Abstract
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermartingale property of the trace exponential proved within the framework of stochastic calculus. We provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales.
This is a preview of subscription content, access via your institution.
Notes
Let us note that this definition does not imply that a purely discontinuous martingale is the sum of its jumps: for example a compensated Poisson process \(N_t-\lambda t\) is a purely discontinuous martingale that has a continuous component.
References
Ahlswede, R., Winter, A.: Strong converse for identification via quantum channels. Inf. Theory IEEE Trans. 48(3), 569–579 (2002)
Bandeira, A.S.: Concentration inequalities, scalar and matrix versions. Lecture Notes. Available at http://math.mit.edu/~bandeira (2015)
Bhatia, R.: Matrix Analysis. Springer, New York (1997)
Brémaud, P.: Point Processes and Queues: Martingale Dynamics: Martingale Dynamics. Springer, New York (1981)
Bunea, F., She, Y., Wegkamp, M.H.: Optimal selection of reduced rank estimators of high-dimensional matrices. Ann. Stat. 39(2), 1282–1309 (2011)
Candès, E.J., Li, X., Ma, Y., John, W.: Robust principal component analysis? J. ACM 8, 1–37 (2009)
Christofides, D., Markström, K.: Expansion properties of random cayley graphs and vertex transitive graphs via matrix martingales. Random Struct. Algorithms 32(1), 88–100 (2008)
Gaïffas, S., Guilloux, A.: High-dimensional additive hazards models and the lasso. Electron. J. Stat. 6, 522–546 (2012)
Gittens, A.: The spectral norm error of the naive nystrom extension. arXiv preprint arXiv:1110.5305 (2011)
Golub, G.H., Van Loan, C.F.: Matrix Computations. JHU Press, Baltimore (2013)
Gross, D.: Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57, 1548–1566 (2011)
Hansen, N.R., Reynaud-Bouret, P., Rivoirard, V.: Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21(1), 83–143 (2015)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, New York (1987)
Johnson, C.R., Horn, R.A.: Topics in Matrix Analysis. Cambridge University, Cambridge (1991)
Koltchinskii, V.: Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems: Saint-Flour XXXVIII-2008, 2033rd edn. Springer, New York (2011)
Koltchinskii, V., Lounici, K., Tsybakov, A.B.: Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Stat. 39(5), 2302–2329 (2011)
Latala, R.: Some estimates of norms of random matrices. Proc. Am. Math. Soc. 133, 1273–1282 (2005)
Lieb, E.H.: Convex trace functions and the Wigner–Yanase–Dyson conjecture. Adv. Math. 11(3), 267–288 (1973)
Liptser, R.S., Shiryayev, A.N.: Theory of Martingales. Springer, New York (1989)
Mackey, L., Jordan, M.I., Chen, R.Y., Farrell, B., Tropp, J.A.: Matrix concentration inequalities via the method of exchangeable pairs. Ann. Probab. 42(3), 906–945 (2014)
Mackey, L.W., Jordan, M.I., Talwalkar, A.: Divide-and-conquer matrix factorization. In: NIPS, pp. 1134–1142 (2011)
Massart, P.: Concentration Inequalities and Model Selection, 1896th edn. Springer, New York (2007)
Minsker, S.: On some extensions of Bernstein’s inequality for self-adjoint operators. arXiv preprint arXiv:1112.5448 (2011)
Negahban, S., Wainwright, M.J.: Restricted strong convexity and weighted matrix completion: optimal bounds with noise. J. Mach. Learn. Res. 13(1), 1665–1697 (2012)
Oliveira, R.I.: Concentration of the adjacency matrix and of the laplacian in random graphs with independent edges. arXiv preprint arXiv:0911.0600 (2009)
Oliveira, R.I.: Sums of random hermitian matrices and an inequality by Rudelson. Electron. Commun. Probab. 15(203–212), 26 (2010)
Paulsen, V.I., Bollobás, B., Fulton, W., Katok, A., Kirwan, F., Sarnak, P.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)
Petz, D.: A survey of certain trace inequalities. Funct. Anal. Oper. Theory 30, 287–298 (1994)
