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April, 1986 Improved Erdos-Renyi and Strong Approximation Laws for Increments of Renewal Processes
J. Steinebach
Ann. Probab. 14(2): 547-559 (April, 1986). DOI: 10.1214/aop/1176992530


Let $X_1, X_2,\cdots$ be an i.i.d. sequence with $EX_1 = \mu > 0, \operatorname{var}(X_1) = \sigma^2 > 0, E \exp(sX_1) < \infty, |s| < s_1$, and partial sums $S_0 = 0, S_n = X_1 + \cdots + X_n$. For $t \geq 0$, put $N(t) = \max \{n \geq 0: S_0,\ldots, S_n \leq t\}$, i.e., $L(t) = N(t) + 1$ denotes the first-passage time of the random walk $\{S_n\}$. Starting from some analogous results for the partial sum sequence, this paper studies the almost sure limiting behaviour of $\sup_{0 \leq t \leq T - K_T} (N(t + K_T) - N(t))$ as $T \rightarrow \infty$, under various conditions on the real function $K_T$. Improvements of the Erdos-Renyi strong law for renewal processes (resp. first-passage times) are obtained as well as strong invariance principle type versions. An indefinite range between strong invariance and strong noninvariance is also treated.


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J. Steinebach. "Improved Erdos-Renyi and Strong Approximation Laws for Increments of Renewal Processes." Ann. Probab. 14 (2) 547 - 559, April, 1986.


Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0601.60029
MathSciNet: MR832023
Digital Object Identifier: 10.1214/aop/1176992530

Primary: 60F15
Secondary: 60F10 , 60F17 , 60G17 , 60K05

Keywords: Erdos-Renyi strong laws , Increments of renewal processes , large deviations , strong approximations , Strong invariance principles

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 2 • April, 1986
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