Showing papers in "Annals of Probability in 1982"
TL;DR: In this paper, the zero range process is constructed when the stochastic matrix determining the one particle motion satisfies a mild assumption, and the set of invariant measures for this process is described in the following two cases: a) the system is attractive and $p(x, y)$ is recurrent.
Abstract: On a countable set of sites $S$, the zero range process is constructed when the stochastic matrix $p(x, y)$ determining the one particle motion satisfies a mild assumption. The set of invariant measures for this process is described in the following two cases: a) The system is attractive and $p(x, y)$ is recurrent. b) The system is attractive, $p(x, y)$ corresponds to a simple random walk on the integers and the rate at which particles leave any site is bounded.
317 citations
TL;DR: In this paper, a simple proof of the central limit theorem for stationary random fields under mixing conditions is given, generalizing some results obtained by more complicated methods, e.g. Bernstein's method.
Abstract: A simple proof of a central limit theorem for stationary random fields under mixing conditions is given, generalizing some results obtained by more complicated methods, e.g. Bernstein's method.
307 citations
TL;DR: In this paper, sufficient conditions are obtained for averages of the extreme values of a sequence of nonnegative i.i.d. random variables of the form: (X n+1-i, n} - X n-k_n,n) = 1, \cdots, k_n + 1.
Abstract: Let $X_1, X_2, \cdots$, be a sequence of nonnegative i.i.d. random variables with common distribution $F$, and for each $n \geq 1$ let $X_{1n} \leq \cdots \leq X_{nn}$ denote the order statistics based on $X_1, \cdots, X_n$. Necessary and sufficient conditions are obtained for averages of the extreme values $X_{n+1-i, n}i = 1, \cdots, k_n + 1$ of the form: $k^{-1}_n \sum^{k_n}_{i = 1} (X_{n+1-i, n} - X_{n-k_n,n})$, where $k_n \rightarrow\infty$ and $n^{-1}k_n \rightarrow 0$, to converge in probability or almost surely to a finite positive constant. In the process, characterizations are given of the classes of distributions with regularly varying upper tails and of distributions with "exponential-like" upper tails.
303 citations
TL;DR: In this article, the theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial, Dirichlet and the multivariate hypergeometric.
Abstract: : The theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial, Dirichlet and the multivariate hypergeometric. Some conditions of negative dependence that are satisfied by these distributions and which have practical meaning are introduced. Preservation results for some of these concepts are derived. Useful inequalities for some widely used distributions are obtained. Results of Mallows (1969) that apply to the multinomial distributions are extended to more distributions. Examples are listed. (Author)
260 citations
TL;DR: In this paper, a set of points in Euclidean space is partitioned into the groups that minimize the within groups sum of squares, and conditions are found to assure asymptotic normality of the vector of means of the groups.
Abstract: A set of $n$ points in Euclidean space is partitioned into the $k$ groups that minimize the within groups sum of squares. Under the assumption that the $n$ points come from independent sampling on a fixed distribution, conditions are found to assure asymptotic normality of the vector of means of the $k$ groups. The method of proof makes novel application of a functional central limit theorem for empirical processes--a generalization of Donsker's theorem due to Dudley.
253 citations
TL;DR: In this article, the authors studied the local behavior of empirical processes for independent identically distributed random variables on the real line and obtained best rates of convergence for various types of density estimators as well as error estimates for the Bahadur representation of the quantile process.
Abstract: In this paper we study the local behavior of empirical processes for independent identically distributed random variables on the real line. The results are applied to get best rates of convergence for various types of density estimators as well as error estimates for the Bahadur representation of the quantile process obtained by Kiefer.
224 citations
TL;DR: In this article, it was shown that normal variables are associated if and only if their correlations are nonnegative, i.e., they are associated with normal variables only if they are correlated.
Abstract: It is shown that normal variables are associated if and only if their correlations are nonnegative.
222 citations
TL;DR: In this paper, the authors derived a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function and applied it to get optimal window-widths with respect to almost sure uniform convergence.
Abstract: In this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function. Extensions to higher derivatives are included. The results are applied to get optimal window-widths with respect to almost sure uniform convergence.
