Showing papers in "Annals of Probability in 1984"
TL;DR: In this article, the elements of an arbitrary max-stable sequence are exhibited as functionals of a 2-dimensional Poisson point process, and the result is extended to a continuous time maxstable process that is continuous in probability.
Abstract: The elements of an arbitrary max-stable sequence are exhibited as functionals of a 2-dimensional Poisson point process. The result is extended to a continuous time max-stable process that is continuous in probability. We define an analogue of a stochastic integral appropriate for this context.
590 citations
TL;DR: In this paper, a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions was provided, and a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets.
Abstract: In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\mathscr{F}$ From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition) The case of unbounded $\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case The results for empiricals also yield some new CLT's in $C\lbrack 0, 1\rbrack$ and $D\lbrack 0, 1\rbrack$
478 citations
TL;DR: A self-contained survey of most of the results known about oriented percolation can be found in this article, along with a discussion of some of the most important topics. But this survey is limited to oriented percoding.
Abstract: This paper is a self-contained survey of most of the results known about oriented percolation. The table of contents below should give an idea of the topics which will be covered. A more detailed account can be found in the first section.
433 citations
TL;DR: In this article, the authors proved large deviation theorems for a general class of random vectors taking values in √ R √ d$ and in certain infinite dimensional spaces based on convexity methods.
Abstract: This paper proves large deviation theorems for a general class of random vectors taking values in $\mathbb{R}^d$ and in certain infinite dimensional spaces. The proofs are based on convexity methods. As an application, we give a new proof of the large deviation property of the empirical measures of finite state Markov chains (originally proved by M. Donsker and S. Varadhan). We also discuss a new notion of stochastic convergence, called exponential convergence, which is closely related to the large deviation results.
422 citations
TL;DR: In this paper, it was shown that the limiting conditional distribution of (either) $X_i$ is characterized as a member of the exponential family determined by the unconditional distribution $P_X, while (X_1, \cdots, X_n) are conditionally asymptotically quasi-independent.
Abstract: Known results on the asymptotic behavior of the probability that the empirical distribution $\hat P_n$ of an i.i.d. sample $X_1, \cdots, X_n$ belongs to a given convex set $\Pi$ of probability measures, and new results on that of the joint distribution of $X_1, \cdots, X_n$ under the condition $\hat P_n \in \Pi$ are obtained simultaneously, using an information-theoretic identity. The main theorem involves the concept of asymptotic quasi-independence introduced in the paper. In the particular case when $\hat P_n \in \Pi$ is the event that the sample mean of a $V$-valued statistic $\psi$ is in a given convex subset of $V$, a locally convex topological vector space, the limiting conditional distribution of (either) $X_i$ is characterized as a member of the exponential family determined by $\psi$ through the unconditional distribution $P_X$, while $X_1, \cdots, X_n$ are conditionally asymptotically quasi-independent.
380 citations
TL;DR: Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal as mentioned in this paper showed that the unconditional constant of a monotone basis of $L^p(0, 1)$ is Ω(p + 1) − 1.
Abstract: Let $p^\ast$ be the maximum of $p$ and $q$ where $1 0$. This improves an earlier inequality of the author by giving the best constant and conditions for equality. The inequality holds with the same constant if $\varepsilon$ is replaced by a real-valued predictable sequence uniformly bounded in absolute value by 1, thus yielding a similar inequality for stochastic integrals. The underlying method rests on finding an upper or a lower solution to a novel boundary value problem, a problem with no solution (the upper is not equal to the lower solution) except in the special case $p = 2$. The inequality above, in combination with the work of Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal, implies that the unconditional constant of a monotone basis of $L^p(0, 1)$ is $p^\ast - 1$. The paper also contains a number of other sharp inequalities for martingale transforms and stochastic integrals. Along with other applications, these provide answers to some questions that arise naturally in the study of the optimal control of martingales.
356 citations
TL;DR: In this paper, the authors consider a sequence of r.l.v.'s and obtain functions f_\beta(n)$ such that f_n\|_β = o(f_β(n))$ for some ε > 0.
Abstract: Let $(X_n)_{n \in \mathscr{X J}}$ be a sequence of r.v.'s with $E X_n = 0, E(\sum^n_{i = 1} X_i)^2/n \rightarrow \sigma^2 > 0, \sup_{n,m}E(\sum^{m + n}_{i = m + 1} X_i)^2/n 1$, or $\alpha(k) = O(b^{-k})$ for some $b > 1$, we obtain functions $f_\beta(n)$ such that $\|X_n\|_\beta = o(f_\beta(n))$ for some $\beta \in (2, \infty\rbrack$ is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if $\|X_n\|_\beta = O(f_\beta(n))$.
