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In this article, we propose space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Poisson and Skellam processes by independent \alpha-stable subordiantor and tempered stable subordiantor, respectively. Further we derive the marginal probabilities, Levy measure, governing difference-differential equations of the introduced processes. At last, we give the algorithm to simulate the sample paths of these processes.

We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an n\times nrandom symmetric matrix A We establish universality in noise for this AMP in the nlimit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.

We prove a scaling limit result for random walk on certain random planar maps with its natural time parametrization. In particular, we show that for \gamma \in (0,2) the random walk on the mated-CRT map with parameter \gammaconverges to \gammaLiouville Brownian motion, the natural quantum time parametrization of Brownian motion on a \gammaLiouville quantum gravity (LQG) surface. Our result applies if the mated-CRT map is embedded into the plane via the embedding which comes from SLE ...

We consider a 2D stochastic wave equation driven by a Gaussian noise, which is temporally white and spatially colored described by the Riesz kernel. Our first main result is the functional central limit theorem for the spatial average of the solution. And we also establish a quantitative central limit theorem for the marginal and the rate of convergence is described by the total-variation distance. A fundamental ingredient in our proofs is the pointwise L^pestimate of Malliavin derivative, wh...

We are interested in the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters. We argue that the vertices for which the eigenvector has the largest and the smallest entries, respectively, are unusually strongly connected to their own cluster and more reliably classified than the rest. This can be regarded as a discrete ...

We analyze the fluctuations of incomplete Ustatistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled and centered version of the U-statistic converges to a normal random variable. Our method of proof relies on a martingale CLT. A possible application -- a CLT for the hitting time for random walk on random graphs -- will be presented in \cite{LoTe20b}

In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges obtained in a recent work by Elad Altman and Zambotti to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.

We prove explicit L^pbounds for second order Riesz transforms of the sub-Laplacian in the Lie groups \mathbb H \mathbb{SU}(2)and \mathbb{SL}(2)

We compute the kinetic energy of the Langevin particle using different approaches. We build stochastic differential equations that describe this physical quantity based on both the Ito and Stratonovich stochastic integrals. It is shown that the Ito equation possesses a unique solution whereas the Stratonovich one possesses infinitely many, all but one absent of physical meaning. We discuss how this fact matches with the existent discussion on the Ito vs Stratonovich dilemma and the apparent pref...

We propose non-asymptotic controls of the cumulative distribution function P(|X_{t}|\ge \varepsilon) for any t>0 \varepsilon>0and any L\'evy process Xsuch that its L\'evy density is bounded from above by the density of an \alphastable type L\'evy process in a neighborhood of the origin. The results presented are non-asymptotic and optimal, they apply to a large class of L\'evy processes.

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