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Random Walks with Stochastically Bounded Increments: Renewal Theory
Author(s) -
Alsmeyer Gerold
Publication year - 1995
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19951750103
Subject(s) - mathematics , random walk , bounded function , renewal theory , subsequence , combinatorics , type (biology) , stability (learning theory) , discrete mathematics , distribution (mathematics) , upper and lower bounds , mathematical analysis , statistics , ecology , machine learning , computer science , biology
Summary. This paper develops renewal theory for a rather general class of random walks S N including linear submartingales with positive drift. The basic assumption on S N is that their conditional increment distribution functions with respect to some filtration ℱ ℕ are bounded from above and below by integrable distribution functions. Under a further mean stability condition these random walks turn out to be natural candidates for satisfying Blackwell‐type renewal theorems. In a companion paper [2], certain uniform lower and upper drift bounds for S N , describing its average growth on finite remote time intervals, have been introduced and shown to be equal in case the afore‐mentioned mean stability condition holds true. With the help of these bounds we give lower and upper estimates for H * U(B ), where U denotes the renewal measure of S N , H a suitable delay distribution and B a Borel subset of IR. This is then further utilized in combination with a coupling argument to prove the principal result, namely an extension of Blackwell's renewal theorem to random walks of the previous type whose conditional increment distribution additionally contain a subsequence with a common component in a certain sense. A number of examples are also presented.