278 citations
TL;DR: In this article, it is shown that certain measure-valued stochastic processes describing the time evolution of systems of weakly interacting particles converge in the limit, when the particle number goes to infinity, to a deterministic nonlinear process.
Abstract: It is shown that certain measure-valued stochastic processes describing the time evolution of systems of weakly interacting particles converge in the limit, when the particle number goes to infinity, to a deterministic nonlinear process.
229 citations
TL;DR: In this paper, sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class of functions are proved under several kinds of conditions on the functions.
Abstract: Sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class $\mathscr{F}$ of functions are proved under several kinds of conditions on $\mathscr{F}$. These bounds are used to establish laws of the iterated logarithm for this supremum and to obtain rates of convergence in total variation for empirical processes on the integers.
218 citations
TL;DR: In this article, it was shown that for every i.i.d. (i.i) > 0, the threshold stopping rule for the distribution of the distribution is 2EX(n, t(m) = n/t(m), where t(c) is the threshold for the smallest i and t(n) is defined as the threshold.
Abstract: Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.
201 citations
TL;DR: In this article, the large deviation principle for stochastic processes with stationary and independent increments has been studied under the weak$^\ast$-topology, where the moment generating function of the increments is assumed to lie in the space of functions of bounded variation.
Abstract: Let $\mathscr{X}$ be a topological space and $\mathscr{F}$ denote the Borel $\sigma$-field in $\mathscr{X}$. A family of probability measures $\{P_\lambda\}$ is said to obey the large deviation principle (LDP) with rate function $I(\cdot)$ if $P_\lambda(A)$ can be suitably approximated by $\exp\{-\lambda \inf_{x\in A}I(x)\}$ for appropriate sets $A$ in $\mathscr{F}$. Here the LDP is studied for probability measures induced by stochastic processes with stationary and independent increments which have no Gaussian component. It is assumed that the moment generating function of the increments exists and thus the sample paths of such stochastic processes lie in the space of functions of bounded variation. The LDP for such processes is obtained under the weak$^\ast$-topology. This covers a case which was ruled out in the earlier work of Varadhan (1966). As applications, the large deviation principle for the Poisson, Gamma and Dirichlet processes are obtained.
TL;DR: In this paper, the authors derived sharp finite sample estimates and exact almost sure limit results for local deviations of multivariate empirical processes, which are useful for obtaining, e.g., exact convergence rates of multiivariate kernel density estimators.
Abstract: We derive sharp finite sample estimates and exact almost sure limit results for local deviations of multivariate empirical processes. These are useful for obtaining, e.g., exact convergence rates of multivariate kernel density estimators. It is also indicated how local properties of multivariate empirical processes may be used to study various problems in nonparametric multivariate analysis.
TL;DR: The validity of May's criteria for these systems under more restrictive hypotheses remains open question as discussed by the authors, and the validity of these criteria for more general distributions of random matrices is also open.
Abstract: Let $A(1), A(2), \cdots$ be a sequence of independent identically distributed (iid) random real $n \times n$ matrices and let $x(t) = A(t)x(t - 1), t = 1, 2, \cdots$ Define $\bar{\lambda}_n = \sup\{\lim_{t \uparrow \infty}\|x(t)\|^{1/t}: 0
eq x(0) \in R^n\}$ where $\|\cdot\|$ denotes, eg the Euclidean norm, providing the limit exists almost surely (as) and is nonrandom, and define $\underline\lambda_n$ analogously with sup replaced by inf If all $n^2$ entries of each $A(t)$ are iid standard symmetric stable random variables of exponent $\alpha$, then $\underline\lambda_n = \overline\lambda_n = \lambda_n(\alpha)$ In the standard normal case $(\alpha = 2), \lambda_n(2) = (2 \exp\lbrack\psi(n/2)\rbrack)^{1/2}$, where $\psi$ is the digamma function, and $n^{-1/2}\lambda_n(2) \rightarrow 1$; for $0 1)$ are investigated for more general distributions of $A(t)$ We obtain, for example, the general bound, $\overline\lambda_n \leq \{r\lbrack E(A(1)^T A(1))\rbrack\}^{1/2}$, where $A^T$ is the transpose of $A$ and $r$ denotes the spectral radius In the case of independent entries of mean zero and common variance $s^2/n$, this leads to $\lim \sup_n \overline\lambda_n \leq s$ If the entries of $A(t)$ are iid and distributed as $W/n^{1/2}$ where $W$ is independent of $n$, has mean zero, variance $s^2$ and satisfies $E(\exp\lbrack iuW\rbrack) = O(|u|^{-\delta})$ as $|u| \uparrow \infty$ for some $\delta > 0$, then $\lim \inf_n\bar\lambda_n \geq s$ These conditions for the asymptotic stability or instability of a product of random matrices are of the form originally proposed by May for differential equations governed by a single random matrix We give counterexamples to show that May's criteria for the system of linear ordinary differential equations that he considered are not valid in the generality originally proposed, nor are they valid for the related system of difference equations considered by Hastings The validity of May's criteria for these systems under more restrictive hypotheses remains an open question
TL;DR: In this article, the authors study the set of confluences in the plane where all paths meet and also the set $M_0$ of times the paths meet in a plane.
