Showing papers in "Probability Theory and Related Fields in 1974"

On extreme values in stationary sequences

Abstract: In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are: (i) To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences. (ii) To obtain limiting laws of the form $$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$ where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results. (iii) As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d. (iv) To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.

Pointwise convergence in terms of expectations

Abstract: This paper is concerned with the connection between almost sure convergence of a sequence of random variables and convergence of certain related expectations. Theorems of the kind we are interested in were proved by Meyer I-7, p. 232] and Mertens [6, p. 47-1 in the continuous-parameter case, and by Baxter 1-11 in the discrete-parameter case. For example, Baxter's theorem is the following: Let (X~)~_>_I be a sequence of random variables with values in a compact metric space S, and let the set F of bounded stopping times be directed by the obvious ordering. Then (X~)n_>l converges almost surely if and only if the generalized sequence (6 ~b (X~))~r of expectations converges for every real-valued continuous function q5 on S. In the present paper we generalize this theorem in two ways: we replace S by an arbitrary complete separable metric space, and we use as few test functions ~b as possible. IfS is the real line, the single test function ~b (x) = x suffices (Theorem 2); for any complete separable metric space, a countable set of functions suffices (Theorem 3); and for a separable Banach space, there is a countable set of convex functions which suffices (Theorem 4). We have included a different proof of the key step in Baxter's proof (Corollary 1), in order to make the present paper selfcontained.

Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes

Abstract: In this paper, we find the limit set of a sequence (2 log n)−1/2X n (t), n≧3) of Gaussian processes in C [0,1], where the processes X n (t) are defined on the same probability space and have the same distribution. Our result generalizes the theorems of Oodaira and Strassen, and we also apply it to obtain limit theorems for stationary Gaussian processes, moving averages of the type \(\int\limits_0^t {f\left( {t - s} \right)dW\left( s \right)} \), where W(s) is the standard Wiener process, and other Gaussian processes. Using certain properties of the unit ball of the reproducing kernel Hubert space of X n (t), we derive the usual law of the iterated logarithm for Gaussian processes. The case of multidimensional time is also considered.

H p -spaces of martingales, 0<p≦1

Abstract: For 0

The maximum term of uniformly mixing stationary processes

Abstract: Let {X n } be a uniformly (or strongly) mixing stationary process and let Z n =max(X 1, X 2,..., X n ). For ξ>0, let c n (ξ)=inf {xeR: n P(X 1>x)≦ξ}. Under a condition which holds for all ϕ-mixing processes, necessary and sufficient conditions are given for P(Zn≦cn(ξ)) to converge to each possible limit. Some conditions for convergence of P(Zn≦dn) for any sequence d n are also obtained.

Uniform Dimension Results for Processes with Independent Increments

Abstract: Let X,(to) be a process in R a with stationary independent increments and let X(E, 09) denote the image under Xt(to) of a time set E. It is shown that dim X(E, og) 1 in R l is found to be a(1-1/~).

Positivity of the K-entropy on non-abelian K-flows

Abstract: A non-commutative extension of certain aspects of classical probability theory is presented in such a manner that the notion of Kolmogorov entropy can be extended to a large class of non-classical dynamical systems. In particular, the generalized K-entropy so defined is shown to be strictly positive on the class of non-abelian K-flows.

A Note on Exchangeable Sequences.

Abstract: : If (X sub 1),(X sub 2),... is an exchangeable sequence taking values in a complete, separable metric space, then there is a random variable M such that: (i) given M, (X sub 1),(X sub 2),... are conditionally independent and identically distributed; (ii) if (X sub 1),(X sub 2),... are conditionally independent and identically distributed given the random object Y, then a version of M is measurable with respect to the sigma-field spanned by Y; (iii) the sigma-field spanned by M coincides as a measure algebra with the invariant, tail, and exchangeable sigma-fields of the process (X sub 1),(X sub 2),... , even though none of the latter three is countably generated. (Author)

Preservation of rates of convergence under mappings

Abstract: For appropriate metrics characterizing various modes of stochastic convergence, it is shown that rates of convergence are preserved by a large class of functions. For example, the extensions of a Lipschitz function on a separable metric space S to the space of all probability measures on S with the Prohorov metric and to the space of all S-valued random variables with the usual metric associated with convergence in probability inherit the Lipschitz property. Consequently, just as with the continuous mapping theorem associated with ordinary convergence, new rate of convergence theorems can sometimes be obtained from old ones by applying appropriate mappings.

Path properties of processes with independent and interchangeable increments

Abstract: In [14] random processes (r.pr.) with interchangeable increments (ich.incr.) were examined with respect to canonical representations and convergence in distribution. The main purpose of the present paper is to investigate the path properties of such pr. Apart from the extensive litterature on independent (ind.) incr. pr. and from the results in [14], quite few such properties seem to be known (and then only in particular cases), including Biihlmann's proof [4] of the pointwise a.s. continuity, Takacs' generalizations [22] of the "ballot" theorem and Hagberg's extensions [13] of Sparre Andersen's combinatorial results. As shown in [_4, 14], ich. incr. pr. on infinite intervals are merely mixtures of pr. with stationary ind. incr. On finite intervals, however, the class of ich.incr.pr, is much more extensive. In fact (Th. 2.1 in [14]), any pr. of this type on [0, 1] which is continuous in probability is equivalent to apr. in D [0, 1] of the form

On the tails of infinitely divisible distributions

Abstract: Similar theorems are proved in [4] and [2]. Ruegg supposes T(x) to be of the form T(x)=O(exp{--axb}) or T(x)=O(exp{--ax(logx)b}). In his proofs he makes extensive use of the theory of entire functions. Horn considers T(x) satisfying T(x) = O (exp { x M(x)}), where M (x) is non-negative, continuous and ultimately increasing. His proofs depend on straightforward but rather intricate inequalities for the two-sided Laplace transform. In our proof, it seems, more efficient use is made of the infinite divisibility condition. Ruegg's theorems 1 and 2, in fact, hold for all d.f.'s having non-vanishing entire characteristic function (c. f.'s). In Section 3 we prove that an i.d.d.f, with x 2 log T(x)~ ~ , is degenerate by proving that i~ts variance must be zero, as in the case of i.d.d.f. 's on a finite interval (cf. [1], p. 117). In Section 4 we give a very simple proof of a variant of Theorem 1 for distributions on the non-negative integers.