Martingale Inequalities Research Articles

Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing

In this work, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength $n$ when the transmission rates approach the channel capacity at a rate slower than $1/\sqrt{n}$, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality.

Modular inequalities in martingale Orlicz–Karamata spaces

AbstractIn this paper, some new martingale inequalities in the framework of Orlicz‐Karamata spaces are provided. More precisely, we establish modular martingale inequalities associated with concave functions and slowly varying functions.

Asymptotic Approximation of Optimal Portfolio for Small Time Horizons

We consider the problem of portfolio optimization in a simple incomplete market and under a general utility function. By working with the associated Hamilton--Jacobi--Bellman partial differential equation (HJB PDE), we obtain a closed-form formula for a trading strategy which approximates the optimal trading strategy when the time horizon is small. This strategy is generated by a first order approximation to the value function. The approximate value function is obtained by constructing classical sub- and super-solutions to the HJB PDE using a formal expansion in powers of horizon time. Martingale inequalities are used to sandwich the true value function between the constructed sub- and super-solutions. A rigorous proof of the accuracy of the approximation formulas is given. We end with a heuristic scheme for extending our small-time approximating formulas to approximating formulas in a finite time horizon.

TARGETED SEQUENTIAL DESIGN FOR TARGETED LEARNING INFERENCE OF THE OPTIMAL TREATMENT RULE AND ITS MEAN REWARD.

This article studies the targeted sequential inference of an optimal treatment rule (TR) and its mean reward in the non-exceptional case, i.e., assuming that there is no stratum of the baseline covariates where treatment is neither beneficial nor harmful, and under a companion margin assumption. Our pivotal estimator, whose definition hinges on the targeted minimum loss estimation (TMLE) principle, actually infers the mean reward under the current estimate of the optimal TR. This data-adaptive statistical parameter is worthy of interest on its own. Our main result is a central limit theorem which enables the construction of confidence intervals on both mean rewards under the current estimate of the optimal TR and under the optimal TR itself. The asymptotic variance of the estimator takes the form of the variance of an efficient influence curve at a limiting distribution, allowing to discuss the efficiency of inference. As a by product, we also derive confidence intervals on two cumulated pseudo-regrets, a key notion in the study of bandits problems. A simulation study illustrates the procedure. One of the corner-stones of the theoretical study is a new maximal inequality for martingales with respect to the uniform entropy integral.

Burkholder’s inequalities associated with Orlicz functions in rearrangement invariant spaces

In this paper, we prove martingale inequalities associated with Orlicz functions in the framework of rearrangement invariant spaces. More precisely, let $$\Phi $$ be an Orlicz function and let X be a rearrangement invariant spaces. We establish the new moment Burkholder inequalities when the Simonenko indices of $$\Phi $$ and the Boyd indices of X satisfy some natural conditions. Our approach mainly relies on a new distribution estimate of the Davis type decomposition.

Near-Optimal and Practical Jamming-Resistant Energy-Efficient Cognitive Radio Communications

This paper studies the problem of jamming-resistant spectrum aggregation and access (SAA) for energy-efficiency (EE) cognitive radio communications. We consider various jamming behaviors, where jammers may attack all available channels with arbitrarily changing strategies over time, attack a subset of the channels at certain time slots, or have different intelligence, i.e., oblivious or adaptive adversary, and so on. Without any priori knowledge about the channels and jammers, it is very challenging to design an efficient and practical jamming-resistant SAA algorithm to reach the optimal EE goal. In this paper, we utilize the advanced martingale concentration inequalities in an multi-armed bandits-based online learning framework to facilitate the optimal detection of various jamming behaviors. We first define a novel EE model for discontiguous orthogonal frequency division multiplexing to facilitate scalable SAA over distributed spectrum pools in practice. Then, the jamming-resistant dynamic channel access problem is formulated as a regret minimization problem. Meanwhile, an online stochastic gradient descent with bandit feedback procedure is adopted to allocate the transmit power. The proposed algorithm can autonomously detect the environmental features and find a near-optimal solution in each attacking scenario. Our algorithm is implemented with low complexity and with multiple users under some practical jamming scenarios. Extensive numerical studies show that under some practical jamming scenarios, our algorithm has an EE improvement of 45.3% over a fixed learning period, and an improvement of 82.5% in terms of learning duration compared with existing approaches.

Noncommutative Burkholder/Rosenthal inequalities associated with convex functions

Nous prouvons des inégalités de martingales non commutatives associées à des fonctions convexes. Plus précisément, nous obtenons des analogues des inégalités de Burkholder non commutatives et des inégalités de Rosenthal non commutatives pour des $\Phi$-moments associés à toute fonction convexe $\Phi$ dont les indices de Matuzewska–Orlicz $p_{\Phi}$ et $q_{\Phi}$ vérifient $1<p_{\Phi}\leq q_{\Phi}<2$ ou $2<p_{\Phi}\leqq_{\Phi}<\infty$. Ces résultats généralisent les inégalités de Burkholder/Rosenthal non commutatives obtenues par Junge et Xu. L’ingrédient clé de notre approche est une version simultanée de l’inégalité de Burkholder récemment demontrée dans le cas des espaces $L_{p}$ non commutatifs pour $1<p<2$.

Martingale weak Orlicz–Karamata–Hardy spaces associated with concave functions

In this paper, we introduce the weak Orlicz–Karamata–Hardy spaces associated with concave functions and establish the atomic decompositions. As applications, we prove several martingale inequalities and obtain a duality theorem.

Concentration inequalities for matrix martingales in continuous time

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.

Martingale inequalities of type Dzhaparidze and van Zanten

ABSTRACTFreedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by , Dzhaparidze and van Zanten [On Bernstein-type inequalities for martingales. Stoch Process Appl. 2001;93:109–117] have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that can be refined to . Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason's inequality [Tightened exponential bounds for discrete-time conditionally symmetric martingales with bounded jumps. Statist Probab Lett. 2013;83:1928–1936] to martingales with possibly unbounded differences and establish the connection between Sason's inequality and De la Peña's inequality [A general class of exponential inequalities for martingales and ratios. Ann Probab. 1999;27(1):537–564]. An application to self-normalized deviations is given.

Independent sums of [formula omitted] and [formula omitted

We construct a new idempotent Fourier multiplier on the Hardy space on the bidisc, which could not be obtained by applying known one dimensional results. The main tool is a new L1 equivalent of the Stein martingale inequality which holds for a special filtration of periodic subsets of T with some restrictions on the functions involved. We also identify the isomorphic type of the range of the associated operator as the independent sum of dyadic Hn1, which is known to be a complemented and invariant subspace of dyadic H1.

Optimal Skorokhod embedding given full marginals and Azéma–Yor peacocks

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock (“process increasing in convex order,” by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1–44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509–536] peacock under their “increasing mean residual value” condition. We also discuss the associated martingale inequality.

An elementary proof of the A 2 bound

A martingale transform $ T$, applied to an integrable locally supported function $ f$, is pointwise dominated by a positive sparse operator applied to $ \lvert f\rvert $, the choice of sparse operator being a function of $ T$ and $ f$. As a corollary, one derives the sharp $ A_p$ bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-$L^1$ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the $ A_2$ bounds in that setting.

Hardy–Lorentz spaces for B-valued martingales

In this paper we study the Hardy–Lorentz spaces for Banach space valued martingales. The dual spaces are characterized and several martingale inequalities are established. The proofs mainly depend on the classical tool of atomic decompositions. As usual, these conclusions are closely related to the geometrical properties of the underlying Banach spaces.

Parameter estimation for partially observed nonlinear stochastic system

This paper is concerned with the problem of parameter estimation for a partially observed linear stochastic system. The state estimator is obtained by using the continuous-time Kalman linear filtering theory. The likelihood function is given based on the innovation theorem and Girsanov theorem, the parameter estimator and error of estimation are derived. The strong consistency of the parameter estimator and the asymptotic normality of the error of estimation are proved by applying ergodic theorem, maximal inequality for martingale, Borel-Cantelli lemma and the central limit theorem for stochastic integrals.

Weighted weak-type inequality for martingales

Weighted weak-type inequality for martingales

Quasi sure exponential stabilization of nonlinear systems via intermittent &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ G $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Brownian motion

This paper focuses on the quasi sure exponential stabilization of nonlinear systems. By virtue of exponential martingale inequality under \begin{document}$ G $\end{document} -framework and intermittent \begin{document}$ G $\end{document} -Brownian motion (in short, \begin{document}$ G $\end{document} -ISSs), we establish the sufficient conditions to guarantee quasi surely exponential stability. The efficiency of the proposed results is illustrated by the memristor-based Chua's oscillator.

Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps

The main aim of this article is to examine almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. On the one hand, some sufficient criteria ensure almost sure stabilization of the unstable deterministic system by applying exponential martingale inequality with jumps. On the other hand, sufficient conditions of destabilization are provided under which the system is stable by the well-known strong law of large numbers of local martingale and Poisson process. Both the sample Lyapunov exponents are closely related to the control period \begin{document}$ T $\end{document} and noise width \begin{document}$ \theta $\end{document} . As for applications, the well-known Lorenz chaotic systems and nonlinear Lienard equation with jumps are discussed. Finally, two simulation examples demonstrating the effectiveness of the results are provided.

Atomic decompositions of vector-valued weak Musielak–Orlicz martingale spaces and applications

In this paper, some vector-valued weak martingale Hardy spaces of Musielak–Orlicz type are introduced, and several atomic decompositions are established for them. As applications of the atomic decompositions, a sufficient condition for sublinear operators defined on vector-valued weak Musielak–Orlicz martingale spaces to be bounded is given. Using the sufficient condition, some martingale inequalities are deduced. These results closely depend on the geometrical properties of the Banach space in which the martingales take values. The results obtained here generalize the corresponding known results of scalar-valued and vector-valued weak martingale Hardy spaces.

Deviation inequalities for martingales with applications

Using changes of probability measure developed by Grama and Haeusler (2000) [18], we extend the deviation inequalities of Lanzinger and Stadtmüller (2000) [26] and Fuk and Nagaev (1971) [15] to the case of martingales. Our inequalities recover the best possible decaying rate in the independent case. In particular, these inequalities improve the results of Lesigne and Volný (2001) [27] under a stronger condition that the martingale differences have bounded conditional moments. Applications to linear regressions with martingale difference innovations, weak invariance principles for martingales and self-normalized deviations are provided. In particular, we establish a type of self-normalized deviation bounds for parameter estimation of linear regressions. Such type bounds have the advantage that they do not depend on the distribution of the regression random variables.