We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.
Martingale representation theorem for set-valued martingales was proposed by M. Kisielewicz [J. Math. Anal. Appl. 2014]. We shall prove that the result holds only for very special case: the set-valued martingale degenerates to the point-valued one. A revised representation theorem for a special kind of non-degenerate set-valued martingales is presented.
In this paper, we will study concentration inequalities for Banach space-valued martingales. Firstly, we prove that a Banach space $X$ is linearly isomorphic to a $p$-uniformly smooth space ($1<p\leq 2$) if and only if an Azuma-type inequality holds for $X$-valued martingales. This can be viewed as a generalization of Pinelis' work on Azuma inequality for martingales with values in $2$-uniformly smooth space. Secondly, Azuma-type inequality for self-normalized sums will be presented. Finally, some further inequalities for Banach space-valued martingales, such as moment inequalities for double …
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of normalized maxima of partial sums.
In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued Lévy process (which is not required to satisfy any moment condition), with respect to a cylindrical Lévy processe…
We prove maximal inequalities for $L_q$-valued martingales obtained by stochastic integration with respect to compensated random measures. A version of these estimates for integrals with respect to compensated Poisson random measures were first obtained by Dirksen (arXiv:1208:3885) using arguments based on inequalities for sums of independent Banach-space-valued random variables, geometric properties of Banach spaces, and decoupling inequalities. Our proofs are completely different and rely almost exclusively on classical stochastic analysis for real semimartingales.
This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $ϕ$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapień, McConnell, and Woyczyński. The vast majority of the assertions presented…
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (2001) and Fan et al. (2012). In the multidimensional ca…
Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437--456]. In these frameworks, a collection of bounded $X$-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinate-wise order. Moreover, a necessary and sufficient condition is presented for the collection of $\mathcal{E}$-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded $\mathcal{E}$-martingales is a…
In this paper we consider (probability-)measure valued processes, which we call MVMs, which have a natural martingale structure. Following previous work of Eldan and Cox-Källblad, these processes are known to have a close connection to the solutions to the Skorokhod Embedding Problem. In this paper, we consider properties of these processes, and in particular, we are able to show that the MVMs connected to the Bass and Root embeddings have natural measure-valued analogues which also possess natural optimality properties. We also introduce a new MVM which is a generalisation of both the Bass an…
Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [21]. We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck, Jourdain, Margheriti, and Pammer [5], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w…
We show that the canonical decomposition (comprising both the Meyer-Yoeurp and the Yoeurp decompositions) of a general $X$-valued local martingale is possible if and only if $X$ has the UMD property. More precisely, $X$ is a UMD Banach space if and only if for any $X$-valued local martingale $M$ there exist a continuous local martingale $M^c$, a purely discontinuous quasi-left continuous local martingale $M^q$, and a purely discontinuous local martingale $M^a$ with accessible jumps such that $M = M^c + M^q + M^a$. The corresponding weak $L^1$-estimates are provided. Important tools used in the…
We prove sharp maximal inequalities for $L^q$-valued stochastic integrals with respect to any Hilbert space-valued local martingale. Our proof relies on new Burkholder-Rosenthal type inequalities for martingales taking values in an $L^q$-space.
We provide an analogue of Gundy's decomposition for L1-bounded non-commutative martingales. An important difference from the classical case is that for any L1-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1,1) boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence (x_n) in a normed space X is called…
Let (f_n) be a mean zero vector valued martingale sequence. Then there exist vector valued functions (d_n) from [0,1]^n such that int_0^1 d_n(x_1,...,x_n) dx_n = 0 for almost all x_1,...,x_{n-1}, and such that the law of (f_n) is the same as the law of (sum_{k=1}^n d_k(x_1,...,x_k)) . Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales. This paper may be found at http://math.missouri.edu/~stephen/preprints
In this paper we consider Meyer-Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$, any $X$-valued $L^p$-martingale $M$ has a unique decomposition $M = M^d + M^c$ such that $M^d$ is a purely discontinuous martingale, $M^c$ is a continuous martingale, $M^c_0=0$ and \[ \mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_{p,X} \mathbb E \|M_{\infty}\|^p. \] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-l…
Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given.
The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with t…
This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of $S=\sum_{i=1}^n X_i$ given some $σ$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0 $ as the vector $S$. Assuming that the conditional variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, an…
In this paper non-asymptotic exponential and moment estimates are derived for tail of distribution for discrete time martingale and martingale transform by means of martingale differences in the terms of moments and tails of distributions of summands and multipliers. We show also the exactness of obtained estimations and consider some applications in the theory of limit theorem for Banach space valued martingales.
