# Volume 52, № 9, 2000

### Anatolii Vladimirovich Skorokhod (On His 70th Birthday)

Korolyuk V. S., Kovalenko I. N., Portenko N. I., Samoilenko A. M., Sytaya G. N., Yadrenko M. I.

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1155-1157

### Regularized Brownian Motion on the Siegel Disk of Infinite Dimension

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1158-1165

We construct a process of Brownian motion on the Siegel disk of infinite dimension.

### On the Asymptotic Properties of Solutions of Linear Stochastic Differential Equations in $R^d$

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1166-1175

We investigate necessary and sufficient conditions for the almost-sure boundedness of normalized solutions of linear stochastic differential equations in $R^d$ their almost-sure convergence to zero. We establish an analog of the bounded law of iterated logarithm.

### Stochastic Flow and Noise Associated with the Tanaka Stochastic Differential Equation

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1176-1193

We study the properties of the noise (in the Tsirelson sense) that is generated by the solutions of the well-known Tanaka equation.

### Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1194-1204

We consider the structure of orthogonal polynomials in the space *L* _{2}(*B*, μ) for a probability measure μ on a Banach space *B*. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (*B*, μ).

### A Remark on the Characterization of the Global Behavior of a Process with Independent Increments

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1205-1207

We show that the analysis of the global behavior of a process with independent increments in terms of the existence of the stationary distribution of the corresponding storage process leads to results that differ from the classical ones.

### On the Extrapolation of Entire Functions Observed in a Gaussian White Noise

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1208-1218

We solve the problem of extrapolation of an analytic function of a certain class in the case where its values are observed in a white noise whose intensity is not high.

### Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System $(≤ λ)/G/m$

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1219-1225

We consider a queuing system (≤ λ)/*G*/*m*, where the symbol (≤ λ) means that, independently of prehistory, the probability of arrival of a call during the time interval *dt*does not exceed λ*dt*. The case where the queue length first attains the level *r*≥ *m*+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity μ_{1}(*t*) of the flow of homogeneous events associated with the monotone refusals of the system, namely, μ_{1}(*t*) = *O*(λ^{ r+ 1}α_{1} ^{ m− 1}α_{ r− m+ 1}), where α_{ k }is the *k*th moment of the service-time distribution.

### Nonlinear Transformations of Smooth Measures on Infinite-Dimensional Spaces

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1226-1250

We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.

### Qualitative Analysis of Systems of Itô Stochastic Differential Equations

Kulinich G. L., Pereguda О. V.

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1251-1256

For inhomogeneous systems of Itô stochastic differential equations, we introduce the notion of local invariance of surfaces and the notion of local first integral. We obtain results that give the general possibility of finding invariant surfaces and functionally independent first integrals of stochastic differential equations.

### Properties of the Likelihood Ratio for Counting Processes in the Problem of Estimation of Unknown Parameters

Lin'kov Yu. N., Nikolaeva O. A.

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1257-1268

We obtain an asymptotic decomposition of the logarithm of the likelihood ratio for counting processes in the case of similar hypotheses. We establish the properties of the normalized likelihood ratio in the problem of estimation of an unknown parameter.

### On the Upper Limit of a Random Sequence and the Law of the Iterated Logarithm

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1269-1271

We obtain some results concerning the upper limit of a random sequence and the law of the iterated logarithm for sums of independent random variables.

### A Probabilistic Representation for the Solution of One Problem of Mathematical Physics

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1272-1282

We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.

### Multivariate Sobel–Uppuluri–Galambos-Type Bounds

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1283-1293

We improve the known upper and lower bounds for the probability of the fact that exactly *k* _{i}events should occur in a group consisting of *n* _{i}events simultaneously for all *i*= 1, 2, ..., *d*.

### On Randomly Perturbed Linear Oscillating Mechanical Systems

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1294-1303

We prove that the amplitudes and the phases of eigenoscillations of a linear oscillating system perturbed by either a fast Markov process or a small Wiener process can be described asymptotically as a diffusion process whose generator is calculated.

### On Sums of Overlapping Products of Independent Bernoulli Random Variables

Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1304-1309

We find the exact distribution of an arbitrary remainder of an infinite sum of overlapping products of a sequence of independent Bernoulli random variables.