Strong limit theorems in the multi-color generalized allocation scheme

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.Comment: 11 page

On the number of empty cells in the allocation scheme of indistinguishable particles

The allocation scheme of $n$ indistinguishable particles into $N$ different cells is studied. Let the random variable $\mu_0(n,K,N)$ be the number of empty cells among the first $K$ cells. Let $p=\frac{n}{n+N}$. It is proved that $\frac{\mu_0(n,K,N)-K(1-p)}{\sqrt{ K p(1-p)}}$ converges in distribution to the Gaussian distribution with expectation zero and variance one, when $n,K, N\to\infty$ such that $\frac{n}{N}\to\infty$ and $\frac{n}{NK}\to 0$. If $n,K, N\to\infty$ so that $\frac{n}{N}\to\infty$ and $\frac{NK}{n}\to \lambda$, where 0<\lambda<\infty, then $\mu_0(n,K,N)$ converges in distribution to the Poisson distribution with parameter $\lambda$. Two applications of the results are given to mathematical statistics. First, a method  is offered to test the value of $n$. Then, an analogue of the run-test is suggested with an application in signal processing

Inequalities and limit theorems for random allocations

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

The conditional maximum of Poisson random variables

© 2017 Taylor & Francis Group, LLC 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

LIMIT THEOREMS FOR SUMS OF INDEPENDENT RANDOM ELEMENTS WITH RANDOM PARAMETERS

The investigation objects are the sums and other functions from random elements depending upon the random parameters - random processes observed in the random time moments and also the sums of the independent random elements with random coeffieients. The limit theorems for sums and other functions from the random values depending upon the random parameters - observations of the random process in the random moments have been obtained at sufficient-general suppositions. Their generalizations for the Banach-valued random elements have been obtained. The uinctional limit theorems from such random values have been obtained. The evaluations of the convergence rate in these theorems have been found.Available from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

On the Number of Colored Balls from a Fixed Set in a Multi-Color Urn Scheme without Return

We consider random vectors that consist of numbers of colored balls belonging to a fixed value set in a multi-color urn scheme without return.We proved, that under certain conditions, random vectors, consisting of centered and normed elements of these vectors, converge in distribution to a random vector made up of independent Gaussian random variables with means 0 and variances of 1. We also obtained limit theorems for the functions of these random vectors.Applications of these theorems to estimate probabilities of type I errors of the $\chi^2$-test and to estimate probabilities of type I errors and type II errors of some statistical tests are given

Almost sure limit theorems for the Pearson statistic

Almost sure versions of limit theorems by Kruglov for the Pearson χ 2-statistic are obtained. Key words and phrases: almost sure limit theorem, functional limit theorem, Pearson’s χ 2-statistic

On the number of empty cells on the number of empty cells in non-homogeneous allocation scheme

In non-homogeneous allocation scheme of distinguishable particles by dierent cellswe formulate the conditions under which the number of empty cells from the rst K cellsconverges to a Poisson random variable and the conditions under which the number of givenvalue cells from the rst K cells converges to a Gaussian random variable. These results weapply to study the type I error and the type II error in some analog of the empty box test

Inequalities and limit theorems for random allocations

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