Showing papers in "Annals of Probability in 1989"

Two Moments Suffice for Poisson Approximations: The Chen-Stein Method

Abstract: Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable result is due to Chen (1975). The method also provides an upper bound on the total variation distance to the Poisson distribution, and succeeds in cases where third and higher moments blow up. This paper presents Chen's results in a form that is easy to use and gives a multivariable extension, which gives an upper bound on the total variation distance between a sequence of dependent indicator functions and a Poisson process with the same intensity. A corollary of this is an upper bound on the total variation distance between a sequence of dependent indicator variables and the process having the same marginals but independent coordinates.

Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$

Abstract: Let $\eta_t$ be the basic voter model on $\mathbb{Z}^d$ and let $\eta^{(N)}_t$ be the voter model on $\Lambda(N)$, the torus of side $N$ in $\mathbb{Z}^d$. Unlike $\eta_t, \eta^{(N)}_t$ (for fixed $N$) gets trapped with probability 1 as $t \rightarrow\infty$ at all 0's or all 1's. We examine the asymptotic growth of these trapping or consensus times $\tau^{(N)}$ as $N \rightarrow\infty$. To do this we obtain limit theorems for coalescing random walk systems on the torus $\Lambda(N)$, including a new hitting time limit theorem for (noncoalescing) random walk on the torus.

Coupling Methods for Multidimensional Diffusion Processes

Abstract: In this paper, coupling methods for diffusion processes are studied mainly to obtain upper bound estimates in two different probability metrics. We use the martingale approach and explore the construction of explicit coupling operators which are sometimes optimal. The paper presents some criteria for the success of coupling and for the finiteness of the moments of the coupling times. Rates of convergence in various metrics are also studied.

On Normal Approximations of Distributions in Terms of Dependency Graphs

Abstract: L'auteur interprete les bornes de l'erreur, introduites par Stein, dans l'approximation normale de sommes de variables aleatoires dependantes en termes de graphes de dependance. Ceci mene a des ameliorations d'un theoreme de limite centrale de Petrovskaya et Leontovich et de recentes applications de Baldi et Rinatt. En particulier, on obtient des bornes pour des taux de convergence. On etudie ensuite l'approximation normale du nombre de maximums locaux d'une fonction aleatoire sur un graphe

Sums of Independent Random Variables in Rearrangement Invariant Function Spaces

Abstract: Let $X$ be a quasinormed rearrangement invariant function space on (0, 1) which contains $L_q(0, 1)$ for some finite $q$. There is an extension of $X$ to a quasinormed rearrangement invariant function space $Y$ on $(0, \infty)$ so that for any sequence $(f_i)^\infty_{i = 1}$ of symmetric random variables on (0,1), the quasinorm of $\sum f_i$ in $X$ is equivalent to the quasinorm of $\sum\mathbf{f}_i$ in $Y$, where $(\mathbf{f}_i)^\infty_{i = 1}$ is a sequence of disjoint functions on $(0, \infty)$ such that for each $i, \mathbf{f}_i$ has the same decreasing rearrangement as $f_i$. When specialized to the case $X = L_q(0, 1)$, this result gives new information on the quantitative local structure of $L_q$.

A Central Limit Theorem for Two-Dimensional Random Walks in Random Sceneries

Abstract: Let $S_n, n \in \mathbb{N}$, be a recurrent random walk on $\mathbb{Z}^2 (S_0 = 0)$ and $\xi(\alpha), \alpha \in \mathbb{Z}^2$, be i.i.d. $\mathbb{R}$-valued centered random variables. It is shown that $\sum^n_{i = 1}\xi(S_i)/ \sqrt{n \log n}$ satisfies a central limit theorem. A functional version is presented.

Integration by Parts and Time Reversal for Diffusion Processes

Abstract: In this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property, assuming the existence of a density at every time $t$. The proofs are based on techniques of the stochastic calculus of variations.

