Volume 30, Issue 4

Let ( ξ ( i ), η ( i )) ∈ ℝ d +1 , 1 ≤ i < ∞, be independent identically distributed random vectors, η ( i ) be nonnegative random variables, the vector ( ξ (1), η (1)) satisfy the Cramer condition. On the base of renewal process, N T = max{ k : η (1) + … + η ( k ) ≤ T } we define the generalized renewal process Z T = ∑ i=1NT$\begin{array}{} \sum_{i=1}^{N_T} \end{array}$ ξ ( i ). Put I Δ T ( x ) = { y ∈ ℝ d : x j ≤ y j < x j + Δ T , j = 1, …, d }. We find asymptotic formulas for the probabilities P ( Z T ∈ I Δ T ( x )) as Δ T → 0 and P ( Z T = x ) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of ( ξ (1), η (1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

We study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.

Let b , n be two positive integers such that b ≥ 2, and S ( b , n ) be the numerical semigroup generated by {bn+1+i+bn+i− 1b− 1∣ i∈ N}$\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$. Applying two order relations, we give formulas for computing the embedding dimension, the Frobenius number, the type and the genus of S ( b , n ).

We obtain a criterion for the minimal logarithmic growth rate for an arbitrary set with a given set of operations defined on it, i.e., we describe all finite sets A with operations on them such that the growth rate differs by at most a constant from the logarithmic growth rate to base ∣ A ∣.

A functional system of Boolean vector functions with a naturally defined superposition operation is considered. It is shown that each closed class of vector functions with α - or δ -functions as components has a finite basis.