## 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

### MSC:

 60F15 Strong limit theorems 60F10 Large deviations
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