Showing papers in "Annals of Probability in 1983"
TL;DR: In this paper, the authors give an entropic criterion for triviality of the boundary and prove an analogue of Shannon's theorem for entropy, obtain a boundary triviality criterion in terms of the limit behavior of convolutions and prove a conjecture of Furstenberg about existence of a non-degenerate measure with trivial boundary on any amenable group.
Abstract: The paper is devoted to a study of the exit boundary of random walks on discrete groups and related topics. We give an entropic criterion for triviality of the boundary and prove an analogue of Shannon's theorem for entropy, obtain a boundary triviality criterion in terms of the limit behavior of convolutions and prove a conjecture of Furstenberg about existence of a nondegenerate measure with trivial boundary on any amenable group. We directly connect Kesten's and Folner's amenability criteria by consideration of the spectral measure of the Markov transition operator. Finally we give various examples, some of which disprove some old conjectures.
677 citations
TL;DR: In this paper, a simple exclusion interaction is considered where each particle independently attempts to execute a simple symmetric random walk, but any jump which would carry a particle to an already occupied site is suppressed.
Abstract: Consider a system of particles moving on the integers with a simple exclusion interaction: each particle independently attempts to execute a simple symmetric random walk, but any jump which would carry a particle to an already occupied site is suppressed. For the system running in equilibrium, we analyze the motion of a tagged particle. This solves a problem posed in Spitzer's 1970 paper "Interaction of Markov Processes." The analogous question for systems which are not one-dimensional, nearest-neighbor, and either symmetric or one-sided remains open. A key tool is Harris's theorem on positive correlations in attractive Markov processes. Results are also obtained for the rightmost particle in the exclusion system with initial configuration $Z^-$, and for comparison systems based on the order statistics of independent motions on the line.
232 citations
TL;DR: In this article, it was shown that the Banach space of all continuous functions is tight in the topological dual of the Schwartz space of rapidly decreasing functions, and that the space of right continuous mappings of continuous functions to the Schwartz spaces is also tight.
Abstract: Let $C_\mathscr{I}' = C(\lbrack 0, 1\rbrack; \mathscr{I}')$ be the space of all continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$, where $\mathscr{L}'$ is the topological dual of the Schwartz space $\mathscr{I}$ of all rapidly decreasing functions. Let $C$ be the Banach space of all continuous functions on $\lbrack 0, 1\rbrack$. For each $\varphi \in \mathscr{I}, \Pi_\varphi$ is defined by $\Pi_\varphi:x \in C_\mathscr{I}' \rightarrow x. (\varphi) \in C$. Given a sequence of probability measures $\{P_n\}$ on $C_\mathscr{I}'$ such that for each $\varphi \in \mathscr{I}, \{P_n\Pi^{-1}_\varphi\}$ is tight in $C$, we prove that $\{P_n\}$ itself is tight in $C_\mathscr{I}'.$ A similar result is proved for the space of all right continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$.
213 citations
TL;DR: In this paper, the limiting behavior of sums of functions of nearest neighbor distances is studied for an m-dimensional sample, and a central limit theorem and moment bounds for such sums, and an invariance principle for the empirical process of nearest neighbour distances are both established.
Abstract: : The limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample. A central limit theorem and moment bounds for such sums, and an invariance principle for the empirical process of nearest neighbor distances are both established. As a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances.
181 citations
TL;DR: In this paper, Royden and Tsuji modified the Nash-Williams criterion to give a necessary and sufficient condition for a reversible countable state Markov chain to be transient, which can be used as a substitute for the criterion of this paper.
Abstract: An old argument of Royden and Tsuji is modified to give a necessary and sufficient condition for a reversible countable state Markov chain to be transient. This Royden criterion is quite convenient and can, on occasion, be used as a substitute for the criterion of Nash-Williams [6]. The result we give here yields a very simple proof that the Nash-Williams criterion implies recurrence. The Royden criterion also yields as a trivial corollary that a recurrent reversible random walk on a state space $X$ remains recurrent when it is constrained to run on a subset $X'$ of $X$. An apparently weaker criterion for transience is also given. As an application, we discuss the transience of a random walk on a horn shaped subset of $\mathbb{Z}^d$.
152 citations
TL;DR: In this paper, a one-to-one correspondence between Markov chains and a set of "entrance laws" associated with Markov specifications on regular infinite trees is established.
