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A Supplement to the Einmahl-Li Results on Two-Sided Iterated Logarithm Type Behavior for I.I.D. Random Variables
Ist Teil von
Stochastic analysis and applications, 2008-09, Vol.26 (5), p.1095-1110
Ort / Verlag
Philadelphia, PA: Taylor & Francis Group
Erscheinungsjahr
2008
Link zum Volltext
Quelle
Taylor & Francis
Beschreibungen/Notizen
For a sequence of i.i.d. mean 0 random variables {X, X
n
; n ≥ 1} with partial sums
, necessary and/or sufficient conditions are provided for {X, X
n
; n ≥ 1} to enjoy iterated logarithm type behavior of the form
almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases X
2
< ∞ and X
2
= ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an "integral test" involving whether
is finite or infinite where Lx = log
e
(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).