Alexey Chuprunov

ALMOST SURE LIMIT THEOREMS FOR THE PEARSON STATISTIC

Almost sure versions of limit theorems by Kruglov for the Pearson

Convergence of random step lines to Ornstein-Uhlenbeck-type processes

\chi^2

An analogue of the generalised allocation scheme: limit theorems for the number of cells containing a given number of particles

-statistic are obtained.

INTEGRAL ANALOGUES OF ALMOST SURE LIMIT THEOREMS

The paper deals with random step-line processes defined by sums of independent identically distributed random variables multiplied by independent indicators. These processes describe some models in which random variables are replaced with other ones. We prove the convergence in distribution of such processes to the weighted Ornstein-Uhlenbeck process.

An exponential inequality and strong limit theorems for conditional expectations

An analogue of the generalised allocation scheme is considered. In the definition of the original scheme, the equality is replaced by an inequality. Strong laws of large numbers and local limit theorems are obtained for the number of cells containing r particles.

The conditional maximum of Poisson random variables

SummaryAn integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.

STRONG LIMIT THEOREMS IN THE MULTI-COLOUR GENERALIZED ALLOCATION SCHEME

An exponential inequality for the tail of the conditional expectation of sums of centered independent random variables is obtained. This inequality is applied to prove analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers for conditional expectations. As corollaries we obtain certain strong theorems for the generalized allocation scheme and for the nonuniformly distributed allocation scheme.

Inequalities and limit theorems for random allocations

ABSTRACT The conditional maxima of independent Poisson random variables are studied. A triangular array of row-wise independent Poisson random variables is considered. If condition is given for the row-wise sums, then the limiting distribution of the row-wise maxima is concentrated onto two points. The result is in accordance with the classical result of Anderson. The case of general power series distributions is also covered. The model studied in Theorems 2.1 and 2.2 is an analogue of the generalized allocation scheme. It can be considered as a non homogeneous generalized scheme of allocations of at most n balls into N boxes. Then the maximal value of the contents of the boxes is studied.

A central limit theorem for random fields.

The generalized allocation scheme is studied. Its extension for coloured balls is defined. Some analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers are obtained for the number of boxes containing fixed numbers of balls.

An Almost Sure Functional Limit Theorem for the Domain of Geometric Partial Attraction of Semistable Laws

Random allocations of balls into boxes are considered. Properties of the number of boxes containing a fixed number of balls are studied. A moment inequality is obtained. A merge theorem with Poissonian accompanying laws is proved. It implies an almost sure limit theorem with a mixture of Poissonian laws as limiting distribution. Almost sure versions of the central limit theorem are obtained when the parameters are in the central domain.