Recht, B.: A simpler approach to matrix completion. J. Mach. Learn. Res. 12, 3413–3430 (2011)
Reynaud-Bouret, P.: Compensator and exponential inequalities for some suprema of counting processes. Stat. Probab. Lett. 76(14), 1514–1521 (2006)
Rohde, A., Tsybakov, A.B.: Estimation of high-dimensional low-rank matrices. Ann. Stat. 39(2), 887–930 (2011)
Seginer, Y.: The expected norm of random matrices. Combin. Probab. Comput. 9, 149–166 (2000)
Tropp, J.A.: Freedman’s inequality for matrix martingales. Electron. Commun. Probab. 16, 262–270 (2011)
Tropp, J.A.: User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12(4), 389–434 (2012)
van de Geer, S.: Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Stat. 23(5), 1779–1801 (1995)
Acknowledgements
We gratefully acknowledge the anonymous reviewers of the first version of this paper for their helpful comments and suggestions. The authors would like to thank Carl Graham and Peter Tankov for various comments on our paper. This research benefited from the support of the Chair “Markets in Transition”, under the aegis of “Louis Bachelier Finance and Sustainable Growth” laboratory, a joint initiative of École Polytechnique, Université d’Évry Val d’Essonne and Fédération Bancaire Francaise, and of the Data Initiative of Ecole Polytechnique.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Tools for the study of matrix martingales in continuous time
In this section we give tools for the study of matrix martingales in continuous time. We proceed by steps. The main result of this section, namely Proposition 1, proves that the trace exponential of a matrix martingale is a supermartingale, when properly corrected by terms involving quadratic covariations.
1.1 Appendix A.1: A first tool
We give first a simple lemma that links the largest eigenvalues of random matrices to the trace exponential of their difference.
Lemma A.1
Let \(\varvec{X}\) and \(\varvec{Y}\) be two symmetric random matrices such that
for some \(k > 0\). Then, we have
for any \(x > 0\).
Proof
Using the fact that [28]
along with the fact that \({\varvec{Y}}\preccurlyeq \lambda _{\max }({\varvec{Y}}) {\varvec{I}}\), one has
where we set \(E = \{ \lambda _{\max }({\varvec{X}}) \ge \lambda _{\max }({\varvec{Y}}) + x \}\). Now, since \(\lambda _{\max }({\varvec{M}}) \le {{\mathrm{tr}}}{\varvec{M}}\) for any symmetric positive definite matrix \({\varvec{M}}\), we obtain
so that taking the expectation on both sides proves Lemma A.1. \(\square \)
1.2 Appendix A.2: Various definitions and Itô’s Lemma for functions of matrices
In this section we describe some classical notions from stochastic calculus [13, 19] and extend them to matrix semimartingales. Let us recall that the quadratic covariation of two scalar semimartingales \(X_t\) and \(Y_t\) is defined as
It can be proven (see e.g. [13]) that the non-decreasing process \([X, X]_t\), often denoted as \([X]_t\), does correspond to the quadratic variation of \(X_t\) since it is equal to the limit (in probability) of \(\sum _i (X_{t_i}-X_{t_{i-1}})^2\) when the mesh size of the partition \(\{t_i \}_i\) of the interval [0, t] goes to zero.
If \(X_t\) is a square integrable scalar martingale, then its predictable quadratic variation \(\langle X \rangle _t\) is defined as the unique predictable increasing process such that \(X_t^2 - \langle X \rangle _t\) is a martingale. The predictable quadratic covariation between two square integrable scalar martingales \(X_t\) and \(Y_t\) is then defined from the polarization identity:
A martingale \(X_t\) is said to be continuous if its sample paths \(t\mapsto X_t\) are a.s. continuous, and purely discontinuous Footnote 1 if \(X_0 = 0\) and \(\langle X, Y \rangle _t = 0\) for any continuous martingale \( Y_t\).
