Strong renewal theorems with infinite mean
Strong renewal theorems with infinite mean
复制标题
具有无限均值的强更新定理
DOI：
10.1090/s0002-9947-1970-0268976-9
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发表时间：
1970
影响因子：
1.3
通讯作者：
K. B. Erickson
中科院分区：
文献类型：
--
作者：
K. B. Ericksonc;K. B. Erickson
设F是(0，oo)上的非算术概率分布，1-F(F)在oo按指数a，0<；aáL正则变化，U(T)=J，F“‘(T)为更新函数。本文首先给出了当t-*o，h>；0不变时U(t+h)-U(T)的各种渐近表达式。接下来，我们导出了一大类可积函数z的卷积U*z(T)，t->；oo的渐近关系。所有这些渐近关系都用截断平均函数m(T)=f0[1-F(X)]dx，t Large表示，并且是经典的有限均值分布的强更新定理的自然推广。最后，在本文的最后几节中，我们利用特例a=L，得到了与更新过程相关的某些等待时间分布的极限定理。1.主要定理。设A是集中在[0，oo)(2)上的概率测度，U是定义在任意可测集上的相关更新测度/by(L.i)t/{/}=！>；“·{/}0，其中Fn‘表示F与其自身的”折叠卷积“(P°*是集中在原点的概率测度).对于每个有界的I，级数(1.1)收敛到一个有限数(U的这一性质和其他基本性质见[3，VI.6]；U的概率解释见本文第9节。)我们写U(X)表示U{[0，x]}，此后我们将忽略U(度量)和U(函数)之间的区别。(本公约也适用于其他措施。)本文的主要结果主要讨论了h>；0固定和t-*oo时U(t+h)-U(T)的差值。主要假设P具有形式(1.2)F(T)=t~al(T)，t>；0，由编辑于1969年10月4日收到。A MS科目分类。小学6070、6020、6030；中学4042、4252。
Let F be a nonarithmetic probability distribution on (0, oo) and suppose 1 —F(f) is regularly varying at oo with exponent a, 0<aál. Let U(t) = J, F"'(t) be the renewal function. In this paper we first derive various asymptotic expressions for the quantity U(t+h)— U(t) as t -* oo, h>0 fixed. Next we derive asymptotic relations for the convolution U*z(t), t —>■ oo, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function m(t) = f0 [1 — F(x)] dx, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case a = l to derive some limit theorems for the distributions of certain waiting times associated with a renewal process. 1. Principal theorems. Let A be a probability measure concentrated on [0, oo)(2) and let U be the associated renewal measure defined for any measurable set / by (l.i) t/{/} = !>"•{/} 0 where Fn' denotes the «-fold convolution of F with itself (P°* is the probability measure concentrated at the origin). The series (1.1) converges to a finite number for every bounded I. (For this and other elementary properties of U see [3, VI. 6] ; for a probabilistic interpretation of U see §9 in this paper.) We write U(x) for U{[0, x]} and we shall henceforth ignore the distinction between U the measure and U the function. (This convention applies to other measures as well.) The main results of this paper deal primarily with the differences U(t+h) — U(t) for h>0 fixed, and t -*■ oo. The principal assumption is that Phas the form (1.2) \-F(t) = t~aL(t), t>0, Received by the editors October 4, 1969. A MS Subject Classifications. Primary 6070, 6020, 6030; Secondary 4042, 4252.