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

Frolov, A. N.

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 \(b'_{i}=b'_{i}(c)\), \(i=1,2\).

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60F15	Strong limit theorems
60K05	Renewal theory

Keywords:

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