The notion of predictable quadratic variation can be extended to semimartingales. Indeed, any semimartingale \(X_t\) can be represented as a sum:
where \(X^{c}_t\) is a continuous local martingale, \(X^{d}_t\) is a purely discontinuous local martingale and \(A_t\) is a process of bounded variations. Since in the decomposition (24), \(X^{c}_t\) is unambiguously determined, \(\langle X^c \rangle _t\) is therefore well defined [13]. Within this framework, one can prove (see e.g. [13]) that if \(X_t\) and \(Y_t\) are two semimartingales, then:
All these definitions can be naturally extended to matrix valued semimartingales. Let \({\varvec{X}}_t\) be a \(p \times q\) matrix whose entries are real-valued square-integrable semimartingales. We denote by \(\langle {\varvec{M}} \rangle _t\) the matrix of entry-wise predictable quadratic variations. The predictable quadratic covariation of \({\varvec{X}}_t\) is defined with the help of the vectorization operator \(\mathrm {vec}: \mathbb {R}^{p \times q} \rightarrow \mathbb {R}^{pq}\) which stacks vertically the columns of \({\varvec{X}}\), namely if \({\varvec{X}}\in \mathbb {R}^{p \times q}\) then
We define indeed the predictable quadratic covariation matrix \(\langle \mathrm {vec}{\varvec{X}} \rangle _t\) of \({\varvec{X}}_t\) as the \(pq \times pq\) matrix with entries
for \(1 \le i, j \le pq\), namely such that \(\mathrm {vec}({\varvec{X}}_t) \mathrm {vec}({\varvec{X}}_t)^\top - \langle \mathrm {vec}{\varvec{X}} \rangle _t\) is a martingale. The matrices of quadratic variations \([{\varvec{X}}]_t\) and quadratic covariations \([\mathrm {vec}{\varvec{X}}]_t\) are defined along the same line.
Then according to Eq. (25), we have:
and
An important tool for our proofs is Itô’s lemma, that allows one to compute the stochastic differential \(d F({\varvec{M}}_t)\) where \(F : \mathbb {R}^{p \times q} \rightarrow \mathbb {R}\) is a twice differentiable function. We denote by \(\frac{d F}{d \mathrm {vec}({\varvec{X}})}\) the pq-dimensional vector such that
The second order derivative is the \(pq \times pq\) symmetric matrix given by
A direct application of the multivariate Itô Lemma ([19] Theorem 1, p. 118) writes for matrix semimartingales as follows.
Lemma A.2
(Itô’s Lemma) Let \(\{{\varvec{X}}_t\}_{t \ge 0}\) be a \(p \times q\) matrix semimartingale and \(F : \mathbb {R}^{p \times q} \rightarrow \mathbb {R}\) be a twice continuously differentiable function. Then
As an application, let us apply Lemma A.2 to the function \(F({\varvec{X}})= {{\mathrm{tr}}}\exp ({\varvec{X}})\) that acts on the set of symmetric matrices. This result will be of importance for the proof of our results.
Lemma A.3
(Itô’s Lemma for the trace exponential) Let \(\{{\varvec{X}}_t\}\) be a \(d \times d\) symmetric matrix semimartingale. The Itô formula for \(F({\varvec{X}}_t) = {{\mathrm{tr}}}\exp ({\varvec{X}}_t)\) gives
where \(\langle {\varvec{X}}_{\bullet , i}^c \rangle _t\) denotes the \(d \times d\) predictable quadratic variation of the continuous part of the i-th column \(({\varvec{X}}_t)_{\bullet , i}\) of \({\varvec{X}}_t\).
Proof
An easy computation gives
for any symmetric matrices \({\varvec{X}}\) and \({\varvec{H}}\). Note that \({{\mathrm{tr}}}(e^{\varvec{X}}{\varvec{H}}) = (\mathrm {vec}{\varvec{H}})^\top \mathrm {vec}(e^{\varvec{X}}) \), and we have from [14] Exercise 25 p. 252 that
where the Kronecker product \({\varvec{I}}\otimes e^{\varvec{X}}\) stands for the block matrix
This entails that
Hence, using Lemma A.2 with \(F({\varvec{X}}) = {{\mathrm{tr}}}e^{{\varvec{X}}}\) we obtain
Since \(\mathrm {vec}({\varvec{Y}})^\top \mathrm {vec}({\varvec{Z}}) = {{\mathrm{tr}}}({\varvec{Y}}{\varvec{Z}})\), one gets
To conclude the proof of Lemma A.3, it remains to prove that
First, let us write
where \({\varvec{E}}^{i, j}\) is the \(d \times d\) matrix with all entries equal to zero excepted for the (i, j)-entry, which is equal to one. Since
for any matrices \({\varvec{A}}, {\varvec{B}}, {\varvec{C}}, {\varvec{D}}\) with matching dimensions (see for instance [14]), we have
since \({{\mathrm{tr}}}{\varvec{E}}^{i, j} = 0\) for \(i \ne j\) and 1 otherwise. This concludes the proof of Lemma A.3. \(\square \)
1.3 Appendix A.3: Proof of Proposition 1
Define for short
Since \({\varvec{A}}_t\) and \(\langle {\varvec{Y}}_{\bullet , j}^c \rangle _t\) for \(j=1, \ldots , d\) are FV processes, then
and in particular \(\langle {\varvec{Y}}_{\bullet , j}^c \rangle = \langle {\varvec{X}}_{\bullet , j}^c \rangle \) for any \(j=1, \ldots , d\). Using Lemma A.3, one has that for all \(t_1 < t_2\):
where we used (29) together with the fact that \(\Delta {\varvec{X}}_t = \Delta {\varvec{Y}}_t\), since \({\varvec{A}}_t\) and \(\langle {\varvec{Y}}_{\bullet , j}^c \rangle _t\) are both continuous.