Kolmogorov's contributions to information theory and algorithmic complexity

Abstract: Briefly, information theory says a random object X - p(x) has complexity (entropy) H = - Xp(x)log p(x), with the attendant interpretation that H bits are sufficient to describe X on the average. Algorithmic complexity says an object x has a complexity K(x) equal to the length of the shortest (binary) program that describes x. It is a beautiful fact that these ideas are much the same. In fact, it is roughly true that EK(X) - H. Moreover, if we let Pu(x) = Pr{U prints x} be the probability that a given computer U prints x when given a random program, it can be shown that log(1/Pu(x)) - K(x) for all x, thus establishing a vital link between the " universal" probability measure Pu and the " universal" complexity K. More on this later. The relationship of these ideas to probability theory was summed up in Kolmogorov's 1983 paper which was based on his 1970 talk in Nice. Perhaps only

Notes on the Wasserstein Metric in Hilbert Spaces

Abstract: Let $(X, Y)$ be a pair of Hilbert-valued random variables for which the Wasserstein distance between the marginal distributions is reached We prove that the mapping $\omega \rightarrow (X(\omega), Y(\omega))$ is increasing in a certain sense Moreover, if $Y$ satisfies a nondegeneration condition, we can take $X = T(Y)$ with $T$ monotone in the sense of Zarantarello We apply these results to obtain a proof of the central limit theorem (CLT) in Hilbert spaces which does not make use of the CLT for real-valued random variables

Hungarian constructions from the nonasymptotic viewpoint

Abstract: Let $x_1, \ldots, x_n$ be independent random variables with uniform distribution over $\lbrack 0, 1\rbrack$, defined on a rich enough probability space $\Omega$. Denoting by $\hat{\mathbb{F}}_n$ the empirical distribution function associated with these observations and by $\alpha_n$ the empirical Brownian bridge $\alpha_n(t) = \sqrt n(\hat{\mathbb{F}}_n(t) - t)$, Komlos, Major and Tusnady (KMT) showed in 1975 that a Brownian bridge $\mathbb{B}^0$ (depending on $n$) may be constructed on $\Omega$ in such a way that the uniform deviation $\|\alpha_n - \mathbb{B}^0\|_\infty$ between $\alpha_n$ and $\mathbb{B}^0$ is of order of $\log(n)/\sqrt n$ in probability. In this paper, we prove that a Poisson bridge $\mathbb{L}^0_n$ may be constructed on $\Omega$ (note that this construction is not the usual one) in such a way that the uniform deviations between any two of the three processes $\alpha_n, \mathbb{L}^0_n$ and $\mathbb{B}^0$ are of order of $\log(n)/\sqrt n$ in probability. Moreover, we give explicit exponential bounds for the error terms, intended for asymptotic as well as nonasymptotic use.

The erdos-renyi strong law for pattern matching with a given proportion of mismatches

Abstract: 1. Informal introduction. The Erdos-Bnyi law [Erdijs and R6nyi (1970)l for coin tossing is a strong law for the behavior of the length of long success rich runs. For the length R, of the longest run of pure heads in n tosses, the result is that with probability l,lin~~+~ RJlog,,,(n) = 1, where p = flheads) > 0. A more general result is quoted as formula (1) in Section 2 for RZ, the longest head rich run in which the fraction of heads is at least a > p. Note that R, = Ri. In many practical situations, such as manufacturing or roulette, observations are taken sequentially and each can be classified as success or failure. For these cases, it is possible to use the Erdijs-Wnyi law to test the hypothesis that the success probability is p. From another point of view, the Erdijs-Bnyi law can be used to recognize patterns of unusually long runs of succe88e8 (or failures). Our interest is in the recognition of unusually long patterns or words common to two random sequences. The patterns are unknown prior to an examination of the sequences. The motivation for this work is the comparison of DNA sequences, which can be modeled as sequences of i.i.d. or Markov distributed letters. Evolution operates to conserve, although imperfectly, patterns important to biological function. It is a task of biology to discover these patterns and their function. In earlier work, we generalized the Erdiie-Bnyi law for pure head runs (a = 1) to exact matching patterns between two sequences. In this paper we extend those results to include matchings of quality a between p and 1. The examples below illustrate the natural analogs of RZ studied in this paper. Two words form a “quality a matching” if they have the same length and the

