A Law of the Iterated Logarithm for Randomly Stopped Sums of Heavy Tailed Random Vectors

 Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further that is a sequence of positive integer valued random variables such that for some for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

Strong and conditional invariance principles for samples attracted to stable laws

Summary. We prove almost sure convergence of a representation of normalized partial sum processes of a sequence of i.i.d. random variables from the domain of attraction of an α-stable law, α<2. We obtain an explicit form of the limit in terms of the LePage series representation of stable laws. One consequence of these results is a conditional invariance principle having applications to option pricing as well as to resampling by signs and permutations. Received: 11 April 1994 / In revised form: 5 November 1996     

Convergence rates in the strong laws of asymptotically negatively associated random fields

In this paper, a notion of negative side p-mixing (p -mixing) which can be regardedas asymptotic negative association is defined, and some Rosenthal type inequalities for p -mix-ing random fields are established. The complete convergence and almost sure summability onthe convergence rates with respect to the strong law of large numbers are also discussed for p--mixing random fields. The results obtained extend those for negatively associated sequences andp“ -mixing random fields.     

Valleys and the Maximum Local Time for Random Walk in Random Environment

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ^*(n) = max _x ξ(n, x). It is known that lim sup is a positive constant a.s. We prove that lim inf is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.     

Laplace approximations for sums of independent random vectors

Let X _i , i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a mapping B→R. Under a central limit theorem assumption, an asymptotic evaluation of Z _n = E (exp (n Φ (∑ _{i =1} ⁿ X _i /n))), up to a factor (1 + o(1)), has been gotten in Bolthausen [1]. In this paper, we show that the same asymptotic evaluation can be gotten without the central limit theorem assumption. Received: 19 September 1997 / Revised version:22 April 1999     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Iterates of expanding maps

The iterates of expanding maps of the unit interval into itself have many of the properties of a more conventional stochastic process, when the expanding map satisfies some regularity conditions and when the starting point is suitably chosen at random. In this paper, we show that the sequence of iterates can be closely tied to an m-dependent process. This enables us to prove good bounds on the accuracy of Gaussian approximations. Our main tools are coupling and Stein's method. Received: 27 June 1997 / Revised version: 21 September 1998     

Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces

Summary.   Let X,X ₁,X ₂,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ ^d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ ^d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S _N ] of the normalized sums S _N =N ^−1/2(X ₁+⋯+X _N ) and show that

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤ^ν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2. Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001     

The perimeter of uniform and geometric words: a probabilistic analysis

Abstract

Let a word be a sequence of n i.i.d. integer random variables. The perimeter P of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of P. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of P is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motion.     

Complete convergence for arrays

Let {(X _nk , 1≤k≤n),n≥1}, be an array of rowwise independent random variables. We extend and generalize some recent results due to Hu, Móricz and Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means.     

Asymptotic independence of maximum waiting times for increasing alphabet

For everyk≥1 consider the waiting time until each pattern of lengthk over a fixed alphabet of sizen appears at least once in an infinite sequence of independent, uniformly distributed random letters. Lettingn→∞ we determine the limiting finite dimensional joint distributions of these waiting times after suitable normalization and provide an estimate for the rate of convergence. It will turn out that these waiting times are getting independent. Research supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905.     

一类独立随机场的强定律的收敛率

§ 1　IntroductionDefinition1 .[1 ] A field{ Xi,i∈Nd} is called negatively associated(NA) if for every pair ofdisjoint subsets T1 ,T2 of Nd,Cov(f1 (Xi,i∈ T1 ) ,f2 (Xj,j∈ T2 ) )≤ 0 ,whenever f1 and f2 are coordinatewise increasing.Definition2 .[1 ] A field{ Xi,i∈Nd} is calledρ* -mixing ifρ* (s) =sup{ (ρ(S,T) ;S,T N,dist(S,T)≥ s}→ 0 (s→∞ ) ,whereρ(S,T) =sup{ |E(f -Ef) (g -Eg) |/‖ f -Ef‖2 ‖ g -Eg‖2 ,f∈ L2 (σ(S) ) ,g∈ L2 (σ(T) ) } .Definition 3.[1 ] A field { Xi…     

On the number of lattice points between two enlarged and randomly shifted,copies of an oval

Summary Let A be an oval with a nice boundary in ²,R a large positive number,c>0 some fixed number and a uniformly distributed random vector in the unit square [0,1]². We are interested in the number of lattice points in the shifted annular region consisting of the difference of the sets {(R+c/R)A–} and {(R–c/R)A–}. We prove that whenR tends to infinity, the expectation and the variance of this random variable tend to 4c times the area of the set A, i.e. to the area of the domain where we are counting the number of lattice points. This is consistent with computer studies in the case of a circle or an ellipse which indicate that the distribution of this random variable tends to the Poisson law. We also make some comments about possible generalizations.     

Two renewal theorems for general random walks tending to infinity

Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S _n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S _n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S _n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given. Received: 18 September 1995/In revised form: 28 February 1996     

A bounded N-tuplewise independent and identically distributed counterexample to the CLT

Summary. A sequence of random variables X ₁,X ₂,X ₃,… is said to be N-tuplewise independent if X i ₁,X i ₂,…,X i _N are independent whenever (i ₁,i ₂,…,i _N ) is an N-tuple of distinct positive integers. For any fixed N∈ℤ⁺, we construct a sequence of bounded identically distributed N-tuplewise independent random variables which fail to satisfy the central limit theorem. Received: 17 May 1996 / In revised form: 28 January 1998     

Stable windings on hyperbolic surfaces

Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T ¹ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t ^{−1/(2δ−1)}, for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves. Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997