The Golden-Thompson’s inequality, see [3], states that \({{\mathrm{tr}}}e^{{\varvec{A}}+ {\varvec{B}}} \le {{\mathrm{tr}}}( e^{\varvec{A}}e^{\varvec{B}})\) for any symmetric matrices \({\varvec{A}}\) and \({\varvec{B}}\). Using this inequality we get
Since \({\varvec{Y}}_t\) and \({\varvec{U}}_t - {\varvec{A}}_t\) are matrix martingales, \(e^{{\varvec{X}}_{t^-}}\) is a predictable process with locally bounded entries and \(L_t \ge 0\), the r.h.s of the last equation corresponds to the variation between \(t_1\) and \(t_2\) of a non-negative local martingale, i.e., of a supermartingale. It results that \(\mathbbm {E}[L_{t_2} - L_{t_1}| \mathscr {F}_{t_1}] \le 0\), which proves that \(L_t\) is also a supermartingale. Using this last inequality with \(t_1 = 0\) and \(t_2 = t\) gives \(\mathbbm {E}[L_{t}] \le d\). This concludes the proof of Proposition 1.
1.4 Appendix A.4: Bounding the odd powers of the dilation operator
The process \(\{ {\varvec{Z}}_t \}\) is not symmetric, hence following [34], we will force symmetry in our proofs by extending it in larger dimensions, using the symmetric dilation operator [27] given, for a matrix \({\varvec{X}}\), by
The following Lemma will prove useful:
Lemma A.4
Let \({\varvec{X}}\) be some \(n \times m\) matrix and \(k \in \mathbb {N}\). Then
Proof
The first equality results from a simple algebra. It can be rewritten as:
where
Since \(({\varvec{X}}{\varvec{X}}^\top )^{k} \succcurlyeq \varvec{0} \) and
we obtain that \({\varvec{A}}\otimes ({\varvec{X}}{\varvec{X}}^\top )^{k} \succcurlyeq \varvec{0}\), since the eigenvalues of a Kronecker product \({\varvec{A}}\otimes {\varvec{B}}\) are given by the products of the eigenvalues of \({\varvec{A}}\) and \({\varvec{B}}\), see [10]. This leads to:
Using the fact that [28]
for any real matrices \({\varvec{A}}, {\varvec{B}}, {\varvec{C}}\) (with compatible dimensions), we have:
Along the same line, one can establish that:
The square root of the product of the two inequalities provides the desired result. \(\square \)
Appendix B: Proof of Theorem 1
Let us recall the definition (30) of the dilation operator. Let us point out that \(\mathscr {S}({\varvec{X}})\) is symmetric and satisfies \(\lambda _{\max }(\mathscr {S}({\varvec{X}})) = \Vert \mathscr {S}({\varvec{X}})\Vert _{{{\mathrm{op}}}} = \Vert {\varvec{X}}\Vert _{{{\mathrm{op}}}}\). Note that \(\mathscr {S}({\varvec{Z}}_t)\) is purely discontinuous, so that \(\langle \mathscr {S}({\varvec{Z}})_{\bullet , j}^c \rangle _t = {\varvec{0}}\) for any j. Recall that we work on events \(\{ \lambda _{\max }({\varvec{V}}_t) \le v \}\) and \(\{ b_t \le b\}\).
