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The present paper studies the connection between the local limit theorem for lattice random variables and the corresponding symmetrized random variables. An example is constructed to show that the local limit theorem holds for some sequence of symmetrized random variables, but fails to hold for the original random variables.

This paper gives an overview of multivariate local limit theorems for integer random vectors and provides the sufficient condition for local limit theorems in ${\bf R}^n$.

Let $\xi $ be a vector-valued random variable in $\mathbf{R}^s$ and a corresponding density function $p_\xi(x)$ be "close" to the "standard" normal density. Under this condition an inequality for a characteristic function is proved. The inequality obtained is of interest for the problem of a lower estimator of the rate of convergence in the local limit theorem for densities. An analogous inequality for a lattice distribution was investigated in [N. G. Gamkrelidze, Litovsk. Mat. Sb., 7 (1967), pp. 405--408 (in Russian)] and was given in [V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, Berlin, New York, 1975] and [Yu. V. Prohorov and Yu. A.~Rozanov, Probability Theory: Basic Concepts, Limit Theorems, and Random Processes, Springer-Verlag, Berlin, New York, 1969].

A differential game of evasion in which the state is governed by an nth order differential equation is studied. The equation is linear in the state and its derivatives. Control is exercised through a nonlinear forcing term. A sufficient condition for evasion is given and the evasion strategy is obtained as a generalized solution of a Volterra integral equation. The generalized solution is approximated by ordinary measurable functions by constructing a sliding regime.

Let $\xi _1 ,\xi _2 , \cdots $ be integer valued independent random variables and $S_n = \xi _1 + \cdots + \xi _1 $. We introduce the function \[\sum\limits_m {|{\bf P}\{ S_n = m\} - {\bf P}\{ S_n = m - 1\} |} \]as a measure of the “smoothness” of the distribution Sn. Some properties of this function are investigated.

Yu. V. Prokhorov made a conjecture that the two statements are equivalent: (a) the local limit theorem is applicable to a sequence of integer-valued random variables, (b) these variables obey the central limit theorem and their sums are asymptotically uniformly distributed modulo h for arbitrary h.We give an example showing that the above conjecture does not hold true.