Weak convergence of random polygonal lines to a Gaussian process

We obtain a limit theorem of convergence in distribution for random polygonal lines defined by sums of independent random variables with replacements. In a particular case, the limit is the Gaussian Ornstein-Uhlenbeck process.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 33–44, January–March, 2005.Translated by V. Mackeviius     

The variance matrix of a multivariate random process of a stochastic differential system

We show the possibility of determining the variance matrix of a multivariate random process of a stochastic differential system by passing from integral dependence to a rather simple algebraic expression. We determine a procedure for realizing the relation thereby obtained on a computer.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 102–106.     

Uniform convergence of fuzzy random renewal process

Recently, Zhao et al. (in Fuzzy Optimization and Decision Making 2007 6, 279–295) presented a fuzzy random elementary renewal theorem and fuzzy random renewal reward theorem in the fuzzy random process. In this paper, we study the convergence of fuzzy random renewal variable and of the total rewards earned by time t with respect to the extended Hausdorff metrics d _∞ and d ₁. Using this convergence information and applying the uniform convergence theorem, we provide some new versions of the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem.     

On the path regularity of a stochastic process in a hilbert space,defined by the ito integral

Poisson Processes, by J.F.C. Kingman. Clarendon Press (Oxford University Press), Oxford (1993), 104 pp. $ 39,95 ISBN 0-19-853693-3     

Weak convergence of compound stochastic process,I

Compound stochastic processes are constructed by taking the superpositive of independent copies of secondary processes, each of which is initiated at an epoch of a renewal process called the primary process. Suppose there are M possible k-dimensional secondary processes {ξ^v(t):t?0}, v=1,2,…,M. At each epoch of the renewal process {A(t):t?0} we initiate a random number of each of the M types. Let m_l:l?1} be a sequence of M-dimensional random vectors whose components specify the number of secondary processes of each type initiated at the various epochs. The compound process we study is

, where the ξ^v_lj() are independent copies of ξ^v,m_lv is the vth component of m and {τ_l:l?1} are the epochs of the renewal process. Our interest in this paper is to obtain functional central limit theorems for {Y(t):t?0} after appropriately scaling the time parameter and state space. A variety of applications are discussed.     

The various aggregates of random polygons determined by random lines in a plane

The probability distributions (defined in an ergodic sense) of various aggregates of random convex polygons determined by the standard isotropic Poisson line process in the plane are investigated.     

The multi-stage dynamic stochastic decision process with unknown distribution of the random utilities

We consider a decision maker who performs a stochastic decision process over a multiple number of stages, where the choice alternatives are characterized b     

A simple characterization of almost uniform convergence by stochastic convergence

Motivated by Egorov's theorem and the characterization of the equivalence of P-stochastic convergence and P-almost convergence by the property of the probability distribution P to be purely atomic and concentrated on a countable number of pairwise disjoint P-atoms (cf. [1], p. 68), it is proved that P-stochastic resp. P-almost convergence is equivalent to P-almost uniform convergence (cf. [2], p. 89/90) if and only if P is purely atomic and concentrated on a finite number of pairwise disjoint P-atoms. Furthermore, this property of P is equivalent to the condition that any P-stochastic convergent sequence admits a P-almost uniform convergent subsequence. Finally a proof is given that P is purely atomic and concentrated on a finite number of pairwise disjoint P-atoms if and only if there does not exist a purely finitely additive {0,1}-valued probability charge, which vanishes for all P-zero sets.     

The separability of the Hilbert space generated by a stochastic process

The separability of the Hilbert space generated by a stochastic process is one of the basic assumptions in the time-spectral analysis of stochastic processes. This assumption is either presupposed explicitly or, more often, obtained as a consequence of the assumption of existence of left and right limits of the process for any value of the time parameter. In this paper it is shown that the existence of a left limit only, for each value of the time parameter, is a sufficient condition for the separability of the Hilbert space generated by the process.     

Analytic stochastic process solutions of second-order random differential equations

In this work, trigonometric stochastic processes arise as mean square solutions of random differential equations, using a random Fröbenius method. Important operational properties of the trigonometric stochastic processes are established.     

On the convergence rate of a recursively defined sequence

Consider the following recursively defined sequence: $\tau _1 = 1,\sum\limits_{j = 1}^n {\frac{1} {{\sum\nolimits_{s = j}^n {\tau _s } }}} = 1forn \geqslant 2, $ , which originates from a heat conduction problem first studied by Myshkis (1997). Chang, Chow, and Wang (2003) proved that $\tau _n = \log n + O(1) for large n.$ . In this note, we refine this result to $\tau _n = \log n + \gamma + O\left( {\frac{1} {{\log n}}} \right). $ . where γ is the Euler constant.     

The maximum likelihood method for censoring of observations by a random interval

The article investigates the maximum likelihood estimators of an unknown parameter using a sample censored by a random interval, when both the distribution of the observed random variable and the distribution of the interval ends depend on the same parameter . Asymptotic normality and consistency of these estimators is proved.Translated from Statisticheskie Metody, pp. 19–30, 1980.     

The uniform convergence of the sum of independent random variables

LetX _i (i=1, 2, ...) be the independent random variables on the probability space (, ,P). In this paper we will show that the necessary and sufficient condition of the uniform convergence of X _i is the convergence of m _i ⁽¹⁾ and m _i ⁽⁰⁾ , wherem _i ⁽¹⁾ , m _i ⁽⁰⁾ denote the 1-quantile and 0-quantile ofX _i respectively.     

Kruglov operator and operators defined by random permutations

The Kruglov property and the Kruglov operator play an important role in the study of geometric properties of r. i. function spaces. We prove that the boundedness of the Kruglov operator in an r. i. space is equivalent to the uniform boundedness on this space of a sequence of operators defined by random permutations. It is also shown that there is no minimal r. i. space with the Kruglov property.