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Improved Erdos-Renyi and Strong Approximation Laws for Increments of Renewal Processes

J. Steinebach
The Annals of Probability
Vol. 14, No. 2 (Apr., 1986), pp. 547-559
Stable URL: http://www.jstor.org/stable/2243926
Page Count: 13
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Abstract

Let X1, X2,⋯ 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 S0 = 0, Sn = X1 + ⋯ + Xn. For t ≥ 0, put N(t) = max {n ≥ 0: S0,..., Sn ≤ t}, i.e., L(t) = N(t) + 1 denotes the first-passage time of the random walk {Sn}. 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 → ∞, under various conditions on the real function KT. 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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