Abstract: Let $S$ and $A$ be countable sets and let $\mathscr{G}(\Pi)$ be the set of Markov random fields on $S^A$ (with the $\sigma$-field generated by the finite cylinder sets) corresponding to a specification $\Pi$, Markov with respect to a tree-like neighbour relation in $A$. We define the class $\mathscr{M}(\Pi)$ of Markov chains in $\mathscr{G}(\Pi)$, and generalise results of Spitzer to show that every extreme point of $\mathscr{G}(\Pi)$ belongs to $\mathscr{M}(\Pi)$. We establish a one-to-one correspondence between $\mathscr{M}(\Pi)$ and a set of "entrance laws" associated with $\Pi$. These results are applied to homogeneous Markov specifications on regular infinite trees. In particular for the case $|S| = 2$ we obtain a quick derivation of Spitzer's necessary and sufficient condition for $|\mathscr{G}(\Pi)| = 1$, and further show that if $|\mathscr{M}(\Pi)| > 1$ then $|\mathscr{M}(\Pi)| = \infty$.
150 citations
TL;DR: For a class of expansive transformations of the unit interval, the authors showed a local limit theorem for the process where f is a real bounded variation function, and showed that the speed in the central limit theorem is O(1/n) n.
Abstract: For a class of expansive transformations of the unit interval, we show a local limit theorem for the process $(f \circ T^n)_{n \in N}$, where $f$ is a real bounded variation function. We show also, that the speed in the central limit theorem is $1/\sqrt n$.
135 citations
TL;DR: In this paper, the univariate conditions of Gnedenko characterizing domains of attraction for univariate extreme value distributions are generalized to higher dimensions, and random variables with a multi-dimensional extreme value distribution are associated.
Abstract: The univariate conditions of Gnedenko characterizing domains of attraction for univariate extreme value distributions are generalized to higher dimensions. In addition, it is shown that random variables with a multivariate extreme value distribution are associated. Applications are given to a number of parametric families of joint distributions with given marginal distributions.
103 citations
TL;DR: In this paper, it was shown that the concave majorant of the empirical process is the inverse of a pure jump process with independent nonstationary increments and that Brownian motion can be generated by the latter process and Brownian excursions between values of the process at successive jump times.
Abstract: Let $S_t$ be a version of the slope at time $t$ of the concave majorant of Brownian motion on $\lbrack 0, \infty)$. It is shown that the process $S = \{1/S_t: t > 0\}$ is the inverse of a pure jump process with independent nonstationary increments and that Brownian motion can be generated by the latter process and Brownian excursions between values of the process at successive jump times. As an application the limiting distribution of the $L_2$-norm of the slope of the concave majorant of the empirical process is derived.
102 citations
TL;DR: In this paper, it was proved that the value of this game is approximately of order $2^{d/2}$ steps as $d \rightarrow \infty.
Abstract: Consider the number of steps needed by algorithms to locate the minimum of functions defined on the $d$-cube, where the functions are known to have no local minima except the global minimum. Regard this as a game: one player chooses a function, trying to make the number of steps needed large, while the other player chooses an algorithm, trying to make this number small. It is proved that the value of this game is approximately of order $2^{d/2}$ steps as $d \rightarrow \infty$. The key idea is that the hitting times of the random walk provide a random function for which no algorithm can locate the minimum within $2^{d(1/2 - \varepsilon)}$ steps.
93 citations
TL;DR: In this article, a strong law of large numbers for random sets whose values are compact convex subsets of a Banach space was proved for convex sets with cardinality 2.
Abstract: In this paper we prove a strong law of large numbers for random sets whose values are compact convex subsets of a Banach space.
TL;DR: In this article, the mixing and dependence conditions commonly used in extreme value theory for stationary sequences were established for the normalized partial sums under the mixing conditions. But the mixing condition was not considered in this paper.
Abstract: Let $\{X_n\}$ be a stationary sequence of random variables whose marginal distribution $F$ belongs to a stable domain of attraction with index $\alpha, 0 < \alpha < 2$. Under the mixing and dependence conditions commonly used in extreme value theory for stationary sequences, nonnormal stable limits are established for the normalized partial sums. The method of proof relies heavily on a recent paper by LePage, Woodroofe, and Zinn which makes the relationship between the asymptotic behavior of extreme values and partial sums exceedingly clear. Also, an example of a process which is an instantaneous function of a stationary Gaussian process with covariance function $r_n$ behaving like $r_n \log n \rightarrow 0$ as $n \rightarrow \infty$ is shown to satisfy these conditions.