134 citations
TL;DR: In this paper, the convergence rates for martingale difference arrays are derived in central limit theorems for Martingale Difference Arrays, and it is shown that the rates which are obtained are the exact ones under the stated conditions.
Abstract: Convergence rates are derived in central limit theorems for martingale difference arrays. The rates depend heavily on the behavior of the conditional variances and on moment conditions. It is also shown that the rates which are obtained are the exact ones under the stated conditions.
130 citations
TL;DR: In this article, the strength of a bundle of parallel fibres is asymptotically normally distributed in a series-parallel model consisting of a long chain of bundles arranged in series.
Abstract: A classical result due to Daniels is that the strength of a bundle of parallel fibres is asymptotically normally distributed. Extensions of this result are obtained and applied to a series-parallel model consisting of a long chain of bundles arranged in series. This model is of importance in studying the reliability of fibrous materials. Improved approximations are also obtained which reduce the error associated with Daniels' approximation both for the single bundle and for the series-parallel system.
126 citations
TL;DR: In this article, weak invariance principles for mixing sequences of $L_2$-integrable random variables under the condition that the variance of the sum of $n$ random variables is asymptotic to $sigma^2n$ where π^2 > 0.
Abstract: In this note we prove weak invariance principles for some classes of mixing sequences of $L_2$-integrable random variables under the condition that the variance of the sum of $n$ random variables is asymptotic to $\sigma^2n$ where $\sigma^2 > 0$. One of the results is simultaneously an extension to nonstationary case of a theorem of Ibragimov and an improvement of the $\varphi$-mixing rate used by McLeish in his invariance principle for nonstationary $\varphi$-mixing sequences.
TL;DR: In this paper, the authors show how the methods developed in the Gaussian case can be extended to the general non-Gaussian case, and apply some of the results to specific examples of nonGaussian processes, and classes of processes containing a Gaussian subclass.
Abstract: Let $X(t), -\infty 0$, put $L_t(u) =$ Lebesgue measure of $\{s: 0 \leq s \leq t, X(s) > u\}$ and $M(t) = \max(X(s): 0 \leq s \leq t)$. For several years the author has studied the limiting properties of these random variables in the case where $X(t)$ is a Gaussian process and under two kinds of limiting operations: i) $t$ fixed and $u \rightarrow \infty$; ii) $t \rightarrow \infty$ and $u = u(t) \rightarrow \infty$ as a function of $t$. The purpose of this paper is to show how the methods developed in the Gaussian case can be extended to the general, not necessarily Gaussian case. This is illustrated by applications of some of the results to specific examples of non-Gaussian processes, and classes of processes containing a Gaussian subclass.
TL;DR: In this paper, the first two moments of derivatives or differences of the function were shown to be a regularity condition for the variance of a function of a random variable, which satisfies the usual regularity conditions.
Abstract: Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.
TL;DR: In this article, implicit universal constants for extremal distributions for which equality is attained (or nearly attained) are given in implicit form, where the universal constants are defined in both implicit and explicit form.
Abstract: Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if $X_1, X_2, \cdots$ are i.i.d. non-negative random variables and if $T_n$ is the set of stop rules for $X_1, \cdots, X_n$, then $E(\max\{X_1, \cdots, X_n\}) \leq a_n \sup\{EX_t: t \in T_n\}$, and the bound $a_n$ is best possible. Similar universal constants $0 < b_n < \frac{1}{4}$ are found so that if the $\{X_i\}$ are i.i.d. random variables taking values only in $\lbrack a, b\rbrack$, then $E(\max\{X_1, \cdots, X_n\}) \leq \sup\{EX_t: t \in T_n\} + b_n(b - a)$, where again the bound $b_n$ is best possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.
TL;DR: In this paper, the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion was obtained by a modification of the proof of a similar result by Kuelbs and Philipp.
Abstract: We obtain the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion. This is achieved by a modification of the proof of a similar result by Kuelbs and Philipp (1980) on $\phi$-mixing Banach space valued random variables. As by-products we get almost sure invariance principles for sums of absolutely regular sequences of random variables with values in a Banach space and necessary and sufficient conditions for the almost sure invariance principle for sums of independent, identically distributed random variables.
