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Almost Sure Convergence of Delayed Renewal Processes
Author(s) -
Steinebach J.
Publication year - 1987
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-36.3.569
Subject(s) - law of the iterated logarithm , renewal theory , logarithm , independent and identically distributed random variables , sequence (biology) , iterated function , mathematics , moment (physics) , convergence (economics) , process (computing) , combinatorics , discrete mathematics , computer science , random variable , statistics , mathematical analysis , physics , economics , genetics , economic growth , operating system , classical mechanics , biology
Consider a renewal process { N ( t )} t⩾0 associated with a sequence { X t } t−1, 2, … of non‐negative and independent, identically distributed ‘failure times’. Appropriate generalized moment conditions are presented which are necessary and sufficient for certain ‘law of iterated logarithm’ type results on the ‘delayed renewal process’, that is, { N ( t ) – N ( t – a t )} t⩾0 , where a t → ∞ (as t → ∞) at a given rate. Auxiliary tools are recent strong invariance principles for renewal processes and corresponding relationships for the ‘delayed sums’ of their failure times.