A uniform limit theorem for predictive distributions

Modeling Sequences of Long Memory Non-negative Covariance Stationary Random Variables

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

The stochastic process {Xn} satisfyingXn+1= max{Yn+1+ αβ Xn,βXn} where {Yn} is a stationary sequence of non-negative random variables and, 0<β<1, can be regarded as a simple thermal energy storage model with controlled input. Attention is mostly confined to the study ofμ=EXwhere the random variableXhas the stationary distribution for {Xn}. Even for special cases such as i.i.d.Ynorα= 0, little explicit information appears to be available on the distribution ofXorμ. Accordingly, bounding techniques that have been exploited in queueing theory are used to studyμ. The various bounds are illustrated numerically in a range of special cases.

Let ${X_n;n\geq 0}$ be a sequence of random variables. We consider its geometrically weighted series $\xi(\beta)=\sum_{n=0}^\infty \betaX_n$ for $0<\beta < 1$. This paper proves that $\xi (\beta)$ can be approximated by $\sum_{n=0}^\infty \beta^n Y_n$ under some suitable conditions, where ${Y_n; n \geq 0}$ is a sequence of independent normal random variables. Applications to the law of the iterated logarithm for $\xi(\beta)$ are also discussed.

A sequence of random variables, $Y_0,Y_1,Y_2,\ldots\,$, is called standard if there exists a one-sided isomorphism between it and a sequence of independent random variables. In this paper it is demonstrated that the sequence arising from the past of the $T,T^{-1}$ map is not standard.

An elementary proof of de Finetti’s theorem

As we have seen, a sequence of random variables can describe how some process evolves in time. For example, the gambler’s accumulated winnings at time n, that is, after the nth play of a game of chance, is given by the random variable Sn = Xl +...Xn, where the X’s are independent random variables representing the gambler’s winnings at each play of the game. For convenience, we can think of the subscript n of any such sequence of random variables as denoting the nth moment of time, and the value of the random variable Sn as the measurement made at that moment. The sequence therefore represents a “picture” of the evolution of the process at the times 1, 2 ,... .

CHAPTER 6 - ESTIMATION OF PARAMETERS OF DISTRIBUTIONS

In this paper, we study the complete moment convergence for the partial sum of linear processes with random coefficients to form where is a sequence of random variables with zero means and stochastically dominated by a random variable ε and is a sequence of random variables, also and satisfying the Rosenthal-type maximal inequality.

This dissertation broadly deals with two areas of probability theory and investigates how methods from nonstandard analysis may provide new perspectives in these topics. In particular, we use nonstandard analysis to prove new results in the topics of limiting spherical integrals and of exchangeability. In the former area, our methods allow us to represent finite dimensional Gaussian measures in terms of marginals of measures on hyperfinite-dimensional spheres in a certain strong sense, thus generalizing some previously known results on Gaussian Radon transforms as limits of spherical integrals. This first area has roots in the kinetic theory of gases, which is also described. In the latter area, we prove a new generalization of de Finetti's theorem for exchangeable random variables, a theorem important for the foundations of Bayesian statistics. In particular, we extend the de Finetti--Hewitt--Savage theorem to certain general sequences of exchangeable random variables taking values in any Hausdorff space. Under mild distributional conditions, our work expresses a sequence of exchangeable random variables taking values in any Hausdorff space as a mixture of sequences of iid random variables. Prior to this work, this result was known for random variables taking values in a Polish space. Hence, the current work has removed the need to have any assumptions on the state space, and shown that it is the underlying distribution of the random variables that is important. We prove several preparatory results in nonstandard and topological measure theory along the way, a highlight being a new generalization of Prokhorov's theorem.

Introduction. These results are generalizations of a theorem of Chung [I ] on the limiting distribution of the number of crossings of a value, c, by the successive partial sums of a sequence of independent random variables, and of theorems by Chung and Erdos [3 ] on the lower limits of such sums. Two of these theorems concern the case of independent, equidistributed random variables whose distributions have an absolutely continuous component (Case A). The others are for binomial variates (Case B). We extend the results of Case A to sums of independent random variables whose distributions need not be the same, and those of Case B to sums of independent and equidistributed random variables of the lattice type. Let {X } be a sequence of independent random variables whose c.d.f.'s, { Fn(x) }, need not be the same. We use the usual notations:

Until now, we mostly dealt with a single random variable at a time and used it to model a random quantity, such as the waiting time for a customer to enter a store or the number of deaths due to horse kicks. Occasionally the problem gave rise to a sequence of independent random variables like in the coupon collector’s problem. We studied tools that help us with the analysis of single random variables and sequences of independent random variables. However, in applications we frequently encounter sequences of random variables that are not independent. Andrey Markov was one of the firsts who gave a systematic analysis of such sequences in the early 1900s. Today we call these sequences Markov chains.

Let {T(n), n greater-than-or-equal-to 1} be an arbitrary sequence of nonlattice random variables and let {S(n), n greater-than-or-equal-to 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic R(n) = T(n)/S(n) based on simple conditions on the moment generating functions of T(n) and S(n). When S(n) = n, our main result reduces to that of Chaganty and Sethuraman[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when T(n) and S(n) are both lattice random variables. We call our theorems large deviation local limit theorems for R(n), since the conditions of our theorems imply that R(n) --&gt; c in probability for some constant c. We present some examples to illustrate our theorems.

Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the $\alpha$-stable CLT ($1<\alpha\le2$). The results give invariants for relative isomorphism of these.

SummaryGnedenko's classical work [1] on the limit of the distribution of the maximum of a sequence of independent random variables is extended to the distribution of the maximum of a sequence of random variables defined on a finite Markov chain.