Related papers: On a formulation of the central limit theorem

A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…

We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is)…

A central limit theorem for arrays of symmetric row-wise exchangeable random variables is presented. The result is valid for finite and infinite extendable and non-extendable sequences. Unlike most reported versions of the central limit…

The Central Limit Theorem states that, in the limit of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to a stable distribution. The…

Since the appearance of H. Robbins article (1948), the central limit theorems for random sums have been studied for about 70 years. The central limit theorems for random sums of independent random variables play a very important role in…

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each,…

General Central limit theorem deals with weak limits (in type) of sums of row-elements of array random variables. In some situations as in the invariance principle problem, the sums may include only parts of the row-elements. For strictly…

It is well-known that the central limit theorem holds for partial sums of a stationary sequence $(X_i)$ of $m$-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if…

The Rotar central limit theorem is a remarkable theorem in the non-classical version since it does not use the condition of asymptotic infinitesimality for the independent individual summands, unlike the theorems named Lindeberg's and…

We derive new bounds of the remainder in a combinatorial central limit theorem without assumptions on independence and existence of moments of summands. For independent random variables our theorems imply Esseen and Berry-Esseen type…

A Central Limit Theorem is proved for linear random fields when sums are taken over finite disjoint union of rectangles. The approach does not rely upon the use of Beveridge Nelson decomposition and the conditions needed are similar to…

We consider a borderline case: the central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the iid case a well-known sufficient condition for this central limit theorem is regular…

This paper addresses the following classical question: giving a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process satisfy the central limit theorem? We study…

A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to the "shrinking operators" $U_r(x):=[\max\{|x|-r,0\}]\cdot x/|x|,\ r \ge 0$. For…

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and…

We give a simple and general central limit theorem for a triangular array of m-dependent variables. The result requires only a Lindeberg condition and avoids unnecessary extra conditions that have been used earlier. The result applies also…

We consider two classical ensembles of the random matrix theory: the Wigner matrices and sample covariance matrices, and prove Central Limit Theorem for linear eigenvalue statistics under rather weak (comparing with results known before)…

We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem,…

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while only imposing minimal moment conditions. Within this result, we address the case…

We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $\xi_j$, perturbed by predictable multiplicative factors $\lambda_j$ with values in intervals $[\underline\lambda_j,\overline\lambda_j]$. It…