Erdős-Rényi laws and the strong invariance principle. (English. Ukrainian original) Zbl 0943.60024

Zinchenko, N. M.

Theory Probab. Math. Stat. 58, 27-33 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 24-30 (1998).

Let \(\{\xi_{i},i\geq 1\}\) be independent random variables, \(S_{m}^{n}=\sum_{i=m+1}^{n}\xi_{i}.\) The author uses the strong invariance principle for the investigation of the asymptotic behaviour of the sum \(S_{n-k_{n}}^{n}\) and the maximum of this sum \(L(N,k_{N})=\max_{0\leq n\leq N-k_{n}}S_{n}^{n+k}.\) Constants \(\gamma_{n}\) are found such that \[ \limsup_{n\to\infty}{1\over\gamma_{n}} S_{n-k_{n}}^{n}=c=\text{const}\not=0 \] with probability one as well as constants \(\delta_{n}\) such that with probability one \[ \limsup_{N\to\infty}L(N,k_{N}){1\over\delta_{N}}=1. \] The case of independent random variables \(\xi_{n}\) with different distributions is considered.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60F15	Strong limit theorems
60G30	Continuity and singularity of induced measures

Keywords:

Erdős-Rényi law; law of large numbers; strong invariance principle; generatrix; sum of independent random variables