TL;DR: In this article, a small deviation theorem of a new form for the functional central limit theorem for partial sums of independent, identically distributed finite-dimensional random vectors was proved, and applied to obtain a functional form of the Chung-Jain-Pruitt law of the iterated logarithm which is also a strong speed of convergence theorem refining Strassen's invariance principle.
Abstract: We prove a small deviation theorem of a new form for the functional central limit theorem for partial sums of independent, identically distributed finite-dimensional random vectors. The result is applied to obtain a functional form of the Chung-Jain-Pruitt law of the iterated logarithm which is also a strong speed of convergence theorem refining Strassen's invariance principle.
TL;DR: In this paper, a percolation construction which allows reducing problems about supercritical contact processes to problems about 1-dependent oriented percolations with density close to 1 was introduced.
Abstract: In this paper we introduce a percolation construction which allows us to reduce problems about supercritical contact processes to problems about 1-dependent oriented percolation with density $p$ close to 1. Using this method we obtain a number of results about the growth of supercritical contact processes and the wet region in oriented percolation. As a corollary to our results we find that the critical probability for oriented site percolation is greater than (>) that for bond percolation.
TL;DR: In this article, a class of empirical processes having the structure of $U$-statistics is considered, and the weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology.
Abstract: A class of empirical processes having the structure of $U$-statistics is considered. The weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology. As an application, a central limit theorem is derived for a very general class of non-parametric statistics.
TL;DR: In this article, the authors studied the Wiener process in terms of the zeros of the Airy function and the positive stable density of exponent 2/3 and found the distribution of the supremum of a certain stationary, mean zero, periodic Gaussian process, which corresponds to the limiting distribution of an optimal test statistic for the isotropy of a set of directions.
Abstract: Let $W(t), 0 \leq t \leq 1$, be the Wiener process tied down at $t = 0, t = 1; W(0) = W(1) = 0$. We find the distribution of $\sup_{0 \leq t \leq 1} W(t) - \int^1_0 W(t) dt$ in terms of the zeros of the Airy function and the positive stable density of exponent 2/3. This corresponds to the distribution of the supremum of a certain stationary, mean zero, periodic Gaussian process. It is also the limiting distribution of an optimal test statistic for the isotropy of a set of directions, proposed by G. S. Watson.
TL;DR: In this paper, the authors studied contact path processes, which are similar to branching random walks, in that the normalized size process starting from a singleton is a martingale which converges to a limit.
Abstract: We study some $\{0, 1, \cdots\}^{z^d}$ valued Markov interactions $\eta_t$ called contact path processes. These are similar to branching random walks, in that the normalized size process starting from a singleton is a martingale which converges to a limit $M_\infty$. In contrast to branching, however, $M_\infty$ depends on the spatial dynamics of the path process. The main result is an exact evaluation of the variance of $M_\infty$, achieved by means of the Feynman-Kac formula. The basic contact process of Harris may be viewed as a projection of $\eta_t$; as a corollary to the main result we obtain bounds on the contact process critical value $\lambda^{(d)}_c$ in dimension $d \geq 3$.
TL;DR: In this article, the authors consider the asymptotic behavior of the occupation time field and show that it has a weak limit field with covariance structure, whose covariance structures are explicitly computed.
Abstract: Let $\{\eta^\theta_s(x)\}, s \geq 0, x \in Z^d$ be the basic voter model starting from product measure with density $\theta(0 < \theta < 1).$ We consider the asymptotic behavior, as $t \rightarrow \infty$, of the occupation time field $\{T^x_t\}_{x \in Z^d}$, where $T^x_t = \int^t_0 \eta^\theta_s(x) ds$. Our main result is that, properly scaled and normalized, the occupation time field has a (weak) limit field as $t \rightarrow \infty$, whose covariance structure we compute explicitly. This field is Gaussian in dimensions $d \geq 2$. It is not Gaussian in dimension one, but has an "explicit" representation in terms of a system of coalescing Brownian motions. We also prove that $\lim_{t \rightarrow \infty} T^x_t/t = \theta$ a.s. for $d \geq 2$ (the result is false for $d = 1$). A striking feature of the behavior of the occupation time field is its elaborate dimension dependence.
TL;DR: In this article, it was shown that the system of annihilating simple random walks on the integer lattice with mutual annihilation of any pair of particles which collide is site recurrent, i.e., the initial configuration does not have any fixed sequence of holes with diameters tending to infinity.
