ON GENERALIZED RENEWAL MEASURES AND CERTAIN 1ST PASSAGE TIMES

• 成果类型: Article
• 署名作者: ALSMEYER, G
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989690
• 发表日期: 1992
• 页码: 1229-1247
• 关键词: theorem

摘要

Let X1, X2,... be i.i.d. random variables with common mean mu greater-than-or-equal-to 0 and associated random walk S0 = 0, S(n) = X1 + ... + X(n), n greater-than-or-equal-to 1. For a regularly varying function phi(t) = t(alpha)L(t), alpha > -1 with slowly varying L(t), we consider the generalized renewal function U(phi)(t) = SIGMA(n greater-than-or-equal-to 0) phi(n)P(S(n) less-than-or-equal-to t), t is-an-element-of R, by relating it to the family tau = tau(t) = inf{n greater-than-or-equal-to 1: S(n) > t} t greater-than-or-equal-to 0. One of the major results is that U(phi)(t) < infinity for all t is-an-element-of R, iff phi(t)-1U(phi)(t) approximately 1/(alpha + 1)mu(alpha+1) as t --> infinity, iff E(X1-)2-phi(X1-) < infinity, provided phi is ultimately increasing (double-line arrow pointing right alpha greater-than-or-equal-to 0). A related result is proved for U(phi)(t + h) - U(phi)(t) and U(phi)+(t) = SIGMA(n greater-than-or-equal-to 0)phi(n)P(M(n) less-than-or-equal-to t), where M(n) = max0 less-than-or-equal-to j less-than-or-equal-to n S(j). Our results form extensions of earlier ones by Heyde, Kalma, Gut and others, who either considered more specific functions phi of used stronger moment assumptions. The proofs are based on a regeneration technique from renewal theory and two martingale inequalities by Burkholder, Davis and Gundy.