We want to apply Proposition 1 (see “Appendix A” above) to \({\varvec{Y}}_t = \xi \mathscr {S}({\varvec{Z}}_t) / b\). In order to do so, we need the following Proposition.
Proposition B.1
Let the matrix \({\varvec{W}}_t\) be the matrix defined in Eq. (7). Let any \(\xi \ge 0\) be fixed and consider \(\phi (x) = e^x - x - 1\) for \(x \in \mathbb {R}\). Assume that
for any \(1 \le i, j \le m + n\) and grant also Assumption 1 from Sect. 2.3. Then, the process
admits a predictable, continuous and FV compensator \({\varvec{\Lambda }}_t\) given by Eq. (39) below. Moreover, the following upper bound for the semi-definite order
is satisfied for any \(t > 0\).
This proposition is proved in “Appendix C” below. We use Proposition 1, Eq. (36) and Eq. (23) together with (9) to obtain
for any \(\xi \in [0, 3]\). Using this with Lemma A.1 entails
Note that on \(\{ b_t \le b \}\) we have \(J_{\max } \Vert {\varvec{C}}_s\Vert _\infty \max (\Vert \mathbb {T}_s\Vert _{{{\mathrm{op}}}; \infty },\Vert \mathbb {T}_s^\top \Vert _{{{\mathrm{op}}}; \infty })b^{-1} \le 1\) for any \(s \in [0, t]\). The following facts on the function \(\phi (x)\) hold true (cf. [12, 22]):
-
(i)
\(\phi (x h)\le h^2 \phi (x)\) for any \(h \in [0,1]\) and \(x > 0\)
-
(ii)
\(\phi (\xi ) \le \frac{\xi ^2}{2(1 - \xi / 3)}\) for any \(\xi \in (0, 3)\)
-
(iii)
\(\min _{\xi \in (0, 1/c)} \big ( \frac{a \xi }{1 - c \xi } + \frac{x}{\xi }\big ) = 2 \sqrt{ax} + c x\) for any \(a, c, x > 0\).
Using successively (i) and (ii), one gets, on \(\{ b_t \le b \} \cap \{ \lambda _{\max }({\varvec{V}}_t) \le v \}\), that for \(\xi \in (0,3)\):
where we recall that \({\varvec{V}}_t\) is given by (10). This gives
for any \( \xi \in (0,3)\). Now, by optimizing over \(\xi \) using (iii) (with \(a = v/2b^2\) and \(c = 1/3\)), one obtains
Since \(\lambda _{\max }(\mathscr {S}({\varvec{Z}}_t)) = \Vert \mathscr {S}({\varvec{Z}}_t)\Vert _{{{\mathrm{op}}}}\), this concludes the proof of Theorem 1 when the variance term is expressed using Eq. (10). It only remains to prove the fact that
Since \({\varvec{W}}_s\) is block-diagonal, we have obviously:
From the definition of \({\varvec{Z}}_t\), since the entries of \(\Delta {\varvec{M}}_t\) do not jump at the same time, the predictable quadratic covariation of \(({\varvec{Z}}_t)_{k,j}\) and \(({\varvec{Z}}_t)_{l,j}\) is simply the predictable compensator of \(\sum _{a,b} \sum _{s \le t} (\mathbb {T}_s)_{k,j;a,b} (\mathbb {T}_s)_{l,j} ({\varvec{C}}_s)^2_{a,b} ({\varvec{J}}_{N_s})^2_{a,b} (\Delta {\varvec{N}}_s)_{a,b}\). It results that
An analogous computation for \(\langle {\varvec{Z}}_{j,\bullet } \rangle _t\) leads to the expected result, and concludes the proof of Theorem 1. \(\square \)
Appendix C: Proof of Proposition B.1
Let us first remark that:
Then, from the definition of \({\varvec{U}}_t\) in Eq. (35), we have:
Since \((\Delta {\varvec{Z}}_s (\Delta {\varvec{Z}}_s^\top \Delta {\varvec{Z}}_s)^k)^\top = \Delta {\varvec{Z}}_s^\top (\Delta {\varvec{Z}}_s \Delta {\varvec{Z}}_s^\top )^k\), we need to compute three terms: \((\Delta {\varvec{Z}}_s \Delta {\varvec{Z}}_s^\top )^k\), \((\Delta {\varvec{Z}}_s^\top \Delta {\varvec{Z}}_s)^k\) and \(\Delta {\varvec{Z}}_s^\top (\Delta {\varvec{Z}}_s \Delta {\varvec{Z}}_s^\top )^k\).