Self-normalized laws of the iterated logarithm

Abstract: On demontre une loi du logarithme itere, qui generalise la version classique, pour toutes les distributions dans la classe de Feller. Pour cette demonstration, l'auteur utilise une auto-normalisation convenable pour les sommes partielles de variables aleatoires independantes et identiquement distribuees. Un cas particulier de ces resultats s'applique a toute distribution dans le domaine de l'attraction d'une certaine loi stable

Heat Semigroup on a Complete Riemannian Manifold

Abstract: Let $M$ be a complete Riemannian manifold and $p(t, x, y)$ the minimal heat kernel on $M$. Let $P_t$ be the associated semigroup. We say that $M$ is stochastically complete if $\int_M p(t, x, y) dy = 1$ for all $t > 0, x \in M$; we say that $M$ has the $C_0$-diffusion property (or the Feller property) if $P_tf$ vanishes at infinity for all $t > 0$ whenever $f$ is so. Let $x_0 \in M$ and let $\kappa(r)^2 \geq -\inf\{Ric(x): \rho(x, x_0) \leq r\}$ ($\rho$ is the Riemannian distance). We prove that $M$ is stochastically complete and has the $C_0$-diffusion property if $\int^\infty_c \kappa(r)^{-1} dr = \infty$ by studying the radial part of the Riemannian Brownian motion on $M$.

Strong approximation for multivariate empirical and related processes, via kmt constructions

Abstract: Let $P$ be the Lebesque measure on the unit cube in $\mathbb{R}^d$ and $Z_n$ be the centered and normalized empirical process associated with $n$ independent observations with common law $P$. Given a collection of Borel sets $\mathscr{J}$ in $\mathbb{R}^d$, it is known since Dudley's work that if $\mathscr{J}$ is not too large (e.g., either $\mathscr{J}$) is a Vapnik-Cervonenkis class (VC class) or $\mathscr{J}$ fulfills a suitable "entropy with bracketing" condition), then $(Z_n)$ may be strongly approximated by some sequence of Brownian bridges indexed by $\mathscr{J}$, uniformly over $\mathscr{J}$ with some rate $b_n$. We apply the one-dimensional dyadic scheme previously used by Komlos, Major and Tusnady (KMT) to get as good rates of approximation as possible in the above general multidimensional situation. The most striking result is that, up to a possible power of $\log(n), b_n$ may be taken as $n^{-1/2d}$ which is the best possible rate, when $\mathscr{J}$ is the class of Euclidean balls (this is the KMT result when $d = 1$ and the lower bounds are due to Beck when $d \geq 2$). We also obtain some related results for the set-indexed partial-sum processes.

Weak Convergence of Smoothed and Nonsmoothed Bootstrap Quantile Estimates

Abstract: Under fairly general assumptions on the underlying distribution function, the bootstrap process, pertaining to the sample $q$-quantile, converges weakly in $D_\mathbb{R}$ to the standard Brownian motion. Furthermore, weak convergence of a smoothed bootstrap quantile estimate is proved which entails that in this particular case the smoothed bootstrap estimate outperforms the nonsmoothed one.

Isoperimetry and Integrability of the Sum of Independent Banach-Space Valued Random Variables

Abstract: We develop a new method to study the tails of a sum of independent mean zero Banach-space valued random variables $(X_i)_{i \leq N}.$ It relies on a new isoperimetric inequality for subsets of a product of probability spaces. In particular, we prove that for $p \geq 1,$ $\bigg\|\sum_{i \leq N} X_i\bigg\|_p \leq \frac{Kp}{1 + \log p}\bigg(\bigg\|\sum_{i \leq N} X_i\bigg\|_1 + \|\max_{i \leq N}\|X_i\|\|_p\bigg),$ where $K$ is a universal constant. Other optimal inequalities for exponential moments are obtained.

