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The work is devoted to the investigation of some asymptotic properties of probability distributions generated by semicontinuous random walks. Complete asymptotic expansions of the distributions of the boundary functionals connected with two boundaries (a walk in a strip) are obtained.

For random permanents, we obtain an estimate of the rate of convergence in the central limit theorem.

ForU-statistics taking values in a Hilbert space, we obtain estimates of the rate of convergence in the central limit theorem.

One considers probability distributions related to the Poisson distribution. It is shown that the boundary problems for the Poisson process admit an interpretation in terms of the well-known Abel-Goncharov interpolation problem.

An exact distribution and complete asymptotic expansions are obtained for a statistic studied by Smirnov in connection with the calculation of limiting distributions for the statistics D+(n1, n2), D-(n1, n2, and

Let x1, x2,...,xn and y1, y2,...,yn be the results of two series of independent observations. Let us denote by FR1 (x) and GR2 (y) the empirical distribution functions constructed on the basis of the first and the second sample, respectively. Let us write This paper deals with a complete asymptotic expansion, for the case n1=n, n2=np, n?8 of the probability in a power series 1/vn, where p?1 is a fixed integer, and ?1>0 and ?2> 0 are fixed positive numbers.

The problem of estimating the rate of convergence in the central limit theorem for the U-statistics with values in the Hilbert space is considered. Bounds of the order o(n-1/2) if the kernel F satisfies the nondegenerate condition and E?F?3 < 8.

Estimates of a rate of convergence of orderO(n-1) are obtained for integral functionals of an empirical process.

Asymptotics of distributions for bootstrapped U-statistis is considered. Conditions for convergence to the normal distribution are indicated.

Goodness-of-fit tests of Kolmogorov-Smirnov-Kac type are considered, as well as the asymptotic properties of exact distributions of these tests whose statistics are constructed on the basis of samples of random size.

Let and be two independent repeated samples from distributions with densities and where p(x)=0 for x<0, and p(x)=e-x for x?0. The problem of similar regions for testing the null hypothesis H0:a1=a2 is studied on the assumption that the positive parameters and are unknown.

Let Fn(x) be an empirical distribution function constructed on the basis of a repeated sample from a population with a continuous distribution function F(x). This paper deals with the classical nonparametric Kolmogorov-Smirnov tests For the exact distribution of the two-dimensional vector (Dn+, Dn—), a representation is obtained in the form of a contour integral, as well as a complete asymptotic expansion of this distribution in a power series in 1/vn1, and a general algorithm of construction of asymptotic expansions of distributions of the statistics of Kuiper, Pyke, Brunk, and many others that are closely related to the Kolmogorov-Smirnov tests.

Permanents of matrices with random elements are studied. For random permanents of finitedimensional projection matrices, new limiting results are obtained in different asymptotic schemes. The principal statistical functionals such asU-statistics, symmetric statistics, and others, are represented as functionals of permanent measures.