Abstract: Consider a system of identical particles moving on the integer lattice with mutual annihilation of any pair of particles which collide. Apart from this interference, all particles move independently according to the same random walk $p$. A system will be called site recurrent if a.s. each site is occupied at arbitrarily large times. The following generalization of a conjecture by Erdos and Ney was open: the system of annihilating simple random walks on $Z_2$, starting with all sites except the origin occupied, is site recurrent. We prove, for general $p$ and a reasonably broad class of initial distributions, that the annihilating system is site recurrent. Loosely speaking, this condition is that the initial configuration does not have any fixed sequence of holes with diameters tending to infinity.
TL;DR: In this article, it is shown that there exists a unique dominating point associated with a convex set with non-empty interior, which leads to a representation formula from which sharp asymptotic estimates of the large deviation probabilities can be derived.
Abstract: Let $\mu(\cdot)$ be a probability measure on $\mathbb{R}^d$ and $B$ be a convex set with nonempty interior. It is shown that there exists a unique "dominating" point associated with $(\mu, B)$. This fact leads (via conjugate distributions) to a representation formula from which sharp asymptotic estimates of the large deviation probabilities $\mu^{\ast n}(nB)$ can be derived.
TL;DR: In this article, the authors studied the almost sure behaviour of big increments of the standard Wiener process with local time (occupation density) in the case of W(t) and W(s), and they showed that the results coincide only for Kesten's above mentioned LIL.
Abstract: Let $W(t)$ be a standard Wiener process with local time (occupation density) $L(x, t)$. Kesten showed that $L(0, t)$ and $\sup_x L(x, t)$ have the same LIL law as $W(t)$. Exploiting a famous theorem of P. Levy, namely that $\{\sup_{0 \leq s \leq t} W(s), t \geq 0\} {\underline{\underline{\mathscr{D}}}} \{L(0, t), t \geq 0\}$, we study the almost sure behaviour of big increments of $L(0, t)$ in $t$. The very same increment problems in $t$ of $L(x, t)$ are also studied uniformly in $x$. The results in the latter case are slightly different from those concerning $L(0, t)$, and they coincide only for Kesten's above mentioned LIL.
TL;DR: In this article, the Hartman-Wintner law of the iterated logarithm was shown to hold in the Banach space with a simple exponential inequality, and the same method was used to give a new, simpler proof of a basic result of Kuelbs on the LIL.
Abstract: A new proof of the Hartman-Wintner law of the iterated logarithm is given. The main new ingredient is a simple exponential inequality. The same method gives a new, simpler proof of a basic result of Kuelbs on the LIL in the Banach space setting.
TL;DR: In this paper, it was shown that the invariance result for i.i.d. sequences of random variables can also be obtained for the standard Wiener process under suitable conditions.
Abstract: Let $W(t)$ be a standardized Wiener process. In this paper we prove that $\lim \sup_{T\rightarrow\infty} \max_{a_T \leq t \leq T}\frac{|W(T) - W(T - t)|}{\{2t\lbrack\log(T/t) + \log \log t\rbrack\}^{1/2}} = 1 \text{a.s.}$ under suitable conditions on $a_T$. In addition we prove various other related results all of which are related to earlier work by Csorgo and Revesz. Let $\{X_k\}$ be an i.i.d. sequence of random variables and let $S_N = X_1 + \cdots + X_N$. Our original objective was to obtain results similar to the ones obtained for the Wiener process but with $N$ replacing $T$ and $S_N$ replacing $W(T)$. Using the work of Komlos, Major, and Tusnady on the invariance principle, we obtain the desired results for i.i.d. sequences as immediate corollaries to our work for the Wiener process.
TL;DR: In this paper, it was shown that the uniform empirical distribution function is asymptotically standard normal when centered at the largest concave majorant and normalized by a factor of 2.
Abstract: Let $F_n$ be the Uniform empirical distribution function. Write $\hat F_n$ for the (least) concave majorant of $F_n$, and let $\hat f_n$ denote the corresponding density. It is shown that $n \int^1_0 (\hat f_n(t) - 1)^2 dt$ is asymptotically standard normal when centered at $\log n$ and normalized by $(3 \log n)^{1/2}$. A similar result is obtained in the 2-sample case in which $\hat f_n$ is replaced by the slope of the convex minorant of $\bar F_m = F_m \circ H^{-1}_N$.