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale …
In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given.
Let $X$ be a $\mathbb{C}$-valued random variable with the property that $$X \ \text{ has the same law as }\ \sum_{j\ge1} T_j X_j$$ where $X_j$ are i.i.d.\ copies of $X$, which are independent of the (given) $\mathbb{C}$-valued random variables $ (T_j)_{j\ge1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins' martingale with complex parameter.
A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + σ(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $σ:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and $\mathcal{L}$ the $L^2$-generator of a Lévy process. Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large $t$) at most a precise exponential rate for the $\mathcal{L}$; and grows in time at most a precise…
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustra…
In this paper, we study the connections between three concepts - the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo (bounded mean oscillation) coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these two equivalent conditions are satisfied. Finally, we apply our results on l…
In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of Hukuhara difference between sets, in order to compensate the lack of "inverse" operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental i…
In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).
Consider a real-valued branching random walk in the boundary case. Using the techniques developed by Aïdékon and Shi [5], we give two integral tests which describe respectively the lower limits for the minimal position and the upper limits for the associated additive martingale.
For a Hilbert space valued martingale $(f_n)$ and an adapted sequence of positive random variables $(w_n)$, we show the weighted Davis type inequality \[ \mathbb{E} \Bigl( |f_0| w_0 + \frac{1}{4} \sum_{n=1}^{N} \frac{|df_n|^2}{f^*_n} w_n \Bigr) \leq \mathbb{E} ( f^*_N w^*_N). \] This inequality is sharp and implies several results about the martingale square function. We also obtain a variant of this inequality for martingales with values in uniformly convex Banach spaces.
Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $σ$-field of the invariant events by all reflec…
University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）
We study an equivalence of (i) deterministic pathwise statements appearing in the online learning literature (termed \emph{regret bounds}), (ii) high-probability tail bounds for the supremum of a collection of martingales (of a specific form arising from uniform laws of large numbers for martingales), and (iii) in-expectation bounds for the supremum. By virtue of the equivalence, we prove exponential tail bounds for norms of Banach space valued martingales via deterministic regret bounds for the online mirror descent algorithm with an adaptive step size. We extend these results beyond the line…
Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration of measure on product spaces for separately Lipschitz functions are presented, including ones concerning the central moments of the norm of the sums of independent random vectors in any separable Banach space.
This letter derives some new exponential bounds for discrete time, real valued, conditionally symmetric martingales with bounded jumps. The new bounds are extended to conditionally symmetric sub/ supermartingales, and they are compared to some existing bounds.
It is well-known in quantum information theory that a positive operator valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalised POVM, namely a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator valued function which is measurable with respect to a quantum probability. In the present work, we study quantum random variables…
A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in the 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of the sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on the infinite-dimensional martingales, most of the results seem to be new even for the one-dimensional one…
In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.
For any two real-valued continuous-path martingales $X=\{X_t\}_{t\geq 0}$ and $Y=\{Y_t\}_{t\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of the form $aX+bY$ with $a, b$ real numbers. The best $L^p$ constant is equal to the norm of the operator $aI+bH$ from $L^p$ to $L^p$, where $H$ is the Hilbert transform on the circle or real line. The values of these norms were found by Hollenbeck, Kalton and Verbitsky \cite{HKV}.
For a set-valued stochastic sequence $(G_n)_{n=0}^N$ with relatively open convex values $G_n(ω)$ we give a criterion for the existence of an adapted sequence $(x_n)_{n=0}^N$ of selectors, admitting an equivalent martingale measure. Mentioned criterion is expressed in terms of supports of the regular conditional upper distributions of the elements $G_n$. This result is a refinement of the main result of author's previous paper (Teor. Veroyatnost. i Primen., 2005, 50:3, 480--500), where the sets $G_n(ω)$ were assumed to be open and where were asked if the openness condition can be relaxed.
Freedman's inequality is a martingale counterpart to Bernstein's inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman's inequality that provides tail bounds for the maximum singular value of a matrix-valued martingale. This note describes a different proof of the matrix Freedman inequality that depends on a deep theorem of Lieb from matrix analysis. This argument delivers sharp constan…
In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{$\star$} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E γ([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ γ([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general U…