Abstract: We envision a network of $N$ paths in the plane determined by $N$ independent, two-dimensional Brownian motions $W_i(t), t \geq 0, i = 1, 2, \cdots, N$. Our problem is to study the set of "confluences" $z$ in $\mathbb{R}^2$ where all $N$ paths meet and also the set $M_0$ of $N$-tuples of times $\mathbf{t} = (t_1, \cdots, t_N)$ at which confluences occur: $M_0 = \{\mathbf{t}: W_1(t_1) = \cdots = W_N(t_N)\}$. The random set $M_0$ is analyzed by constructing a convenient stochastic process $X$, which we call "confluent Brownian motion", for which $M_0 = X^{-1}(0)$ and using the theory of occupation densities. The problem of confluences is closely related to that of multiple points for a single process. Some of our work is motivated by Symanzik's use of Brownian local time in quantum field theory.
TL;DR: A central limit theorem for families of associated random variables indexed by subsets of the Lambda(N) of the voter model was proved in this article, where the moments of the moments were replaced by conditions on the voters' moments, which is an extension of the Newman-Wright invariance principle.
Abstract: We prove a central limit theorem for families $\{X_n(N): \mathbf{n} \in \Lambda(N)\}$ of associated random variables indexed by subsets $\Lambda(N)$ of $\mathbb{Z}^d$, as $N \rightarrow \infty$; this is an extension of the Newman-Wright invariance principle for associated stationary sequences $\{X_n: n \geq 1\}$ satisfying $\sum_n \operatorname{cov}(X_1, X_n) < \infty$, but with the stationarity property replaced by conditions on the moments of the $X$'s. The theorem has applications to the voter model and the percolation model. In the latter case, it provides an extension of a central limit theorem of the authors [4], by reducing the severity of the moment conditions. Also, we prove a central limit theorem for certain non-stationary non-associated families of random variables which arise in percolation theory. This includes, for example, a central limit theorem for the number of open clusters contained within the circuit $\gamma(n)$ of $\mathbb{Z}^2$, where $\{\gamma(n)\}$ is a sequence of circuits which satisfy a regularity condition and whose interiors $\{\overset{\circ}\gamma(n)\}$ satisfy $|\overset{\circ}\gamma(n)| \rightarrow \infty$ as $n \rightarrow \infty$.
TL;DR: In this paper, the authors propose to unify best choice problems under total ignorance of both the candidates, quality distribution and the distribution of the number of candidates, and the result is what they shall call the $e^{-1}$-law.
Abstract: This article tries to unify best choice problems under total ignorance of both the candidates, quality distribution and the distribution of the number of candidates. The result is what we shall call the $e^{-1}$-law because of the multiple role which is played by $e^{-1}$, and this in a more general context as only in the solution of the classical secretary problem. The unification is possible whenever best choice problems can be redefined as continuous time decision problems on conditionally independent arrivals. We shall also give several examples to illustrate how the approach and its implications compare with other models.
TL;DR: Oseledec's Multiplicative Ergodic Theorem is used to give recurrence and transience criteria for random walk in a random environment on the integers in this article, which generalize those given by Solomon in the nearest-neighbor case.
