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We provide an extension of Feller's upper-lower class test for the Hartman-Wintner LIL to the LIL in Euclidean space. We obtain this result as a corollary to a general upper-lower class test for $\Gamma_n T_n$ where $T_n=\sum_{j=1}^n Z_j$…
Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional…
This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ {\it versus} the at most one change in the mean alternative hypothesis. The…
This paper proves several weak limit theorems for the joint version of extreme order statistics and partial sums of independently and identically distributed random variables. The results are also extended to almost sure limit version.
The paper is devoted to the investigation of Esscher's transform on high dimensional Euclidean spaces in the light of its application to the central limit theorem. With this tool, we explore necessary and sufficient conditions of normal…
We generalize Lindeberg's proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions…
We obtain the analogue of the classical result by Erd\"os and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central…
The classical theorem of Erd\H os \& Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under…
We prove entropic and total variation versions of the Erd\H{o}s-Kac limit theorem for the maximum of the partial sums of i.i.d. random variables with densities.
The Generalized Central Limit Theorem is a remarkable generalization of the Central Limit Theorem, showing that the sum of a large number of independent, identically-distributed (i.i.d) random variables with infinite variance may converge…
It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time.…
We give a new, self-contained proof of the multidimensional central limit theorem using the technique of ``doubling variables," which is traditionally used to prove uniqueness of solutions of partial differential equations (PDEs). Our…
We prove a general theorem to bound the total variation distance between the distribution of an integer valued random variable of interest and an appropriate discretized normal distribution. We apply the theorem to 2-runs in a sequence of…
Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend…
We generalise the Erdos-Renyi limit theorem on the maximum of the partial sums of random variables to the case when the number of terms in these sums is randomly distributed. Certain relations between the limiting theorems of this type and…
The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These…
The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${\mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and…
Based on deleting-item central limit theory, the classical Donsker's theorem of partial-sum process of independent and identically distributed (i.i.d.) random variables is extended to incomplete partial-sum process. The incomplete…
We observe a realization of a stationary generalized weighted Voronoi tessellation of the d-dimensional Euclidean space within a bounded observation window. Given a geometric characteristic of the typical cell, we use the minus-sampling…
We consider Canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of R 2 . We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global…