TL;DR: The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world as mentioned in this paper, which is supported by libraries, scholarly societies, publishers,and foundations.
Abstract: Stable URL:http://links.jstor.org/sici?sici=0091-1798%28198308%2911%3A3%3C760%3ATROOOS%3E2.0.CO%3B2-1The Annals of Probability is currently published by Institute of Mathematical Statistics.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/ims.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org.http://www.jstor.orgTue Oct 9 14:25:53 2007
TL;DR: In this paper, the authors established bounds for a quasi-superadditive quasi-additive nonnegative function with index $Q(1 \leq Q 1) = q(1, k) + q(k) = g(k, n) + g(n + 1, k), k + q g(i, k).
Abstract: Let $X_1, \cdots, X_n$ be arbitrary random variables and put $S(i, j) = X_i + \cdots + X_j$ and $M(i, j) = \max \{|S(i, i)|, |S(i, i + 1)|, \cdots, |S(i, j)|\}$ for $1 \leq i \leq j \leq n$ Bounds for $E\{\exp tM (1, n)\}, E M^\gamma(1, n)$ and $P\{M(1, n) \geq t\}$ are established in terms of assumed bounds for $E \{\exp t|S(i, j)|\}, E|S(i, j)|^\gamma$ and $P\{|S(i, j)| \geq t\}$, respectively The bounds explicitly involve a nonnegative function $g(i, j)$ assumed to be quasi-superadditive with index $Q(1 \leq Q \leq 2): g(i, j) + g(j + 1, k) \leq Q g(i, k)$, all $1 \leq i \leq j 1$ include sequences $\{X_i\}$ exhibiting long-range dependence, in particular certain self-similar processes such as fractional Brownian motion
TL;DR: In this paper, it is shown that the random measure is supported on a bounded generalized Cantor set and that this set performs a "wandering" but coherent motion that, if appropriately rescaled, approaches a Brownian motion.
Abstract: Fleming and Viot have established the existence of a continuous-state-space version of the Ohta-Kimura ladder or stepwise-mutation model of population genetics for describing allelic frequencies within a selectively neutral population undergoing mutation and random genetic drift. Their model is given by a probability-measure-valued Markov diffusion process. In this paper, we investigate the qualitative behavior of such measure-valued processes. It is demonstrated that the random measure is supported on a bounded generalized Cantor set and that this set performs a "wandering" but "coherent" motion that, if appropriately rescaled, approaches a Brownian motion. The method used involves the construction of an interacting infinite particle system determined by the moment measures of the process and an analysis of the function-valued process that is "dual" to the measure-valued process of Fleming and Viot.
TL;DR: In this article, the authors proved an arc-sine law for triangular arrays generalizing the Erdos-Kac law for Brownian Motion Processes, and a weak convergence of the distributions of certain functionals of the partial row sums.
Abstract: If $\mu_n,
u$ are probability measures on a separable Banach space, $j_n \rightarrow \infty$ and $\mu^{jn}_n \rightarrow_w
u$ (so $
u$ is necessarily infinitely divisible), then it is possible to construct two row-wise independent triangular arrays $\{X_{nj}\}, \{Y_{nj}\}$ such that $\mathscr{L}(X_{nj}) = \mu_n, \mathscr{L}(Y_{nj}) =
u^{1/jn}$ and $\max_{k \leq jn} \|S_{nk} - T_{nk}\|\rightarrow_\mathrm{P} 0$, where $S_{nk}$ and $T_{nk}$ are the respective partial row sums. Several refinements are proved. These results are applied to establish the weak convergence of the distributions of certain functionals of the partial row sums, improving well-known results of Skorohod. As concrete applications, we prove an arc-sine law for triangular arrays generalizing the Erdos-Kac law and an arc-sine law for strictly stable processes generalizing P. Levy's law for Brownian Motion.
TL;DR: In this paper, it was shown that the Laplace transform of the pinched Wiener process satisfies a simple recurrence for the moments, which seem to be difficult to calculate by direct or by other techniques.
