Volume 35, Issue 3

In a general scheme of allocation of no more than n particles to N cells we prove limit theorems for the random variable η n , N ( K ) which is the number of particles in a given set of K cells. The main result of the paper is Theorem 1. Limit distribution in this theorem depends on s=limKN. $s=\lim \frac{K}{N} .$ If 0 < s < 1, then the limit distribution is that of the minimum of independent Gaussian random variables, and if s = 1, then it is the distribution of the absolute value of a Gaussian random variable taken with the minus sign.

The effect of subgraphs gluing and cloning operations on the graph diameter is studied. A vertex-diameter graph is a graph in which all vertices belong to diametric chains. We study the possibilities of using the vertex-diameter graphs for scaling of graphs with diameter constrains. Examples of scaling of trees, fat trees, and vertex-diameter graphs via cloning and gluing operations are given. We estimate the diameter and complexity of synthesis of such graphs.

Galton-Watson forests formed by a critical branching process starting with N particles are considered. The total number of descendants of the initial particles over all time of evolution is equal to n . Assume that the number of offspring of each particle has the distribution pk=h(k+1)(k+1)τ,k=0,1,2,…,τ∈(2,3), $$p_k=\frac{h(k+1)}{(k+1)^\tau}, \quad k=0,1,2, \ldots, \quad \tau \in(2,3),$$ where the function h ( x ) is slowly varying at infinity. The limit distribution of the maximum size of a tree is found if N , n → ∞ and there exist α > 0 such that n / N τ −1+ α → ∞.

We prove that, for each natural k , any Boolean function can be implemented by a two-pole contact circuit which is k -irredundant and admits a k -diagnostic test of length at most 1 relative to similar-type connected faults of contacts in groups, where each group consists of one closing and one breaking contact.

We study large deviation probabilities for bisexual branching process in a random (i.i.d.) environment. Under several conditions on the mating function (which may depend on the environment) we introduce the associated random walk of the process. We also assume Cramer condition for the step of the walk and moment conditions on the number of descendants of one pair. Under Cramer condition for the steps of the walk and moment conditions on the number of descendants of a pair we find an exact asymptotics of probabilities P (ln N n ∈ [ x , x + Δ n )) as n → ∞ for x / n varying in some domain for all sequences Δ n , tending to zero sufficiently slowly. Similar results we obtain for bisexual branching process with immigration in a random environment.

We study a critical branching random walk on a real line with discrete time controlled by a point process. Sizes of sequential generations form Galton–Watson critical process with one type of particles. The particle coordinates are interpreted as the weights of the vertices on the genealogical tree of the random walk. A reduced tree is obtained after removing branches of the genealogical tree that do not reach the n -th level. The asymptotic behavior of the first two moments of the number of vertices and of the sum of vertex weights over all levels of reduced tree under condition of nonextinction is described. Several limit theorems for the weights of particles in a branching random walk up to n -th generation under condition of nonextinction are proved.