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

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 45:4101–4111, 2017), 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.

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Acknowledgements

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.

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Correspondence to Donald Richards.

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A Martingale Properties of \(\varvec{S_n}\), \(\varvec{M_n}\), and \(\varvec{U_n}\)

A Martingale Properties of \(\varvec{S_n}\), \(\varvec{M_n}\), and \(\varvec{U_n}\)

Let \(\{Y_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) be a sequence of random entities taking values in a finite-dimensional Euclidean space \((\mathbb {R}^d,\Vert \cdot \Vert )\), where \({{\,\mathrm{\mathbb {E}}\,}}(\Vert Y_n\Vert ) < \infty \). The sequence \(\{Y_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) is called a martingale if \(Y_n = {{\,\mathrm{\mathbb {E}}\,}}(Y_{n+1} | Y_1,\ldots ,Y_n)\) for all \(n \ge 1\). If each \(Y_n\) is scalar-valued, then the sequence \(\{Y_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) is called a submartingale if \(Y_n \le {{\,\mathrm{\mathbb {E}}\,}}(Y_{n+1} | Y_1,\ldots ,Y_n)\) for all \(n \ge 1\).

Proposition A.1

Let \(\{X_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) be a sequence of orthogonally invariant, i.i.d. random matrices in \(\mathcal {P}_m(\mathbb {R})\) such that \({{\,\mathrm{\mathbb {E}}\,}}(X_n) = I_n\) for all n. Then, the sequence \(\{S_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) in (3.1) is a martingale, and the sequences \(\{M_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) in (3.2) and \(\{U_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) in (3.3) are submartingales.

Proof

Let \(\mathcal {S}_m(\mathbb {R})\) denote the vector space of \(m \times m\) symmetric matrices, and endow \(\mathcal {S}_m(\mathbb {R})\) with the spectral norm, \(\Vert X\Vert _{\text {sp}} := \max \{|\lambda _1(X)|,\ldots ,|\lambda _m(X)|\}\), the maximum absolute value of the eigenvalues of X. For \(X \in \mathcal {P}_m(\mathbb {R})\), it is evident that \(\Vert X^{1/2}\Vert _{\text {sp}} = \Vert X\Vert _{\text {sp}}^{1/2}\). Also, by [15, p. 343, Theorem 5.6.2], this norm is submultiplicative: \(\Vert XY\Vert _{\text {sp}} \le \Vert X\Vert _{\text {sp}} \cdot \Vert Y\Vert _{\text {sp}}\). Therefore,

$$\begin{aligned} \Vert S_n\Vert _{\text {sp}} \overset{\mathcal {L}}{=}\Vert X_n \circ S_{n-1}\Vert _{\text {sp}}&= \Vert X_n^{1/2} S_{n-1} X_n^{1/2}\Vert _{\text {sp}} \\&\le (\Vert X_n\Vert _{\text {sp}}^{1/2})^2 \cdot \Vert S_{n-1}\Vert _{\text {sp}} = \Vert X_n\Vert _{\text {sp}} \Vert S_{n-1}\Vert _{\text {sp}}. \end{aligned}$$

By iterating this inequality, or by applying (3.7), we obtain \(\Vert S_n\Vert _{\text {sp}} \overset{\mathcal {L}}{\le }\Vert X_1\Vert _{\text {sp}} \cdots \Vert X_n\Vert _{\text {sp}}\).

Since \(X_1,\ldots ,X_n\) are i.i.d., then it follows that

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(\Vert S_n\Vert _{\text {sp}}) \le {{\,\mathrm{\mathbb {E}}\,}}(\Vert X_1\Vert _{\text {sp}} \cdots \Vert X_n\Vert _{\text {sp}}) = (E \Vert X_1\Vert _{\text {sp}})^n. \end{aligned}$$

Also, since \(\Vert X_1\Vert _{\text {sp}} < {\text {tr}}X_1\), then

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(\Vert X_1\Vert _{\text {sp}}) \le {{\,\mathrm{\mathbb {E}}\,}}({\text {tr}}X_1) = {\text {tr}}{{\,\mathrm{\mathbb {E}}\,}}(X_1) = m < \infty ; \end{aligned}$$

therefore, \({{\,\mathrm{\mathbb {E}}\,}}(\Vert S_n\Vert _{\text {sp}}) \le m^n < \infty \).