The Central Limit Theorem for the Right Edge of Supercritical Oriented Percolation

Abstract: On demontre un theoreme central limite pour la croissance des bords droits d'une percolation orientee supercritique. La technique de demonstration utilisee est de trouver des points avec des proprietes d'un type de regeneration appeles «points de ruptures». Cette technique peut etre appliquee a d'autres processus

Flux and Fixation in Cyclic Particle Systems

Abstract: Start by randomly coloring each site of the one-dimensional integer lattice with any of $N$ colors, labeled $0, 1, \ldots, N - 1$. Consider the following simple continuous time Markovian evolution. At exponential rate 1, the color $\xi(y)$ at any site $y$ randomly chooses a neighboring site $x \in \{y - 1, y + 1\}$ and paints $x$ with its color provided $\xi(y) - \xi(x) = 1 \operatorname{mod} N$. Call this interacting process the cyclic particle system on $N$ colors. We show that there is a qualitative change in behavior between the systems with $N \leq 4$ and those with $N \geq 5$. Specifically, if $N \geq 5$ we show that the process fixates. That is, each site is painted a final color with probability 1. For $N \leq 4$, on the other hand, we show that every site changes color at arbitrarily large times with probability 1.

On Independence and Conditioning On Wiener Space

Abstract: Notant I(f) et I(g) des integrales de Wiener-Ito multiples d'ordre p et q, respectivement, on donne une condition necessaire et suffisante sur le couple (f, g) sous laquelle les variables aleatoires I(f) et I(g) sont independantes. Ceci est base sur une caracterisation generale de l'independance de variables aleatoires dans l'espace de Wiener, dans le contexte d'un calcul variationnel stochastique

A Limit Theorem for a Class of Inhomogeneous Markov Processes

Abstract: Let $\{X(t): t \in R^+ \text{or} I^+\}$ be an (aperiodic) irreducible Markov process with a finite state space $S$ and transition rate $q_{ij}(t) = p(i, j)(\lambda(t))^{U(i, j)}$, where $0 \leq U(i, j) \leq \infty$ and $\lambda(t)$ is some suitable rate function with $\lim_{t \rightarrow \infty}\lambda(t) = 0$. We shall show in this article that there are constants $h(i) \geq 0$ and $\beta_i > 0$ such that independent of $X(0), \lim_{t \rightarrow \infty}P(X(t) = i) \div (\lambda(t))^{h(i)} = \beta_i$ for each $i \in S$. The height function $h$ is determined by $(p(i, j))$ and $(U(i, j))$. In particular, a limit distribution exists and concentrates on $\underline{S} = \{i \in S: h(i) = 0\}$.

Further Asymptotic Laws of Planar Brownian Motion

Abstract: The asymptotic distributions for large times of a variety of additive functionals of planar Brownian motion $Z$ are derived. Associated with each point in the plane, and with the point infinity, there is a complex Brownian motion governing the asymptotic behavior of windings of $Z$ close to that point. An independent Gaussian field over the plane governs fluctuations in local occupation times of $Z$, while a further independent family of complex Brownian sheets governs finer features of the windings of $Z$. These results unify and extend earlier results of Kallianpur and Robbins, Spitzer, Kasahara and Kotani, Messulam and the authors.

Exponential $L_2$ Convergence of Attractive Reversible Nearest Particle Systems

Abstract: Nearest particle systems are continuous-time Markov processes on $\{0, 1\}^Z$ in which particles die at rate 1 and are born at rates which depend on their distances to the nearest particles to the right and left There is a natural parametrization of these systems with respect to which they exhibit a phase transition When the process is attractive and reversible, the critical value $\lambda_c$ above which a nontrivial invariant measure exists can be computed exactly This invariant measure is the distribution $v$ of a stationary discrete time renewal process Under a mild regularity assumption, we prove that the following three statements are equivalent: (a) The nearest particle system converges to equilibrium exponentially rapidly in $L_2(v)$ (b) The density of the interarrival times in the renewal process has exponentially decaying tails (c) The nearest particle system is supercritical in the sense that $\lambda > \lambda_c$ Under an additional second-moment assumption, we prove that the critical exponent associated with the exponential convergence is 2 The proof of exponential convergence is based on an unusual comparison of the nearest particle system with an infinite system of independent birth and death chains To carry out this comparison, a new representation is developed for a stationary renewal process with a log-convex renewal sequence in terms of a sequence of iid random variables

Records in a partially ordered set

Abstract: We consider independent identically distributed observations taking values in a general partially ordered set. Under no more than a necessary measurability condition we develop a theory of record values analogous to parts of the well-known theory of real records, and discuss its application to many partially ordered topological spaces. In the particular case of $\mathbb{R}^2$ under a componentwise partial order, assuming the underlying distribution of the observations to be in the domain of attraction of an extremal law, we give a criterion for there to be infinitely many records.

