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Integrals with respect to stationary random measures are considered. A central limit theorem for such integrals is proved. The results are applied to obtain a functional central limit theorem for transformed solutions of the Burgers equation with random initial data.

We prove a theorem on the equivalence of some properties of a random field defined in terms of sample functions. We apply this theorem for studying generalized random fields.

In this paper, the idea of random selection in the theorem on gambling system is extended to Markov chains, by using the notion likelihood ratio and an analytic technique. A strong limit theorem on the relative frequency of ordered couple under random selection is established.

Proof of the central limit theorem for the sample mean can be obtained in the case of independent and identically distributed randomvariables through the moment generating function under some simplifying assumptions. The case for the central limit theorem for the sample mean from finite populations under simple random sample without replacement, the parallel to the simplest case in the standard framework, is not as simple. Most sampling textbooks avoid any technical discussion of the finite population central limit theorem so that there is little appreciation for the conditions that underlay the theorem. Most often the theorem is illustrated with a simulation study; but this can only hint at the technicalities of the theorem. Now, new software has been developed that allows for automatic calculation of moments and cumulants of estimators used in survey sampling, as well as automatic derivation of unbiased or consistent estimators. This software is used to examine the assumptions behind the applications of...

In this paper we study the limiting behavior of sums of dependent random variables under a strong mixing condition. We obtain conditions for which an analog of the Baum-Katz theorem holds and cite an example showing their optimality.

A widely acclaimed achievement of probability has been its success in approximating the distributions of arbitrarily complicated random variables in terms of a rather small number of ‘universal’ distributions. Central limit theorem, which proves convergence to the Gaussian law, is the best known among this type of results. However, for practical purposes, it is much more important to know how accurate such an approximation is, and this is of course a more difficult question to answer. For instance, central limit theorem was known already around 1715 (and in full generality by 1900), whereas the corresponding approximation theorem of Berry and Esseen was only proved in 1941. Stein’s method, introduced in 1972 in [ 60], offers a general means of solving such problems.

The central limit theorem is proved for the integral-type functionals of nonlinear transformations of two-and three-dimensional uniform isotropic Gaussian random fields. A theorem on convergence of finite-dimensional distributions of these functionals to the corresponding distributions of the Wiener process is also established.

A new variant of the CLT is established for random fields defined on ℝ d which are strictly stationary, with a finite second moment and weakly dependent (comprising cases of positive or negative association). The summation domains grow in the van Hove sense. At the same time the indices of observations form more and more dense grids in these domains. Thus the effect of combining two scaling procedures is studied. A statistical version of this CLT is also proved. Some stochastic models in radiobiology based on dependent functional subunits are discussed as well.

The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein's method. The pre-limiting process is then shown, under weak conditions, to converge to a Gaussian limit process. The theorem is used to describe the shape of random permutation tableaux.

A version of central limit is established normalized sums dependent random when a theorem is and conditional are sufficiently The proof ideas from by representing density as integral of score function a translation of distributions. 1980 Subject 60F99.

An approach via generating functions is used to derive multivariate asymptotic distributions for the number of nodes in strata of random forests. For a certain range for the strata numbers we obtain a weak limit theorem to Brownian motion as well. Moreover, a moment convergence theorem for the width of random forests is derived.

A functional central limit theorem is proved for certain random fields whose domain has both a temporal and a spadai component. These processes are made up of dependent summands which are measurable with respect to an increasing filtration. Temporal limit theory for semi-martingales is utilized to provide spatial finite-dimensional convergence. Consequently the limiting random fields have independent increments in time, and can be thought of as evolving random fields. In deriving a tightness result the notion of majorizing measures is employed to allow local spatial variability. Thus a functional central limit theorem for evolving random fields with a rather general dependence structure is given here for the first time. Comparison with results available for empirical processes suggest that this result is close to optimal. Introduction. In this paper we consider conditions for weak convergence of a sequence of random fields which are individually evolving over time. For motivation though, let's first examine a given sequence of random fields at some fixed time point. Let X be an arbitrary index set and for each ? > 1, let {Yn>i(x) : l ? . Let *?(*> = S y*.-w for xsXi l,xejr}isa sequence of random fields. We are interested in general conditions under which S^ converges weakly. We wish to allow dependence among the individual summands, the Y^i's. We do this by considering conditions of the martingale type, involving only conditional first and second moments. (These are more desirable than mixing type conditions which involve the entire distribution.) In addition, we must constrain our indexing family X to satisfy a complexity condition involving the notion of majorizing measures (or, alternatively, metric entropy). There will be a natural topology on X associated with our problem. In two papers Goldie and Greenwood (1986a, b) found conditions for weak convergence of sequences of set-indexed random fields. An unexpectedly difficult aspect of the problem was the characterization of the limiting distribution in

Examples, concepts and tools random line segments, queues and counters vacancy counting and clumping elements of inference. Appendices: direct Radon-Nikodym theorem central limit theorem, Poisson limit theorem, ergodic theorem and law of large numbers Shepp's coverage theorem mean content and mean square content of cells formed by Poisson field of random planes multitype branching processes lattice percolation.

