Showing papers on "Random element published in 1973"

Random difference equations and Renewal theory for products of random matrices

Abstract: where Mn and Qn are random d • d matrices respectively d-vectors and Yn also is a d-vector. Throughout we take the sequence of pairs (Mn, Q~), n >/1, independently and identically distributed. The equation (1.1) arises in various contexts. We first met a special case in a paper by Solomon, [20] sect. 4, which studies random walks in random environments. Closely related is the fact tha t if Yn(i) is the expected number of particles of type i in the nth generation of a d-type branching process in a random environment with immigration, then Yn = (Yn(1) ..... Yn(d)) satisfies (1.1) (Qn represents the immigrants in the nth generation). (1.1) has been used for the amount of radioactive material in a compar tment ([17]) and in control theory [9 a]. Moreover, it is the principal feacture in a model for evolution and cultural inheritance by Cavalli-Sforza and Feldman [2]. Notice also tha t the dth order linear difference equation

Weak convergence of probability measures and random functions in the function space D[0,∞)

Abstract: This paper extends the theory of weak convergence of probability measures and random functions in the function space D[0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞) is a suitable space for the analysis of many processes appearing in applied probability.

Spectra of Random Self Adjoint Operators

Abstract: This survey contains an exposition of the results obtained in the studying the spectra of certain classes of random operators. It consists of three chapters. In the introductory Chapter I we survey some of the pioneering papers (two, in particular), which have sufficient depth of content to suggest the natural problems to be considered in this field. In Chapter II we study the distribution of the eigenvalues for ensembles of random matrices, for instance, the sum of one-dimensional projection operators onto random vectors uniformly and independently distributed over the surface of the -dimensional unit sphere. We show that as , the eigenvalue distribution ceases to be random and can be determined as the solution of a certain functional equation. Chapter III deals with the Schrodinger equation with a random potential. We establish ergodic properties of certain random quantities, constructed from the eigenvalues and eigenfunctions of this equation, and we study the distribution of eigenvalues in the cases when the potential is a Gaussian random field and a homogeneous Markov process.

A theorem about random fields

Abstract: Averintsev [1] and Spitzer [2] proved that the class of Markov fields is identical to the class of Gibbs ensembles when the domain is a finite subset of the cubic lattice and each site may be in either of two given states. Hammersley and Clifford [3] proved the same result for the more general case when the domain is the set of sites of an arbitrary finite graph and the number of possible states for each site is finite. In order to show this, they extended the notion of a Gibbs ensemble to embrace more complex interactions than occur on the cubic lattice. Their method was circuitous and showed merely the existence of a potential function for a Markov field with little indication of its form. In [4], Preston gives a more direct approach to the two-state problem and presents an explicit formula for the potential. We show here that the equivalence of Markov fields and Gibbs ensembles follows immediately from a very simple application of the Mobius inversion theorem of [5] which allows us to construct a natural expression for the potential function of a Markov field. We confine our attention to the set of sites of an arbitrary finite graph and allow each site to be in any one of a countable set of states. The two-state solution of Preston emerges as a corollary.

Potentials for almost Markovian random fields

Abstract: A positive almost Markovian random field is a probability measure on a lattice gas whose finite set conditional probabilities are continuous and positive. We show that each such random field has a potential and in the translation invariant case an absolutely convergent potential. We give a criterion for determining which random fields correspond to pair potentials, or in generaln-body potentials. We show that two translation invariant positive almost Markovian random fields have the same finite set conditional probabilities if and only if one minimizes the specific free energy of the other.

The random coding bound is tight for the average code.

Abstract: The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upperbounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.

Recursive Computation of the Generalized Q Function

Abstract: A recursive method of computing values of the generalized Q function is described. The Q function is also interpreted in terms of noncentral chi-square distributed random variables, and in terms of the difference between Poisson random variables.

Asymptotic properties of Gaussian random fields

Abstract: In this paper we study continuous mean. zero Gaussian random fields X(p) with an N-dimensional parameter and having a correlation function p(p, q) for which 1 p(p, q) is asymptotic to a regularly varying (at zero) function of the distance dis (p, q) with exponent 0 < a < 2. For such random fields, we obtain the asymptotic tail distribution of the maximum of X(p) and an asymptotic almost sure property for X(p) as IPI N. Both results generalize ones previously given by the authors for N = 1.

An invariance principle for mixing sequences of random variables

Abstract: According to Donsker's theorem XN D> W where W is standard Brownian motion on [0, 1]. (see [1, p. 137]). A similar theorem can be formulated in the space C of continuous functions ([1, p. 68]). Donsker's theorem has been generalized in many directions. Two of them will be taken up in the present paper. The first, due to Prohorov E9], deals with independent, not necessarily identically distributed random variables. Prohorov's theorem says that the properly defined random functions XN converge in distribution to standard Brownian motion if, and only if, the ~, satisfy the Lindeberg condition. (For the details see E2, p. 452], El, p. 77, Problem 1 and p. 143, Problem 7].) The second generalization, due to Billingsley El, p. 177] is concerned with strict sense stationary processes satisfying a mixing condition. In the present paper we shall prove Donsker's theorem for not necessarily identically distributed random variables satisfying a mixing condition. For such random variables the second-named author has proved theorems of a somewhat different character E10].