Abstract: Oseledec's Multiplicative Ergodic Theorem is used to give recurrence and transience criteria for random walk in a random environment on the integers. These criteria generalize those given by Solomon in the nearest-neighbor case. The methodology for random environments is then applied to Markov chains with periodic transition functions to obtain recurrence and transience criteria for these processes as well.
TL;DR: In this article, the problem of allocating effort among several competing projects, the states of which evolve according to one-dimensional diffusion processes, is discussed, and it is shown that the play-the-leader policy of continuing the project with the leading Gittins index is optimal, and very explicit computations of the index are offered.
Abstract: We discuss the problem of allocating effort among several competing projects, the states of which evolve according to one-dimensional diffusion processes. It is shown that the "play-the-leader" policy of continuing the project with the leading Gittins index is optimal, and very explicit computations of the index are offered. The question of constructing the diffusions according to the above policy is also addressed.
TL;DR: In this paper, the joint density of Brownian motion, its local time at the origin, and its occupation time were computed, and two derivations of the main result were offered; one is computational, whereas the other uses some of the deep properties of local time.
Abstract: We compute the joint density of Brownian motion, its local time at the origin, and its occupation time of $\lbrack 0, \infty)$. Two derivations of the main result are offered; one is computational, whereas the other uses some of the deep properties of Brownian local time. We use the result to compute the transition probabilities of the optimal process in a stochastic control problem.
TL;DR: In this paper, the asymptotic covariances of the particle number field are computed in the particular model of a zero range jump process in equilibrium, and the main tool in that computation is a general theorem, whose validity is established for the given class of processes, which states that the asyptotic behaviour of any field corresponding to a local function is determined by a suitable projection of that field on the linear (here one-dimensional) space spanned by the fields of conserved quantities.
Abstract: In the particular model of a zero range jump process in equilibrium, the asymptotic covariances of the--spatially and temporally rescaled--particle number field are computed. The main tool in that computation is a general theorem, whose validity is established for the given class of processes, which states that the asymptotic behaviour of the covariances of any field corresponding to a local function is determined by a suitable projection of that field on the linear (here one-dimensional) space spanned by the fields of conserved quantities (here: the particle number).
TL;DR: In this article, the authors show how to use local times to analyze the self-intersections of random fields and compute the Hausdorff dimension of $r$-multiple times for Brownian motion in the plane, Brownian sheets and Levy's multiparameter Brownian motions.
Abstract: We show how to use local times to analyze the self-intersections of random fields. In particular, we compute the Hausdorff dimension of $r$-multiple times for Brownian motion in the plane, Brownian sheets and Levy's multiparameter Brownian motion.
TL;DR: Asymptotic normality is proven for spectral density estimates under strong mixing and a limited number of moment conditions for the process analyzed in this article, which holds for a large class of processes that are not linear and do not require the existence of all moments.
Abstract: Asymptotic normality is proven for spectral density estimates assuming strong mixing and a limited number of moment conditions for the process analyzed. The result holds for a large class of processes that are not linear and does not require the existence of all moments.
TL;DR: In this article, the authors give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias, where the expected number of flips is the measure of efficiency.
Abstract: We give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias. We are interested in efficient algorithms, where the expected number of flips is our measure of efficiency. Other authors have represented algorithms as lattices, but by representing them instead as trees we are able to produce an algorithm more efficient than any previously appearing. We also prove a conjecture of Hoeffding and Simons that there is no optimal algorithm. Further, we consider generalizations where the input is a sequence of iid discrete random variables and the output is a uniform random variable with $N$ possible outcomes. In this setting we provide an algorithm significantly superior to those previously published.
TL;DR: In this paper, the convergence in distribution to Gaussian, generalized Poisson and infinitely divisible laws of the row sums of certain π-mixing triangular arrays of Banach space valued random vectors with stationary rows is studied.
Abstract: This paper deals with the convergence in distribution to Gaussian, generalized Poisson and infinitely divisible laws of the row sums of certain $\phi$ or $\psi$-mixing triangular arrays of Banach space valued random vectors with stationary rows. Necessary and sufficient conditions for convergence in terms of individual r.v.'s are proved. These include sufficient conditions for the convergence to a stable law of the normalized sums of certain $\phi$-mixing, stationary sequences. An invariance principle for stationary, $\phi$-mixing triangular arrays is given.
TL;DR: In this paper, it was shown that all stable densities are Bell-shaped (i.e. their $k$th derivative has exactly $k $ zeros and they are simple) thereby generalizing the well-known property of the normal distribution and associated Hermite polynomials.
Abstract: The central result of this paper consists in proving that all stable densities are bell-shaped (i.e. its $k$th derivative has exactly $k$ zeros and they are simple) thereby generalizing the well-known property of the normal distribution and the associated Hermite polynomials.
TL;DR: In this article, the relation between distributions of random closed sets and their hitting functions is studied, where the hitting functions are defined for Borel sets, and a sequence of random sets converges in distribution iff the corresponding sequence of hitting functions converges on some sufficiently large class of bounded Borel set.
Abstract: We study the relation between distributions of random closed sets and their hitting functions $T$, defined by $T(B) = P\{\varphi \cap B
eq \varnothing\}$ for Borel sets $B$. In particular, a sequence of random sets converges in distribution iff the corresponding sequence of hitting functions converges on some sufficiently large class of bounded Borel sets. This class may be chosen to be countable.
TL;DR: In this article, the authors present a limiting procedure that is based on the conservation laws of the movement of the particles in a system of infinitely many particles in Rd (or Xd) moving according to
Abstract: small volume is so large compared with the distances between molecules that it contains an infinite number of them which are in equilibrium with given characteristics (density, temperature, * * .). Of course these parameters vary with x (say p(x), T(x), * * .). If we look at the system after a time t, the system will be still locally in equilibrium with other local characteristics (p(x, t), T(x, t), * * *) varying slowly in time although each particle individually moves very rapidly. We want to derive the equation of evolution of these macroscopical parameters." However, it is surprising that this derivation is almost never based on the analysis of the movement of the particles but rather on "conservation laws." The limiting procedure that we describe below has been used by various authors [2, 8] in efforts to give rigorous treatment of the program alluded to above. Consider a system of infinitely many particles in Rd (or Xd) moving according
TL;DR: In this paper, it was shown that if the 1st moment of a Dirichlet process exists, then the distribution of the mean is different from that of a normal distribution, except for a degenerate case.
Abstract: Let $P$ be a Dirichlet process with parameter $\alpha$ on $(R, B)$, where $R$ is the real line, $B$ is the $\sigma$-field of Borel subsets of $R$ and $\alpha$ is a non-null finite measure on $(R, B)$. By the use of characteristic functions we show that if $Q(\cdot) = \alpha(\cdot)/\alpha(R)$ is a Cauchy distribution then the mean $\int_R x dP(x)$ has the same Cauchy distribution and that if $Q$ is normal then the distribution of the mean can be roughly approximated by a normal distribution. If the 1st moment of $Q$ exists, then the distribution of the mean is different from $Q$ except for a degenerate case. Similar results hold also in the multivariate case.
TL;DR: The potential theory of L may then be studied with the aid of the associated process ξ: a function h on the state space is (sub)harmonic if h(ξ) is a sub-martingale, the solution to the Dirichlet problem is obtained through the first hitting distributions, and so on.
Abstract: There is a well–known connection between elliptic and parabolic differential operators of second order and diffusion processes. It is possible to construct for certain of these operators L a diffusion process ξ whose characteristic operator is an extension of L (cf. e.g., Dynkin [9], for a survey). The potential theory of L may then be studied with the aid of the associated process ξ: A function h on the state space is (sub)harmonic if h(ξ) is a (sub)martingale, the solution to the Dirichlet problem is obtained through the first hitting distributions, and so on.
TL;DR: In this article, the weak limit of multivariate empirical processes was shown to be λ 2(k-1)-exponential for large λ and appropriate constants (c, C) when k = 2.
Abstract: We consider multivariate empirical processes $X_n(t) := \sqrt n (F_n(t) - F(t))$, where $F_n$ is an empirical distribution function based on i.i.d. variables with distribution function $F$ and $t \in \mathbb{R}^k$. For $X_F$ the weak limit of $X_n$, it is shown that $c(F, k)\lambda^{2(k-1)}e^{-2\lambda^2} \leq P\big\{\sup_t X_F(t) > \lambda\big\} \leq C(k)\lambda^{2(k-1)}e^{-2\lambda^2}$ for large $\lambda$ and appropriate constants $c, C$. When $k = 2$ these constants can be identified, thus permitting the development of Kolmogorov--Smirnov tests for bivariate problems. For general $k$ the bound can be used to obtain sharp upper-lower class results for the growth of $\sup_tX_n(t)$ with $n$.