Using the semigroup product formula of P \text {P} . Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.

The purpose of the present paper is to give a central limit theorem for subadditive process in the sense of J. F. C. Kingman (cf. [3], [4]). Throughout this article (O5 , P) denotes a probability space on which all random variables are defined. Let T be a measure preserving transformation in what follows. According to Kingman, a family (xsy9 s

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.

It is shown in this note that if Q is an algebra of uniformly bounded mean-square continuous real-valued random functions indexed in a compact set T, containing all bounded random variables and separating points of T (i.e., given t1 and t2 in T, there is a random function Xt in Q such that , then given any mean square continuous random function, there is a sequence in Q converging in mean square to the given random function uniformly on T.

The statistics of spectral density of the homogeneous in strong sense random field obtained by time shift are considered.

In [2] a central limit theorem [CLT] for sequences of univariate random linear forms was proved. That result is extended in this note to a multivariate CLT for $q$-dimensional linear forms of constant $q$-vectors with real-valued random weight coefficients. Some applications are indicated in Section 3.

A time series is the realization of a stochastic process. An estimate of the process average can be taken as the equally weighted average of observed values taken from the time series. Normally, the sequence of observations consists of observations taken at equal intervals of time. In certain applications, it is more appropriate to consider a sequence of observations spaced by intervals which are randomly and independently chosen from an exponential distribution. Let $\lbrack x_t, t \varepsilon T\rbrack$ be a real stochastic process that takes on values in a space $X, T$ being restricted to the real line. Let $\Omega$ be a space of points $\omega, F_T = \beta(x_t, t \varepsilon T)$ be the Borel field of $\omega$-sets generated by the class of sets of the form $\lbrack x_t(\omega) \varepsilon A\rbrack$ where $t \varepsilon T$ and $A$ is any Borel set, $P$ be the probability measure of $F_T$-sets, $\alpha$ be the Borel field of $X$-sets and for every real $t, x_t(\cdot)$ be a function from $\Omega$ to $X$ such that $\lbrack x_t(\omega) \varepsilon A\rbrack$ is an $F_T$-set for $A \varepsilon \alpha$. Assume that the process satisfies, for all $t \varepsilon T$, the conditions \begin{align*}\tag{1.1}\quad E\{x_t\} &= \bar x \quad \text{exists and is independent of} \quad t \\ \tag{1.2}E\{(x_t - \bar x^2\} &= \operatorname{var} x \quad \text{exists and is independent of} \quad t \\ \tag{1.3} E\{x_t - \bar x) (x_{t + \tau} - \bar x)\} &= r \quad (\tau) \operatorname{var} x \quad \text{depends only on} \quad \tau \quad \text{and} \\ r \quad (\tau) \text{is harmonizable}. \end{align*} A stochastic process satisfying Conditions (1.1) to (1.3) is often referred to as being stationary in the wide sense [2]. \begin{equation*}\tag{1.4}\lim_{\tau\rightarrow\infty} \tau r(\tau) \rightarrow 0.\end{equation*} Let $\{\xi\}$ be a measurable space of mutually independent random variables $\xi_i$ identically distributed with probability $p$ of having value 1 and $(1 - p)$ of having value 0, all independent of the $x_t$-process. The sample space $\Omega$ contains all possible realizations of the stochastic process. Given a scheme for observing the values of $x_t$ in time and some form of estimate of process mean from the observed values, expected values of such an estimate and of its square can be determined. A measure of the efficiency of the sampling scheme may be defined as the ratio of the variance of the estimator to that which would be obtained if the observations taken were independent of each other. The limit in probability of this measure of sampling efficiency for observations spaced by exponentially distributed mutually independent intervals is obtained simply in terms of the mean observation rate and the spectral density component at zero frequency. The limit theorem considered is a weak version. By augmenting Condition (1.4) with relations between sample size $N$, mean observation rate $\lambda$, length of time series $T$ and sufficiently rapid decay of $r(\tau)$ with $\tau$, sharper probability statements can be made.