A Sharp Deviation Inequality for the Stochastic Traveling Salesman Problem

Abstract: Let $T_n$ denote the length of the shortest closed path connecting $n$ random points uniformly distributed over the unit square. We prove that for some number $K$, we have, for all $t \geq 0$, $P(|T_n - E(T_n)| \geq t) \leq K \exp(-t^2/K).$

The Contact Process on a Finite Set. III: The Critical Case

Abstract: We show that if $\sigma_N$ is the time that the contact process on $\{1, \ldots N\}$ first hits the empty set then for $\lambda = \lambda_c$, the critical value for the contact process on $\mathbb{Z}, \sigma_N/N \rightarrow \infty$ and $\sigma_N/N^4 \rightarrow 0$ in probability as $N \rightarrow \infty$. The keys to the proof are a new renormalized bond construction and lower bounds for the fluctuations of the right edge. As a consequence of the result we get bounds on some critical exponents. We also study the analogous problem for bond percolation in $\{1,\ldots N\} \times \mathbb{Z}$ and investigate the limit distribution of $\sigma_N/E\sigma_N$.

Statistical Mechanics of Crabgrass

Abstract: In this article we consider the asymptotic behavior of the contact process when the range $M$ goes to $\infty$. We show that if $\lambda$ is the total birth rate from an isolated particle, then the critical value $\lambda_c(M) \rightarrow 1$ as $M \rightarrow \infty$. The rate of convergence depends upon the dimension: $\lambda_c(M) - 1 \approx M^{-2/3}$ in $d = 1, \approx (\log M)/M^2$ in $d = 2$, and $\approx M^{-d}$ in $d \geq 3$.

The Correlation Length for the High-Density Phase of Bernoulli Percolation

Abstract: We examine two standard types of connectivity functions in the high-density phase of nearest-neighbor Bernoulli (bond) percolation We show that these two quantities decay exponentially at the same constant rate The reciprocal of this constant defines therefore a correlation length Unfortunately, we cannot prove that this correlation length is finite whenever $p > p_c$, although previous work established this result for $p$ above a threshold which is conjectured to coincide with $p_c$ We examine also a third connectivity function and prove that it too decays exponentially with the same rate as the two standard connectivity functions We establish various useful properties of our correlation length, such a semicontinuity as a function of bond density and convexity in its directional dependence Finally, for bond percolation in two dimensions we show that the correlation length at bond density $p_1 > p_c = \frac{1}{2}$ is exactly half the correlation length at the subcritical bond density $p_2 = 1 - p_1 < p_c$ This sharpens some other exact results for two-dimensional percolation and is the precise analog of known results for the two-dimensional Ising model

Continuity Properties for Random Fields

Abstract: L'auteur donne, pour un champ aleatoire n-dimensionnel donne, une condition simple sur la fonction de covariance qui garantit l'existence d'une version du champ aleatoire dans laquelle les realisations sont partout continues. La demonstration implique une approximation assez delicate du champ aleatoire par des polynomes d'interpellation d'ordre eleve

Windings of Random Walks

Abstract: Let $X_1, X_2, X_3, \ldots$ be a sequence of iid $\mathbb{R}^2$-valued bounded random variables with mean vector zero and covariance matrix identity. Let $S = (S_n; n \geq 0)$ be the random walk defined by $S_n = \sum^n_{i = 1} X_i$. Let $\phi(n)$ be the winding of $S$ at time $n$, that is, the total angle wound by $S$ around the origin up to time $n$. Under a mild regularity condition on the distribution of $X_1$, we show that $2\phi(n)/\log n \rightarrow_d W$ where $\rightarrow_d$ denotes convergence in distribution and where $W$ has density $(1/2)\operatorname{sech}(\pi w/2)$.