbb{E}\left[\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)\right]\hfill \\ -\frac{(m+d+1)}{2}·\mathbb{E}\left[\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right)\right]+\frac{md(m+d+1)}{4}\hfill \end{array}\right\}\hfill \\ \hfill & +{\nu }^{-1}\left\{\begin{array}{c}\mathcal{O}\left(\mathbb{E}\left[\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right){\mathbb{1}}_{\{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}(1/2)\}}\right]\right)\hfill \\ +(m+d)\mathcal{O}\left(\mathbb{E}\left[\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right){\mathbb{1}}_{\{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}(1/2)\}}\right]\right)+\mathcal{O}\left(m(m+d)\right)\hfill \end{array}\right\}\hfill \\ \hfill & +{\nu }^{-2}\left\{\begin{array}{c}\mathcal{O}\left(\mathbb{E}\left[\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^3\right)\right]\right)+(m+d)\mathcal{O}\left(\mathbb{E}\left[\mathrm{tr}\left({\left({\Delta }_X{\Delta }_X^{\top }\right)}^2\right)\right]\right)\hfill \\ +(m+d)\mathcal{O}\left(\mathbb{E}\left[\mathrm{tr}\left({\Delta }_X{\Delta }_X^{\top }\right)\right]\right)+\mathcal{O}\left(md{(m+d)}^2\right)\hfill \end{array}\right\}.\hfill \end{array}$$

(12)

On the right-hand side, the first line is estimated using Lemma A1, and the second line is bounded using Lemma A2. We find

$$\mathbb{E}\left[log\left(\frac{\mathrm{d}{\mathbb{P}}_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}}{\mathrm{d}{\mathbb{Q}}_{\mathrm{\Sigma },\mathsf{\Omega }}}\left(X\right)\right){\mathbb{1}}_{\{X\in B_{\nu ,\mathrm{\Sigma },\mathsf{\Omega }}(1/2)\}}\right]=\mathcal{O}\left(m^3{d}^3{\nu }^{-2}\right).$$

Putting (11) and (12) together in (9) gives the conclusion.    □