By (3.7) and the independence of \(X_{n+1}\) and \(\{X_1,\ldots ,X_n\}\),

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(S_{n+1} | S_1,\ldots ,S_n)&= {{\,\mathrm{\mathbb {E}}\,}}(S_{n}^{1/2} X_{n+1} S_n^{1/2} | S_1, \ldots , S_n) \\&= S_{n}^{1/2} {{\,\mathrm{\mathbb {E}}\,}}(X_{n+1} | S_1,\ldots ,S_n) S_{n}^{1/2} = S_{n}^{1/2} {{\,\mathrm{\mathbb {E}}\,}}(X_{n+1}) S_{n}^{1/2} = S_n. \end{aligned}$$

Therefore, the sequence \(\{S_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) is a martingale.

We now show that \(\{M_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) is a submartingale. First,

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(M_n) = {{\,\mathrm{\mathbb {E}}\,}}\max _{1 \le j \le n} d_R(I_m,X_j) \le \sum _{j=1}^n {{\,\mathrm{\mathbb {E}}\,}}d_R(I_m,X_j) = n {{\,\mathrm{\mathbb {E}}\,}}d_R(I_m,X_1). \end{aligned}$$

By (4.20),

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}d_R(I_m,X_1) = {{\,\mathrm{\mathbb {E}}\,}}\Big (\sum _{j=1}^n [\log \lambda _j(X_1)]^2\Big )^{1/2} \le {{\,\mathrm{\mathbb {E}}\,}}\sum _{j=1}^n |\log \lambda _j(X_1)|. \end{aligned}$$

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 (4.19) and (4.9). Therefore, \({{\,\mathrm{\mathbb {E}}\,}}(M_n) < \infty \).

Also, it is obvious that \(M_{n+1} \ge M_n\) for all \(n \ge 1\). Therefore,

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(M_{n+1} | M_1,\ldots ,M_n) \ge {{\,\mathrm{\mathbb {E}}\,}}(M_n | M_1,\ldots ,M_n) = M_n, \end{aligned}$$

so \(\{M_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) is a submartingale.

As for \(\{U_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\), it follows from (4.15) that

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(U_n) = \int _0^\infty \mathbb {P}(U_n > t) \,\text {d}t \le n v^{-1} [\xi _1(v)]^n, \end{aligned}$$

and therefore

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(U_n) \le n \inf _{v \in {\text {Dom}}(\xi _1)} v^{-1} [\xi _1(v)]^n < \infty . \end{aligned}$$

Further, \(U_{n+1} \ge U_n\) for all \(n \ge 1\) and therefore

$$\begin{aligned} {{\,\mathrm{\mathbb {E}}\,}}(U_{n+1} | U_1,\ldots ,U_n) \ge {{\,\mathrm{\mathbb {E}}\,}}(U_n | U_1,\ldots ,U_n) = U_n, \end{aligned}$$

so \(\{U_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) also is a submartingale. \(\square \)

By Kolmogorov’s inequalities for submartingales [26, p. 314],

$$\begin{aligned} \mathbb {P}(M_n \ge t) \le t^{-1} {{\,\mathrm{\mathbb {E}}\,}}(M_n) \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}(U_n \ge t) \le t^{-1} {{\,\mathrm{\mathbb {E}}\,}}(U_n), \end{aligned}$$

\(t > 0\). Since the sequences \(\{M_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) and \(\{U_n, n \in {{\,\mathrm{\mathbb {N}}\,}}\}\) are nonnegative and increasing, then it follows that Kolmogorov’s inequalities for \(M_n\) and \(U_n\) are the same as Markov’s inequalities.

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Bagyan, A., Richards, D. Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices. J Theor Probab (2022). https://doi.org/10.1007/s10959-022-01189-7

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  • DOI: https://doi.org/10.1007/s10959-022-01189-7

Keywords

  • Orthogonal invariance
  • Riemannian metric
  • Submartingale
  • Symmetric cone
  • Thompson’s metric
  • Wishart distribution

Mathematics Subject Classification (2020)

  • Primary: 60E15
  • 62E15
  • Secondary: 60B20
  • 62E17