TY - JOUR

T1 - Laws of the iterated logarithm for self-normalised levy processes at zero

AU - Buchmann, Boris

AU - Maller, Ross A.

AU - Mason, David M.

N1 - Publisher Copyright:
©2014 American Mathematical Society.

PY - 2015/3/1

Y1 - 2015/3/1

N2 - We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as t↓0) for the “self-normalised” process {Formula presented}, constructed from a Levy process (Xt)t≥0 having quadratic variation process (Vt)t≥0, and an appropriate choice of the constant a. We apply them to obtain LILs when Xt is in the domain of attraction of the normal distribution as t ↓ 0, when Xt is symmetric and in the Feller class at 0, and when Xt is a strictly α-stable process. When Xt is attracted to the normal distribution, an important ingredient in the proof is a Cramer-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.

AB - We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as t↓0) for the “self-normalised” process {Formula presented}, constructed from a Levy process (Xt)t≥0 having quadratic variation process (Vt)t≥0, and an appropriate choice of the constant a. We apply them to obtain LILs when Xt is in the domain of attraction of the normal distribution as t ↓ 0, when Xt is symmetric and in the Feller class at 0, and when Xt is a strictly α-stable process. When Xt is attracted to the normal distribution, an important ingredient in the proof is a Cramer-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.

KW - Cramer bound

KW - Domain of attraction of the normal distribution for small times

KW - Feller stochastic compactness classes

KW - Law of the iterated logarithm for small times

KW - Levy process

KW - Quadratic variation process

KW - Self-normalised process

UR - http://www.scopus.com/inward/record.url?scp=84916611974&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-2014-06112-6

DO - 10.1090/s0002-9947-2014-06112-6

M3 - Article

SN - 0002-9947

VL - 367

SP - 1737

EP - 1770

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -