We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments of martingales. Several statistical applications to autoregressive and branching processes are also provided.

## Abstract

Let (*X*
_{k}) be a sequence of independent r.v.’s such that for some measurable functions *gk* : **R**
^{k} → **R** a weak limit theorem of the form

*G*. By a general result of Berkes and Csáki (“universal ASCLT”), under mild technical conditions the strong analogue

*d*

_{k}) is a logarithmic weight sequence and

*D*

_{N}= ∑

_{k=1}

^{N}

*d*

_{k}. In this paper we extend the last result for a very large class of weight sequences (

*d*

_{k}), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory, are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with, ordinary averages.

## Abstract

Let X_{1},X_{2}, ... be iid random variables, and let **a**
_{n} = (*a*
_{1},*n*, ..., *a*
_{n,n}) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination

*a*

_{1,n}

*X*

_{1}+ ... +

*a*

_{n,n}

*X*

_{n}under the natural negligibility condition lim

_{n→∞}max{|

*a*

_{k,n}|:

*k*= 1, ...,

*n*} = 0. We prove that if

**a**

_{n}, in which the components are of the same magnitude, then the common distribution belongs to

## Abstract

Convergence in Mallows distance is of particular interest when heavy-tailed distributions are considered. For 1≦*α*<2, it constitutes an alternative technique to derive central limit type theorems for non-Gaussian *α*-stable laws. In this note, we further explore the connection between Mallows distance and convergence in distribution. Conditions for their equivalence are presented.

## Abstract

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) *τ*
_{n} = *τ*
_{n}(*ω*) of a plane partition *ω* of the positive integer *n*, assuming that *ω* is chosen uniformly at random from the set of all such partitions. We prove that (*τ*
_{n} − *c*
_{0}
*n*
^{2/3})/*c*
_{1}
*n*
^{1/3} log^{1/2}
*n* converges weakly, as *n* → ∞, to the standard normal distribution, where *c*
_{0} = ζ(2)/ [2ζ(3)]^{2/3}, *c*
_{1} = √(1/3/) [2*ζ*(3)]^{1/3} and *ζ*(*s*) = Σ_{j=1}
^{∞}
*j*
^{−s}.

Let*S*
_{n} be the partial sums of ?-mixing stationary random variables and let*f(x)* be a real function. In this note we give sufficient conditions under which the logarithmic average of*f(S*
_{n}/s_{n}) converges almost surely to ?_{-8}
^{8}
*f(x)d*F*(x)*. We also obtain strong approximation for*H(n)*=?_{k=1}
^{n}
*k*
^{-1}
*f(S*
_{k}/s*k*)=log*n* ?_{-8}
^{8}
*f(x)d*F*(x)* which will imply the asymptotic normality of*H(n)*/log^{1/2}
*n*. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for ?-mixing random variables.

## Abstract

For every*k*≥1 consider the waiting time until each pattern of length*k* over a fixed alphabet of size*n* appears at least once in an infinite sequence of independent, uniformly distributed random letters. Letting*n*→∞ we determine the limiting finite dimensional joint distributions of these waiting times after suitable normalization and
provide an estimate for the rate of convergence. It will turn out that these waiting times are getting independent.

## Abstract

We show that as processes in (*c, d, t*) ∈ *C*(*R*
^{2} × *R*
_{+}
^{1})

*h*→ 0 for Brownian local time

*L*

_{t}

^{x}. Here η(

*x*) is an independent two-sided Brownian motion.

## Abstract

Based on a stochastic extension of Karamata’s theory of slowly varying functions, necessary and sufficient conditions are established for weak laws of large numbers for arbitrary linear combinations of independent and identically distributed nonnegative random variables. The class of applicable distributions, herein described, extends beyond that for sample means, but even for sample means our theory offers new results concerning the characterization of explicit norming sequences. The general form of the latter characterization for linear combinations also yields a surprising new result in the theory of slow variation.

## Abstract

Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations
of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any
one of *n* cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms
and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic
distributions for the total winnings of the *n* players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For
all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that
for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic
expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be
numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations
even for very small *n*.