On the functional law of the iterated logarithm for partially observed sums of random variables

We consider a partial-sum process generated by a sequence of nonidentically distributed independent random variables. Assuming that this process is available for observation along an arbitrary time sequence, we fill the gaps by linear interpolation and prove the functional law of the iterated logarithm (FLIL) for sample paths obtained in this way. Assuming that the V. A. Egorov condition holds, we show that FLIL is valid, while under other conditions sufficient for the usual law of the iterated logarithm FLIL may fail. Bibliography: 16 titles. Translated, fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 73–95.     

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     

Transforms of pseudo-Boolean random variables

As in earlier works, we consider {0,1}ⁿ as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.     

The degree sequence of a random graph. I. The models

We show that the joint distribution of the degrees of a random graph can be accurately approximated by several simpler models derived from a set of independent binomial distributions. On the one hand, we consider the distribution of degree sequences of random graphs with n vertices and ½m edges. For a wide range of values of m, this distribution is almost everywhere in close correspondence with the conditional distribution {(X₁,…,X_n) | ∑ X_i=m}, where X₁,…,X_n are independent random variables, each having the same binomial distribution as the degree of one vertex. We also consider random graphs with n vertices and edge probability p. For a wide range of functions p=p(n), the distribution of the degree sequence can be approximated by {(X₁,…,X>_n) | ∑ X_i is even}, where X₁,…,X_n are independent random variables each having the distribution Binom (n−1, p′), where p′ is itself a random variable with a particular truncated normal distribution. To facilitate computations, we demonstrate techniques by which statistics in this model can be inferred from those in a simple model of independent binomial random variables. Where they apply, the accuracy of our method is sufficient to determine asymptotically all probabilities greater than n^−k for any fixed k. In this first paper, we use the geometric mean of degrees as a tutorial example. In the second paper, we will determine the asymptotic distribution of the tth largest degree for all functions t=t(n) as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 97–117 (1997)     

Coherent random allocations, and the Ewens-Pitman formula

Assume that there is a random number K of positive integer random variables S₁, …, S_K that are conditionally independent given K and all have identical distributions. A random integer partition N = S₁ + S₂ + … + S_K arises, and we denote by P_N the conditional distribution of this partition for a fixed value of N. We prove that the distributions {P_N} _N=1 ^∞ form a partition structure in the sense of Kingman if and only if they are governed by the Ewens-Pitman Formula. The latter generalizes the celebrated Ewens sampling formula, which has numerous applications in pure and applied mathematics. The distributions of the random variables K and S_j belong to a family of integer distributions with two real parameters, which we call quasi-binomial. Hence every Ewens-Pitman distribution arises as a result of a two-stage random procedure based on this simple class of integer distributions. Bibliography: 25 titles. This paper is an edited and actualized version of the unpublished PDMI preprint 21/1995. Further development of the ideas of this work can be found in [21, 25]. A number of detected misprints was fixed without notice, the bibliography was extended beyond the original 19 references, and a few comments were added as footnotes. (Comments by Alexander Gnedin.) __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 127–145.     

On stochastic stationarity of renewal processes

We shall consider point systems inR ₁ which are stationary renewal distributed. We let the points undergo random translations which are assumed to be independent identically distributed random variables with a non-degenerate distribution function. The translations are also independent of the starting positions. It is shown in theorem 3.1 that the only distribution of the points which is conserved after the random translations is the Poisson one. Finally in section 4 we give a characterization of renewal processes on the positive semiaxis in terms of conditional mean values.     

Conditional versions of limit theorems for conditionally associated random variables

Relation between association and conditional association is answered, several examples show that the association of random variables does not imply the conditional association, and vice versa. Several fundamental properties of conditional associated random variables are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional Hájek-Rényi type inequalities, a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established, which are conditional versions of the earlier results for associated random variables, respectively. In addition, some lemmas in the context are of independent interest.     

Stationary distributions of semistochastic processes with disturbances at random times and with random severity

We consider a semistochastic continuous-time continuous-state space random process that undergoes downward disturbances with random severity occurring at random times. Between two consecutive disturbances, the evolution is deterministic, given by an autonomous ordinary differential equation. The times of occurrence of the disturbances are distributed according to a general renewal process. At each disturbance, the process gets multiplied by a continuous random variable (“severity”) supported on [0,1). The inter-disturbance time intervals and the severities are assumed to be independent random variables that also do not depend on the history.We derive an explicit expression for the conditional density connecting two consecutive post-disturbance levels, and an integral equation for the stationary distribution of the post-disturbance levels. We obtain an explicit expression for the stationary distribution of the random process. Several concrete examples are considered to illustrate the methods for solving the integral equations that occur.     

On the strong laws of large numbers for double arrays of random variables in convex combination spaces

Let \mathfrak X{\mathfrak {X}} be a convex combination space as defined by Terán and Molchanov [13]. By using their definition of mathematical expectation of an \mathfrak X{\mathfrak {X}}-valued random variable, we state several new variants of strong laws of large numbers for double arrays of integrable \mathfrak X{\mathfrak {X}}-valued random variables under various assumptions. Some related results in the literature are extended.     

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

We derive exponential bounds for the tail of the distribution of normalized sums of triangular arrays of random variables, not necessarily independent, under the law of ordinary logarithm.

Furthermore, we provide estimates for partial sums of triangular arrays of independent random variables belonging to suitable grand Lebesgue spaces and having heavy-tailed distributions.

     

Limit theorems for random permanents with exchangeable structure

Permanents of random matrices extend the concept of U-statistics with product kernels. In this paper, we study limiting behavior of permanents of random matrices with independent columns of exchangeable components. Our main results provide a general framework which unifies already existing asymptotic theory for projection matrices as well as matrices of all-iid entries. The method of the proofs is based on a Hoeffding-type orthogonal decomposition of a random permanent function. The decomposition allows us to relate asymptotic behavior of permanents to that of elementary symmetric polynomials based on triangular arrays of rowwise independent rv's.     

Complete convergence in mean for double arrays of random variables with values in Banach spaces

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p). In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

Analyticity domains of complex power series with independent random coefficients

We consider a complex power series with independent random coefficients and investigate the possibility of analytic extension of its sum beyond the boundaries of the domain of convergence.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1689–1694, December, 1992.     

The size of random fragmentation trees

We consider the random fragmentation process introduced by Kolmogorov, where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution (the dislocation law); this is repeated recursively for all pieces. More precisely, we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold, and we study the associated random tree. Dean and Majumdar found a phase transition for this process: the number of fragmentations is asymptotically normal for some dislocation laws but not for others, depending on the position of roots of a certain characteristic equation. This parallels the behavior of discrete analogues with various random trees that have been studied in computer science. We give rigorous proofs of this phase transition, and add further details. The proof uses the contraction method. We extend some previous results for recursive sequences of random variables to families of random variables with a continuous parameter; we believe that this extension has independent interest.     

An analytic refinement of the Heyde theorem on linear forms of independent random variables

In terms of the characteristic function of the joint distribution of two linear forms of independent random variables, one refines Heyde's known theorem on the characterization of the Gaussian distribution by the property of symmetry of the conditional distribution of one linear form under a given second form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 54–59, 1988.     

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums , whereX _n,n⩾1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t⩾0 is a process of non-negative integer-valued random variables, independent ofX _n,n⩾1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t⩾0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.     

Convergence and invariance questions for point systems inR 1 under random motion

In section 2 we introduce and study the independence property for a sequence of two-dimensional random variables and by means of this property we define independent motion in section 3. Section 4 is mainly a survey of known results about the convergence of the spatial distribution of the point system as the timet→∞. In theorem 5.1 we show that the only distributions which are time-invariant under given reversible motion of non-degenerated type are the weighted Poisson ones. Lastly in section 6 we study a more general type of random motion where the position of a point after translation is a functionf of its original position and its motion ability. We consider functionsf which are monotone in the starting position. Limiting ourselves to the case when the point system initially is weighted Poisson distributed with independent motion abilities, we prove in theorem 6.1 that this is the case also after the translations, if and only if the functionf is linear in the starting position. In the paper also some implications of our results to the theory of road traffic with free overtaking are given.     

On distributions of random variables chosen at random

The existence of typical distributions for random variables chosen at random from a finite-dimensional random variable vector space of high dimension is established. Possible typical distributions are described, and conditions for the typical distribution to be standard Gaussian are given. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 153–160. Translated by the author.     

On rank correlation measures for non-continuous random variables

For continuous random variables, many dependence concepts and measures of association can be expressed in terms of the corresponding copula only and are thus independent of the marginal distributions. This interrelationship generally fails as soon as there are discontinuities in the marginal distribution functions. In this paper, we consider an alternative transformation of an arbitrary random variable to a uniformly distributed one. Using this technique, the class of all possible copulas in the general case is investigated. In particular, we show that one of its members—the standard extension copula introduced by Schweizer and Sklar—captures the dependence structures in an analogous way the unique copula does in the continuous case. Furthermore, we consider measures of concordance between arbitrary random variables and obtain generalizations of Kendall's tau and Spearman's rho that correspond to the sample version of these quantities for empirical distributions.