## On the Erdős-Rényi law for the renewal processes.(Russian, English)Zbl 1050.60033

Teor. Jmovirn. Mat. Stat. 68, 142-151 (2003); translation in Theory Probab. Math. Stat. 68, 157-166 (2004).
Let $$X_1,X_2,\ldots$$ be a sequence of independent positive random variables with $$EX_{i}=\mu_{i}>0$$, $$\sigma_{i}^2=E(X_{i}-\mu_{i})^2>0$$, $$i=1,2,\ldots$$ Let the following conditions hold true:
(A1) $$\lim_{k\to\infty}\sup_{n}| k^{-1}\sum_{i=n+1}^{n+k}\mu_{i}-\mu|=0$$;
(A2) there exist positive constants $$H$$ and $$d_1,d_2,\ldots$$ such that $$|\log Ee^{z(X_{i}-\mu_{i})}|\leq d_{i}$$, for $$| z|<H$$;
(A3) $$\limsup_{n\to\infty}\sup_{k}B_{nk}^{-1} \sum_{i=n+1}^{n+k}(d_{i}^2+d_{i})<\infty$$, where $$B_{nk}=\sum_{i=n+1}^{n+k}\sigma_{i}^2$$;
(A4) there exist $$\delta>0$$, $$k_0$$ such that $$B_{nk}>\delta k$$ for all $$n$$ and $$k$$.
Let us denote $$S_{n}=X_1+\ldots+X_{n}$$, $$N(t)=\max\{n: S_{n}\leq t\}$$, $\Delta_{T}(b)=\sup_{0\leq t\leq T-b\log T}\{N(t+b\log T)-N(t)\}, \quad \delta_{T}(b)=\inf_{0\leq t\leq T-b\log T}\{N(t+b\log T)-N(t)\},$
where $$b>0$$. The author proves in particular the following results: If conditions (A1)–(A4) are satisfied, then
1) $$\lim_{t\to\infty}N(t)/t=1/\mu$$ a.s.;
2) there exist $$C\geq0$$ and functions $$b_{i}\!: (C,+\infty)\to(0,+\infty)$$, $$i=1,2$$, such that for all $$c>C$$:
$\limsup_{T\to\infty}\Delta_{T}(b_1)/\log T\leq c\leq\liminf_{T\to\infty}\Delta_{T}(b_2)/\log T \text{ a.s.},$
where $$b_{i}=b_{i}(c), i=1,2$$;
3) there exist $$C'\geq0$$ and functions $$b_{i}'\!: (C',+\infty)\to(0,+\infty)$$, $$i=1,2$$ such that for all $$c'>C'$$: $\limsup_{T\to\infty}\delta_{T}(b'_1)/\log T\leq c'\leq\liminf_{T\to\infty}\delta_{T}(b'_2)/\log T \text{ a.s.},$
where $$b'_{i}=b'_{i}(c)$$, $$i=1,2$$.

### MSC:

 60F15 Strong limit theorems 60K05 Renewal theory

### Keywords:

Erdős-Rényi law; renewal processes; law of large numbers