On martingales which are finite sums of independent random variables with time dependent coefficients
On Multi-Dimensional Random Walk Models Approximating Symmetric Space-Fractional Diffusion Processes
Mathematics Subject Classification: 26A33, 47B06, 47G30, 60G50, 60G52, 60G60.In this paper the multi-dimensional analog of the Gillis-Weiss random walk model is studied. The convergence of this random walk to a fractional diffusion process governed by a symmetric operator defined as a hypersingular integral or the inverse of the Riesz potential in the sense of distributions is proved.* Supported by German Academic Exchange Service (DAAD).
On Nonuniform Gaussian Approximation for Random Summation.
On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients.
On random walks with continuous components.
On Schur-convexity of expectation of weighted sum of random variables with applications.
On small deviations of Gaussian processes using majorizing measures
We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences and...
On some general distributions in terms of generalized functions
On some generalized reinforced random walk on integers.
On strictly sparse subsets of a free group.
On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums.
On sums of i.i.d. random variables indexed by N parameters
On sums of independent random variables without power moments.
On the accuracy of approximation of the distribution of negative-binomial random sums by the gamma distribution
The main goal of this paper is to study the accuracy of approximation for the distributions of negative-binomial random sums of independent, identically distributed random variables by the gamma distribution.
On the Asymptotic Behaviour of Convolution Powers of Probabilities on Discrete Groups.
On the asymptotic equidistribution of sums of independent identically distributed random variables
On the Bennett–Hoeffding inequality
The well-known Bennett–Hoeffding bound for sums of independent random variables is refined, by taking into account positive-part third moments, and at that significantly improved by using, instead of the class of all increasing exponential functions, a much larger class of generalized moment functions. The resulting bounds have certain optimality properties. The results can be extended in a standard manner to (the maximal functions of) (super)martingales. The proof of the main result relies on an...
On the convergence of moments in the CLT for triangular arrays with an application to random polynomials
We give a proof of convergence of moments in the Central Limit Theorem (under the Lyapunov-Lindeberg condition) for triangular arrays, yielding a new estimate of the speed of convergence expressed in terms of νth moments. We also give an application to the convergence in the mean of the pth moments of certain random trigonometric polynomials built from triangular arrays of independent random variables, thereby extending some recent work of Borwein and Lockhart.
On the coupling property of Lévy processes
We give necessary and sufficient conditions guaranteeing that the coupling for Lévy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall–Rogers on couplings of random walks. In particular, we obtain that a Lévy process admits a successful coupling, if it is a strong Feller process or if the Lévy (jump) measure has an absolutely continuous component.
On the Dirichlet Problem at Infinity for Manifolds of Nonpositive Curvature.