Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Summary Consider the stationary sequenceX ₁=G(Z ₁),X ₂=G(Z ₂),..., whereG(·) is an arbitrary Borel function andZ ₁,Z ₂,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z ₁ Z _k+1) satisfyingr(0)=1 and _k=1 |r(k)| ^m < , where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X _n }, the integerm is the Hermite rank of the family {I{G(·) x} –F(x):xR}. LetF _n (·) be the empirical distribution function ofX ₁,...,X _n . We prove that, asn, the empirical processn ^1/2{F _n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[–,].Partially supported by NSF Grant DMS-9208067     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

On Berry-Esséen rates,a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean

Let F_n(x) be the empirical distribution function based on n independent random variables X₁,…,X_n from a common distribution function F(x), and let $\text{X}\text{= Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{X}{}_{\mathrm{i}}\text{n}$ be the sample mean. We derive the rate of convergence of $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$ to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for $\text{F}{}_{\mathrm{n}}\text{(}\text{X}\text{)}$.     

Reverse processes and some limit theorems of multitype Galton-Watson processes

We classify the reverse process {X_n} of a multitype Galton-Watson process {Z_n}. In the positive recurrent cases we give the stationary measure for {X_n} explicitly, and in the critical case, supposing that all the second moments of Z₁ are finite, we establish the convergence in law to a gamma distribution. Limit distributions of {Z_cn}, 0 < c < 1, conditioned on Z_n, are also given in the subcritical, supercritical and critical cases, respectively. These extend the previous one-type work of W. W. Esty.     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes for which there is a finite K () such that the conditional distribution of X₁ given the entire past is equal to the conditional distribution of X₁ given only . The least such value of K is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of K, both in the backward sense that we have just described and in the forward sense, where one observes successive values of {Xn} for n?0 and asks for the least value K such that the conditional distribution of Xn+1 given is the same as the conditional distribution of Xn+1 given . We allow for finite or countably infinite alphabet size.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

Hyperamarts: Conditions for regularity of continuous parameter processes

The regularity of trajectories of continuous parameter process (X_t)_t∈R+ in terms of the convergence of sequence E(X_Tn) for monotone sequences (T_n) of stopping times is investigated. The following result for the discrete parameter case generalizes the convergence theorems for closed martingales: For an adapted sequence (X_n)_1≤n≤∞ of integrable random variables, lim X_n exists and is equal to X_∞ and (X_T) is uniformly integrable over the set of all extended stopping times T, if and only if lim E(X_Tn) = E(X_∞) for every increasing sequence (T_n) of extended simple stopping times converging to ∞. By applying these discrete parameter theorems, convergence theorems about continuous parameter processes are obtained. For example, it is shown that a progressive, optionally separable process (X_t)_t∈R+ with E{X_T} < ∞ for every bounded stopping time T is right continuous if lim E(X_Tn) = E(X_T) for every bounded stopping time T and every descending sequence (T_n) of bounded stopping times converging to T. Also, Riesz decomposition of a hyperamart is obtained.     

Amarts: A class of asymptotic martingales a. Discrete parameter

A sequence (X_n) of random variables adapted to an ascending (asc.) sequence $\text{F}$_n of σ-algebras is an amart iff EX_τ converges as τ runs over the set T of bounded stopping times. An analogous definition is given for a descending (desc.) sequence $\text{F}$_n. A systematic treatment of amarts is given. Some results are: Martingales and quasimartingales are amarts. Supremum and infimum of two amarts are amarts (in the asc. case assuming L₁-boundedness). A desc. amart and an asc. L₁-bounded amart converge a.e. (Theorem 2.3; only the desc. case is new). In the desc. case, an adapted sequence such that (EX_τ)_τ∈T is bounded is uniformly integrable (Theorem 2.9). If X_n is an amart such that sup_nE(X_n ? X_n?1)² < ∞, then $\text{X}{}_{\mathrm{n}}\text{n}$ converges a.e. (Theorem 3.3). An asc. amart can be written uniquely as Y_n + Z_n where Y_n is a martingale, and Z_n → 0 in L¹. Then Z_n → 0 a.e. and Z_τ is uniformly integrable (Theorem 3.2). If X_n is an asc. amart, τ_k a sequence of bounded stopping times, k ≤ τ_k, and E(sup_k |X_τk ? X_k?1|) < ∞, then there exists a set G such that X_n → a.e. on G and lim inf X_n = ?∞, lim sup X_n = +∞ on G^c (Theorem 2.7). Let E be a Banach space with the Radon-Nikodym property and separable dual. In the definition of an E-valued amart, Pettis integral is used. A desc. amart converges a.e. on the set {lim sup 6X_n6 < ∞}. An asc. or desc. amart converges a.e. weakly if sup_TE6X_τ6 < ∞ (Theorem 5.2; only the desc. case is new).     

Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Convergence in distribution of quotients of order statistics

Let X₁, X₂,… be i.i.d. random variables with continuous distribution function F < 1. It is known that if 1 - F(x) varies regularly of order - p, the successive quotients of the order statistics in decreasing order of X₁,…,X_n are asymptotically independent, as n→∞, with distribution functions x^kp, k = 1, 2, …. A strong converse is proved, viz. convergence in distribution of this type of one of the quotients implies regular varation of 1 - F(x).     

Sharp estimates for the CDF of quadratic forms of MPE random vectors

In this paper we develop an efficient analytical expansion of the cumulative distribution function (cdf) XBX^t where X=(X₁,…,X_n+1) with n≥2, follows a multivariate power exponential distribution (MPE). Our approach provides a sharp estimate of the cumulative distribution function of a quadratic form of MPE, together with explicit error estimates.