SOME LIMIT-THEOREMS IN LOG DENSITY

I Berkes, H Dehling

    Research output: Contribution to journalArticleAcademicpeer-review

    192 Downloads (Pure)

    Abstract

    Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums S(k) of independent r.v.'s we prove under mild technical conditions that (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)I{S(k)/a(k) is-an-element-of .} --> G(.) (a.s.) if and only if (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)P(S(k)/a(k) is-an-element-of .) --> G(.). A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.

    Original languageEnglish
    Pages (from-to)1640-1670
    Number of pages31
    JournalAnnals of probability
    Volume21
    Issue number3
    DOIs
    Publication statusPublished - Jul-1993

    Keywords

    • PATHWISE CENTRAL LIMIT THEOREM
    • LOG-AVERAGING METHODS
    • STABLE CONVERGENCE
    • STRONG APPROXIMATION
    • LAW OF LARGE NUMBERS

    Fingerprint

    Dive into the research topics of 'SOME LIMIT-THEOREMS IN LOG DENSITY'. Together they form a unique fingerprint.

    Cite this