The distribution of the maximum of a random number of random variables with applications

Abstract: The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group. ASYMPTOTIC DISTRIBUTION; RANDOM NUMBER OF RANDOM VARIABLES; SERVICE TIME; SYSTEM OF A LARGE NUMBER OF COMPONENTS; DEPENDENT RANDOM VARIABLES 1. The main result and applications Let X1,X2,'"-,X, be random variables with the same distribution function F(x). Let nl 0, if t = io 0 and b, be real numbers such that, for n + o , (4) n[1 F(ax + b,)] -~ w(x). Let further v, be a sequence of positive integer valued random variables such that v,/n converges stochastically to a positive random variable v. Then, as tn -+ + c0, (5) lim P[(W, b,,)/a,, < x] = e-w(x) There are several practical situations when our model, described in (1)-(3) and the result of Theorem 1 are needed. We shall describe one through a concrete example; the similarity of several situations to this example is evident. Consider a system of n components which require regular servicing. The number of components to be serviced at a given time is a random variable, i.e., varies from time to time. If v, is the number of components to be serviced then the service is completed in a time period not exceeding a given number T, if, and only if, Wn = max{X1, X2, ...,X,,} does not exceed T, where X1 is the time period required for servicing the jth component. Thus for large n, the conclusion of Theorem 1 gives a good approximation for the time period needed to complete the service. Our assumptions (1)-(3) were made under the guidance of this specific problem. Namely, the assumption of previous models that the service times are stochastically independent is practically never satisfied, even if as many machines are available as there are components to be serviced. To make clear how our assumptions apply to a practical situation, let us specify our system to be an automobile car. Automatic equipment starts servicing all parts virtually at the same time. Because of the relations of the parts, however, it cannot be assumed that service can continue uninterrupted on all parts. As a matter of fact, some This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 05:47:31 UTC All use subject to http://about.jstor.org/terms

Maximal Inequalities and the Law of the Iterated Logarithm

Abstract: A supermartingale maximal inequality is derived A maximal inequality is derived for arbitrary random variables $\{S_n, n \geqq 1\}$ (let $S_0 = 0$) satisfying $E\exp\lbrack u(S_{m + n} - S_m) \rbrack \leqq \exp(Knu^2)$ for all real $u$, all integers $m \geqq 0$ and $n \geqq 1$, and some constant $K$ These two maximal inequalities are used to derive upper half laws of the iterated logarithm for supermartingales, multiplicative random variables, and random variables not satisfying particular dependence assumptions

Solutions of a Class of Random Differential Equations

Abstract: The analysis of a wide class of physical problems involves linear random differential equations—linear differential equations with stochastic coefficients. Here we consider the determination of probability densities for the solution processes of a class of linear random differential equations.This class is characterized by the property that the stochastic coefficients can be modeled by random processes with finite degrees of randomness. The solutions are examined by means of a Liouville-type equation satisfied by their joint probability densities. This technique is then applied to the study of electromagnetic fields in an inhomogeneous medium with special profiles.

Conservation of energy in random media, with application to the theory of sound absorption by an inhomogeneous flexible plate

Abstract: This paper discusses the linear theory of wave propagation through conservative random media. By means of a simple illustrative example taken from acoustics it is verified that, at least uniformly to second order in an expansion parameter associated with the random fluctuations, the net flow of energy is from the mean, or coherent, wave field to the random component of the wave field (in accordance with the second law of thermodynamics). A general formula is derived for the distribution of the random modes of the system responsible for the power flux into the random field. These are demonstrated unambiguously (without recourse to the use of far field asymptotics) to be precisely those propagating modes which satisfy the homogeneous, non-random dispersion relation. The extension of the theory to a wider class of wave propagation problems is then outlined using an approach involving a Lagrangian density of wide generality. Finally the discussion is extended further to cover the case of coupled systems of wave-bearing media. An important analytical feature of such cases is the occurrence of 'branch-cut' integrals in the power flux formula. The situation is illustrated by an investigation of the power extracted from a plane sound wave incident on a flexible plate whose mass density is a random function of position. The division of the scattered power between the acoustic and plate bending modes is obtained, and comparison made with a heuristic argument leading to the same result. The presence of random fluctuations in the properties of an otherwise uniform wavebearing medium gives rise to several important theoretical problems. Broadly speaking these may be divided into two groups. In the first group one is concerned with the elucidation of the characteristics of the mean or coherent part of the field, defined as an average of the wave field taken over an ensemble of statistically equivalent random media. The type of problem encountered here is governed by the size of the region occupied by the random fluctuations in relation to the relevant length scales of the wave field. The important case in which the dimensions of the 'cloud' of inhomogeneities are large enough for it to be assumed infinite has received considerable attention in the literature (see Frisch (I968) for an extensive bibliography). That theory has been approached from widely differing viewpoints by authors working in different branches of theoretical physics

An invariance principle for a class of $d$-dimensional polygonal random functions

Abstract: A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space (d > 1) undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if d > 2, and not being almost surely nonnegative if d = 1. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rose*n, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.

Finite random sums (a historical essay)

Abstract: The study of the stochastic behavior of sums of independent random quantities, the number of which increases without limit, has been a most important problem of the classical theory of probability. However, finite random sums occurring in various fields of application of this theory were also studied again and again by a number of scholars and led to the development of mathematical methods and ideas in the theory of probability proper.

Strong laws of large numbers for normed linear spaces

Abstract: ?Strong laws of large numbers which are useful in the theory and applications of stochastic processes are obtained for sequences of independent random elements in separable normed linear spaces. The hypotheses for these results lie between those for the identically distributed case and the independent non-identieally distributed case. These results and other strong and weak laws of large numbers for separable normed linear spaces can be extended to separable Freshet spa?es. Finally, the results are applied to separable Wiener processes on [0, 1] and on [0, oo).

Random variable generators

Abstract: This note gives APL functions for generating random variables from 21 common statistical distributions: