The central limit theorem and the law of iterated logarithm for empirical processes under local conditions

Summary A CLT and a LIL are proved under weak-L ₂ Gaussian bracketing conditions (weaker than the usual ones). These results have wide applicability and in particular provide an improvement of the Jain-Marcus central limit theorem for C(S)-valued random variables.The research of these authors was carried out while visiting Texas A & M UniversityThe research of these authors has been totally or partially supported by the Danish Natural Science Research Council and by the National Science Foundation Grants numbers DMS-83-18610 and DMS-83-01367     

On the law of the iterated logarithm for Gaussian processes

We present some optimal conditions for the compact law of the iterated logarithm of a sequence of jointly Gaussian processes in different situations. We also discuss the local law of the iterated logarithm for Gaussian processes indexed by arbitrary index sets, in particular for self-similar Gaussian processes. We apply these results to obtain the law of the iterated logarithm for compositions of Gaussian processes. Research partially supported by NSF Grant DMS-93-02583.     

Functional law of iterated logarithm for additive functionals of reversible Markov processes

1.IntroductionandMainResultsAssumethat(Xt),.T(T~NorAl)isaPolishspaceE-valuedMarkovprocess,definedon(fi,F,(R),(ot),(P-c)..E),withitssemigroupoftransitionkernels(Pt).Here(ot)isthesemigroupofshiftsonfisuchthatX.(otw)~X. t(w),Vs,tET;(R)isthenaturalfiltration.Throughoutthispaperweassumethat(Pt)issymmetricandergodicwithrespectto(w.r.t.forshort)aprobabilitymeasurepon(E,e)(eistheBorela--fieldofE),i.e.,.Symmetry:(Ptf,g)~(f,Pig):~isfptgdp,acET,if,gCL'(P);.ErgodicitytFOranyfEL'(P),ifPtf~f…     

On the functional law of the iterated logarithm for trimmed sums

New conditions for the functional law of the iterated logarithm for trimmed sums of symmetric independent identically distributed random variables are obtained. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 119–125. Translated by N. B. Lebedinskaya.     

The rate of convergence in the functional central limit theorem for random quadratic forms with some applications to the law of the iterated logarithm

We prove a convergence rate in the functional central limit theorem for quadratic forms in independent random variables satisfying a fourth moment condition. Using this result we get a law of the iterated logarithm as well as an analogue of Chung's law of the iterated logarithm for random quadratic forms.     

On the law of the iterated logarithm

An analogue of the law of the iterated logarithm for Brownian motion in Banach spaces is proved where the expression √2 loglog s is replaced by a positive non-decreasing function satisfying certain conditions.     

The functional iterated logarithm law for stochastic processes represented by multiple Wiener integrals

Summary In a previous paper the authors obtained a functional law of the iterated logarithm for a class of self-similar processes with stationary increments, which are represented by multiple Wiener integrals. This result is extended to a certain class of processes represented by multiple Wiener integrals which converge to with an appropriate normalization. As an application a functional log log law for nonlinear functionals of some stationary Gaussian processes is given.     

On Strassen's law of the iterated logarithm

Summary Kolmogorov's law of the iterated logarithm has been sharpened by Strassen who proved a more refined theorem by using tools from functional analysis. The present paper gives a classical proof of Strassen's theorem, using a method along the lines of Kolmogorov's original approach. At the same time the result proved here is more general since a) the random variables involved need not have the same distributions, b) the condition of independence is weakened and c) instead of Kolmogorov's growth condition on the random variables, only a mild restriction on their moments of order l3 is needed.     

The functional law of the iterated logarithm for truncated sums

The functional law of the iterated logarithm (FLIL) is obtained for truncated sums $S_n = \sum _{j = l}^n X_j I\{ X_j^{\text{2}} \leqslant b_n \} $ of independent symmetric random variables X_j, 1<-j≤n, b_n≤∞. Considering the random normalization $T_n^{1/{\text{2}}} = \left( {\sum\limits_{j = 1}^n {X_j^{\text{2}} } I\{ X_j^{\text{2}} \leqslant b_n \} } \right)^{1/{\text{2}}} ,$ we obtain an upper estimate in the FLIL, using only the condition that T_n→∞ almost surely. These results are useful in studying trimmed sums. Bibliography: 9 titles.     

A functional law of the iterated logarithm for associated sequences

Recently, a functional central limit theorem and a Berry-Essen Theorem have been demonstrated for classes or associated random variables. Using these results, and similar results for multiplicative sequences, we show a functional law of the iterated logarithm for associated sequences satisfying a rate requirement.     

Asymptotic expansions in the central limit theorem for compound and Markov processes

Summary For independent identically distributed bivariate random vectors (X ₁, Y ₁), (X ₂, Y ₂), ... and for large t the distribution of X ₁ +...+ X _N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y ₁, Y ₂,.... Similar expansions are derived for multivariate X ₁. Furthermore, local asymptotic expansions are valid for the distribution of f(X ₁)+ ...+ f(X _N ) when N is large and nonrandom, and X _i , i=1, 2,..., is a discrete strongly mixing Markov chain.     

On the functional central limit theorem for martingales

Necessary and sufficient conditions for the functional central limit theorem for a double array of random variables are sought. It is argued that this is a martingale problem only if the variables truncated at some fixed point c are asymptotically a martingale difference array. Under this hypothesis, necessary and sufficient conditions for convergence in distribution to a Brownian motion are obtained when the normalization is given (i) by the sums of squares of the variables, (ii) by the conditional variances and (iii) by the variances. The results are proved by comparing the various normalizations with a natural normalization.Research sponsored in part by the Office of Naval Research, Contract N00014-75-C-0809     

A law of the iterated logarithm for processes with independent increments

ＡＬＡＷＯＦＴＨＥＩＴＥＲＡＴＥＤＬＯＧＡＲＩＴＨＭＦＯＲＰＲＯＣＥＳＳＥＳＷＩＴＨＩＮＤＥＰＥＮＤＥＮＴＩＮＣＲＥＭＥＮＴＳＷＡＮＧＪＩＡＧＡＮＧ（汪嘉冈）（ＥａｓｔＣｈｉｎａＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅ＆Ｔｅｃｈｎｏｌｏｇｙ,Ｓｈａｎｇ...     

The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes

We use Nummelin splitting in continuous time in order to prove laws of iterated logarithm for additive functionals of a Harris recurrent Markov process, with deterministic or random renormalization.