Abstract: Let $\tilde{W} = \tilde{W}_t, 0 \leq t \leq 1$, be the pinned Wiener process and let $\xi = \int^1_0|\tilde{W}|$. We show that the Laplace transform of $\xi, \phi(s) = Ee^{-\xi s}$ satisfies \begin{equation*}\tag{*}\int^\infty_0 e^{-us}\phi(\sqrt 2 s^{3/2})s^{-1/2} ds = - \sqrt \pi Ai(u)/Ai'(u)\end{equation*} where $Ai$ is Airy's function. Using $(\ast)$, we find a simple recurrence for the moments, $E\xi^n$ (which seem to be difficult to calculate by direct or by other techniques) namely $E\xi^n = e_n \sqrt \pi(36 \sqrt 2)^{-n}/\Gamma \big(\frac{3n + 1}{2}\big)$ where $e_0 = 1, g_k = \Gamma(3k + \frac{1}{2})/\Gamma(k + \frac{1}{2})$ and for $n \geq 1$, $e_n = g_n + \sum^n_{k=1} e_{n-k}\binom{n}{k} \frac{6k + 1}{6k - 1} g_k.$
TL;DR: In this paper, a simple extension of Slepian's inequality is given which implies the original inequality and the result that positively correlated normal variables are associated, which implies that the original inequalities are associated.
Abstract: : A simple extension of Slepian's inequality is given which implies the original inequality and the result that positively correlated normal variables are associated. (Author)
TL;DR: In this article, sufficient conditions in terms of interpolation variances are given for a Gaussian process to have a jointly continuous local time in the stationary case and a bound for the modulus of continuity in the space variable is also obtained.
Abstract: Sufficient conditions in terms of interpolation variances are given for a Gaussian process to have a jointly continuous local time. In the stationary case these conditions can be verified in terms of the spectral density and are seen to be within logarithmic factors of the best possible conditions. A bound for the modulus of continuity in the space variable is also obtained.
TL;DR: In this paper, a recursive algorithm which includes the multidimensional Robbins-Monro and Kiefer-Wolfowitz processes was studied, and it was shown that the algorithm can be represented as a weighted average of the disturbances.
Abstract: We study a recursive algorithm which includes the multidimensional Robbins-Monro and Kiefer-Wolfowitz processes The assumptions on the disturbances are weaker than the usual assumption that they be a martingale difference sequence It is shown that the algorithm can be represented as a weighted average of the disturbances This representation can be used to prove asymptotic results for stochastic approximation procedures As an example, we approximate the one-dimensional Kiefer-Wolfowitz process almost surely by Brownian motion and as a byproduct obtain a law of the iterated logarithm
TL;DR: In this article, it was shown that the class of all phase type distributions is closed under all three basic operations arising in reliability theory: finite mixtures, finite convolutions, and formation of coherent systems of independent components.
Abstract: Consider three basic operations arising in reliability theory: finite mixtures, finite convolutions, formation of coherent systems of independent components. In this paper it is shown that: (i) The class of all phase type distributions is closed under all three operations; (ii) the class of all phase type distributions having a representation in which the matrix is upper triangular is the smallest class which is closed under all three operations and contains all exponential distributions. The paper includes a section of preliminaries in which the relevant material regarding phase type distributions and reliability theory is summarized.
TL;DR: In this article, the authors proved a general law for weighted sums of the form (1, √ n) = 0 for linear processes with zero mean and a common variance of ϵ 2.
Abstract: Motivated by the problem of establishing laws of the iterated logarithm for least squares estimates in regression models and for partial sums of linear processes, we prove a general $\log \log$ law for weighted sums of the form $\sum^\infty_{i=-\infty} a_{ni}\varepsilon_i$, where the $\varepsilon_i$ are independent random variables with zero means and a common variance $\sigma^2$, and $\{a_{ni}: n \geq 1, -\infty < i < \infty\}$ is a double array of constants such that $\sum^\infty_{i=-\infty} a^2_{ni} < \infty$ for every $n$. Besides applying the general theorem to least squares estimates and linear processes, we also use it to improve earlier results in the literature concerning weighted sums of the form $\sum^n_{i=1} f(i/n)\varepsilon_i$.
