LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

The New Approach to Identification of Parametric and Momentary Damages on the Basis of Lindeberg – Levy’s Model

The new approach for the assessment of reliability condition of the exploited system, based on the appropriate analysis of changes in the current parameters of technical condition aRb and the regulation condition aRc, determined from the compressed condition equation (1 and 2). While analyzing the course of momentary parameters for technical condition aRb and the regulation condition aRc, it was observed that the parametric and momentary damages can be identified on the basis of quantitative relations between momentary threshold value dpr and corresponding momentary permissible value dpr dop, which are calculated from equation (7, 8, 9) resulting from Lindeberg–Levy’s theorem. It is assumed that the damages are prevailing, when for the moment θi: >dpri> dpr dopi. With the number of damages (damage map), reliability parameters for each moment of exploitation of technical object (before the catastrophic damages will occur) can be determined. Parametric damages (expected lifetime E(T) and standard deviation of expected lifetime σE(T) provides the reasonable information for the appropriate planning of the servicing of exploitative objects.

Identification of Parametric and Momentary Damages on the Basis of Lindeberg-Levy’s Model

Article presents the innovative method for the assessment of reliability condition of the exploited system, based on the appropriate analysis of changes in the current parameters of technical condition aRb and the regulation condition aRc, determined from the compressed condition equation (1 and 2). While analyzing the course of momentary parameters for technical condition aRb and the regulation condition aRc, it was observed that the parametric and momentary damages can be identified on the basis of quantitative relations between momentary threshold value dpr and corresponding momentary permissible value dprdop, which are calculated from equation (7, 8, 9) resulting from Lindeberg-Levy’s theorem. It is assumed that the damages are prevailing, when for the moment θi: dpri>dpr dopi. With the number of damages (damage map), reliability parameters for each moment of exploitation of technical object (before the catastrophic damages will occur) can be determined. Parametric damages (expected lifetime E(T) and standard deviation of expected lifetime σE(T) provides the reasonable information for the appropriate planning of the servicing of exploitative objects.

A random walk analogue of Lévy’s Theorem

In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.

On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.

On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.