From Assumption 1, one has, a.s. that the entries of \({\varvec{M}}_t\) cannot jump at the same time, hence
a.s. for any t, \(m \ge 2\) and any indexes \(i_k \in \{ 1, \ldots , p \}\) and \(j_k \in \{ 1, \ldots , q\}\). This entails, with the definition (1) of \(\Delta {\varvec{Z}}_s\), that \((\Delta {\varvec{Z}}_s \Delta {\varvec{Z}}_s^\top )^k\) is given, a.s., by
Let us remark that Eq. (34) entails
for any i, j, so that together with Assumption 1, it is easily seen that the compensator of
is a.s. given by
Following the same arguments as for (38), we obtain that the compensator of
is a.s. given by
Along the same line, one can easily show that the compensator of
reads a.s.:
Finally, we can write, a.s., the compensator of \({\varvec{U}}_t\) as
where
with
One can now directly use Lemma A.4 with \({\varvec{X}}= (\mathbb {T}_s)_{\bullet ;a,b} \mathbbm {E}[({\varvec{J}}_1)_{a,b}^{2k+1}]^{1/(2k+1)} ({\varvec{C}}_s)_{a,b}\) to obtain:
where we used the fact that \(|({\varvec{J}}_1)_{i, j}| \le J_{\max }\) a.s. for any i, j under Assumption 1. Given the fact that
for any a, b, where we used the notations and definitions from Sect. 2.1, we have:
where we recall that \(\phi (x) = e^x - 1 - x\). Hence, we finally get
where \({\varvec{W}}_t\) is given by (7). This concludes the proof of Proposition B.1. \(\square \)
Appendix D: Proof of Theorem 2
The proof follows the same lines as the proof of Theorem 1. We consider as before the symmetric dilation \(\mathscr {S}({\varvec{Z}}_t)\) of \({\varvec{Z}}_t\) (see Eq. (30)) and apply Proposition 1 with \({\varvec{Y}}_t = \xi \mathscr {S}({\varvec{Z}}_t)\) and \(d = m + n\). Since \({\varvec{Z}}_t\) is a continuous martingale, we have \({\varvec{U}}_t = {\varvec{0}}\) (cf. (3)), so that \(\langle {\varvec{U}} \rangle _t = {\varvec{0}}\) and we have \(\langle {\varvec{Z}}^c \rangle _t = \langle {\varvec{Z}} \rangle _t\). So, Proposition 1 gives
From the definition of the dilation operator \(\mathscr {S}\), it can be directly shown that:
where \(\langle {\varvec{Z}}_{\bullet ,j} \rangle _t\) (resp. \(\langle {\varvec{Z}}_{\bullet ,j} \rangle _t\)) is the \(m \times m\) (resp. \(n \times n\)) matrix of the quadratic variation of the j-th column (resp. row) of \({\varvec{Z}}_t\). Since \([{\varvec{M}}^\mathrm {con}]_t = \langle {\varvec{M}}^\mathrm {con} \rangle _t = t {\varvec{I}}\), we have (for the sake of clarity, we omit the subscript t in the matrices):
which gives in a matrix form
One can easily prove in the same way that
Thus,
where \({\varvec{V}}_t\) is given by (13). From (41), it results
Then, using Lemma A.1, one gets
On the event \(\{\lambda _{\max }( {\varvec{V}}_t ) \le v\}\), one gets
Optimizing on \(\xi \), we apply this last result for \(\xi = \sqrt{2x / v}\) and get
Since \( \lambda _{\max }(\mathscr {S}({\varvec{Z}}_t)) = \Vert \mathscr {S}({\varvec{Z}}_t)\Vert _{{{\mathrm{op}}}}\), this concludes the proof of Theorem 2. \(\square \)
Rights and permissions
About this article
Cite this article
Bacry, E., Gaïffas, S. & Muzy, JF. Concentration inequalities for matrix martingales in continuous time. Probab. Theory Relat. Fields 170, 525–553 (2018). https://doi.org/10.1007/s00440-017-0786-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-017-0786-9
Mathematics Subject Classification
- 60B20
- 60G44
- 60H05
- 60G48