TL;DR: In this article, the authors established a link between the canonical measure of a hitting time and the spectral measure of the differential generator of the diffusion, and showed that the derivative of this canonical measure with respect to natural scale (as a function of the point being hit) equals the spectral measures of the generator on a restricted interval.
Abstract: All first hitting times for a one-dimensional diffusion belong to the Bondesson class of infinitely divisible distributions on $\lbrack 0, \infty\rbrack$. A distribution in this class can be conveniently represented in terms of its canonical measure. In this paper we establish a link between the canonical measure of a hitting time and the spectral measure of the differential generator of the diffusion. In particular, it is shown that the derivative of the canonical measure with respect to natural scale (as a function of the point being hit) equals the spectral measure of the differential generator on a restricted interval. The canonical measure is then calculated for several examples arising from the Bessel diffusion process, including the inverse of a gamma variate and the Hartman-Watson mixing distribution.
TL;DR: In this article, the properties of a standard Wiener process were investigated and it was shown that the process is a non-decreasing function of a nondecreasing process.
Abstract: Let $\{W(t), 0 \leq t < +\infty\}$ be a standard Wiener process and $0 < b_t \leq t$ be a nondecreasing function of $t$. The properties of the process $Y_1(t) = b^{-1/2}_t \sup_{0\leq s \leq t - b_t}(W(s + b_t) - W(s))$ are investigated. One of the results says that $\lim_{t\rightarrow\infty}(Y_1(t) - (2 \log tb^{-1}_t)^{1/2}) = 0$ a.s. if $b_t$ is "much less" than $t$. Analogous properties of similar processes are studied.
TL;DR: In this paper, the relationship between the critical behavior of the finite and infinite versions of Spitzer's reversible nearest particle systems is investigated, and it is shown that the critical values for the two systems agree, but that the behavior of two systems at the common critical value can differ.
Abstract: Motivated by several results and open problems concerning Harris' basic contact process, we consider the relationship between the critical behavior of the finite and infinite versions of Spitzer's reversible nearest particle systems. We show that the critical values for the finite and infinite systems agree, but that the behavior of the two systems at the common critical value can differ. The Nash-Williams recurrence criterion for reversible Markov chains is an important tool used in the proofs of the main results, and we give a new treatment of that theory. Finally, we compute several critical exponents for the nearest particle systems.
TL;DR: For a new class of Gaussian function space integrals, the exponential rate of growth or decay as $n \rightarrow \infty$ is determined in this article, where the results are applied to the calculation of specific free energy in statistical mechanics.
Abstract: For a new class of Gaussian function space integrals depending upon $n \in \{1, 2,\cdots\}$, the exponential rate of growth or decay as $n \rightarrow \infty$ is determined. The result is applied to the calculation of the specific free energy in a model in statistical mechanics. The physical discussion is self-contained. The paper ends by proving upper bounds on certain probabilities. These bounds will be used in a sequel to this paper, in which asymptotic expansions and limit theorems will be proved for the Gaussian integrals considered here.
TL;DR: Winkler and Graham as mentioned in this paper showed that even when the hypothesis of the FKG inequality fails, it may be possible to redefine the partial ordering so that the conclusion of the fkG inequality still holds.
Abstract: Consider random variables $x_1, \cdots, x_n$, independently and uniformly distributed on the unit interval. Suppose we are given partial information, $\Gamma$, about the unknown ordering of the $x$'s; e.g., $\Gamma = \{x_1 < x_{12}, x_7 < x_5, \cdots\}$. We prove the "XYZ conjecture" (originally due to Ivan Rival, Bill Sands, and extended by Peter Winkler, R. L. Graham, and other participants of the Symposium on Ordered Sets at Banff, 1981) that $P(x_1 < x_2|\Gamma) \leq P(x_1 < x_2|\Gamma, x_1 < x_3).$ The proof is based on the FKG inequality for correlations and shows by example that even when the hypothesis of the FKG inequality fails it may be possible to redefine the partial ordering so that the conclusion of the FKG inequality still holds.