TL;DR: In this article, it was shown that the voter model is in equilibrium with parameter m(q, t), where t is the solution of the diffusion equation, and m(qs, t) is a deterministic field.
Abstract: We study the voter model on $\mathbb{Z}^d, d \geqq 3$, for a sequence $\mu^\varepsilon$ of initial states which have a gradient in the mean magnetization of the order $\varepsilon, \varepsilon \rightarrow 0$. We prove that the magnetization field $m^\varepsilon(f, t) = \varepsilon^d \sum f(\varepsilon x)\eta(x, \varepsilon^{-2}t)$ tends to a deterministic field $m(f, t) = \int dqf(q)m(q, t)$ as $\varepsilon \rightarrow 0. m(q, t)$ is the solution of the diffusion equation. The fluctuations of $m^\varepsilon(f, t)$ around its mean are given by an infinite dimensional, non-homogeneous Ornstein-Uhlenbeck process. In the limit as $\varepsilon \rightarrow 0$, locally, i.e. around $(\varepsilon^{-1}q, \varepsilon^{-2}t)$, the voter model is in equilibrium with parameter $m(q, t)$.
TL;DR: In this paper, it was shown that if $F$ is an IMRL (increasing mean residual life) distribution on $\lbrack 0, \infty)$ then:
Abstract: It is shown that if $F$ is an IMRL (increasing mean residual life) distribution on $\lbrack 0, \infty)$ then: $\max\{\sup_t |\bar F(t) - \bar G(t)|, \sup_t|\bar F(t) - e^{-t/\mu}|, \sup_t|\bar G(t) - e^{-t/\mu}|, \\ \sup_t |\bar G(t) - e^{-t/\mu_G}|\} = \frac{\rho}{\rho + 1} = 1 - \frac{\mu}{\mu_G}$ where $\bar F(t) = 1 - F(t), \mu = E_FX, \mu_2 = E_FX^2, G(t) = \mu^{-1} \int^t_0 \bar F(x) dx, \mu_G = E_GX = \mu_2/2\mu$, and $\rho = \mu_2/2\mu^2 - 1 = \mu_G/\mu - 1$. Thus if $F$ is IMRL and $\rho$ is small then $F$ and $G$ are approximately equal and exponentially distributed. IMRL distributions with small $\rho$ arise naturally in a class of first passage time distributions for Markov processes, as first illuminated by Keilson. The current results thus provide error bounds for exponential approximations of these distributions.
TL;DR: In this paper, a strictly stationary orthonormal sequence of random variables with a distribution that is not weakly convergent to the standard normal distribution has been studied, where the distribution is defined in terms of the variance of the random variables.
Abstract: For every sequence $(\varepsilon_n)_{n \in N}$ in (0, 1) there exists a strictly stationary orthonormal sequence $(X_n)_{n \in N}$ of random variables with $|P(A \cap B) - P(A)P(B)| \leq \varepsilon_n$ for all $A \in \sigma(X_1, \cdots, X_k), B \in \sigma(X_{k+n}, X_{k+n+1}, \cdots), k \in \mathbb{N}, n \in \mathbb{N}$, such that the distribution of $n^{-1/2} \sum^n_{i=1} X_i$ is not weakly convergent to the standard normal distribution.
TL;DR: In this paper, the Chi-squared approximation of the distribution of a sum of independent random variables is studied and it is shown that the error of approximation is of order $n −1 −1/2 as $n \rightarrow \infty.
Abstract: We suggest several Chi squared approximations to the distribution of a sum of independent random variables, and derive asymptotic expansions which show that the error of approximation is of order $n^{-1}$ as $n \rightarrow \infty$. The error may be reduced to $n^{-3/2}$ by making a simple secondary approximation.
TL;DR: A characterization of independence via uncorrelatedness for families satisfying positive and negative dependence conditions was shown in this article. But this characterization is not applicable to families satisfying strong positive (negative) orthant dependence.
Abstract: A characterization of independence via uncorrelatedness is shown to hold for the families satisfying positive and negative dependence conditions. For the associated random variables, the bounds on covariance functions obtained by Lebowitz (Comm. Math. Phys. $\mathbf{28}$ (1972), 313-321) readily yield such a characterization. An elementary proof for the same characterization is also given for a condition weaker than association, labeled as "strong positive (negative) orthant dependence." This condition is compared with the "linear positive dependence," under which Newman and Wright (Ann. Probab. $\mathbf{9}$ (1981), 671-675) obtained the characterization.