An extended version of the Erdős-Rényi strong law of large numbers. (English) Zbl 0677.60031

Consider an i.i.d. sequence \(X_ 1,X_ 2,..\). with common d.f. F. Assume (i) \(X_ 1\) is nondegenerate with \(0\leq E X_ 1<\infty\) and (ii) sup\(\{\) t: \(\phi (t)=E \exp (t X_ 1)<\infty \}>0.\)
Set \(\omega =\sup \{x:F(x)<1\}\) (\(\in (0,\infty])\), \(\zeta (z)=\sup \{zt- \log \phi (t):t\geq 0\) and \(\phi (t)<\infty \}\) (z\(\geq 0)\), and \(\gamma (x)=\sup \{z:\zeta (z)\leq x\}\) (x\(\geq 0)\). For any nondecreasing integer sequence \(\{\) k(n); \(n=1,2,...\}\) with \(1\leq k(n)\leq n\) let \[ M_ n(k(n))=\max_{0\leq m\leq n-k(n)}\{S_{m+k(n)}-S_ m\}, \] where \(S_ 0=0\), \(S_ m=X_ 1+...+X_ m\). Erdős and Rényi (1970) proved that if \(c(n)=\log n/k(n)\to c\in (0,\infty)\), then \[ (1.1)\quad \lim_{n\to \infty}M_ n(k(n))/k(n)\gamma (c(n))=1\quad a.s. \] Here an interesting extension of the Erdős-Rényi law is presented dealing with “small increments”, i.e. k(n) such that c(n)\(\to \infty\). The results are as follows:
(a) If \(0<\omega <\infty\), then \[ (1.2)\quad \lim_{n\to \infty}M_ n(k(n))/k(n)\gamma (c(n))=\lim_{n\to \infty}M_ n(k(n))/k(n)\omega =1\quad a.s. \] (b) If \(\omega =\infty\), then \[ (1.3)\quad \limsup_{n\to \infty}M_ n(k(n))/k(n)\gamma (c(n))=1\quad a.s. \] Moreover, the lim sup in (1.3) can be replaced by lim if and only if \[ (1.4)\quad \lim_{z\to \infty}\gamma (-\log (1-F(z)))/z=1 \] if and only if \[ (1.5)\quad \lim_{n\to \infty}\max_{1\leq m\leq n}X_ m/\gamma (\log n)=1\quad a.s. \]
Reviewer: J.Steinebach


60F15 Strong limit theorems
60F10 Large deviations
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