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On the Erdős-Rényi maximum of partial sums. (English. Russian original) Zbl 0928.60008
Theory Probab. Appl. 42, No. 2, 254-270 (1997); translation from Teor. Veroyatn. Primen. 42, No. 2, 274-293 (1997).
Let $$\{X_i, i\geq 1\}$$ be i.i.d. r.v.’s satisfying the conditions $$EX_i= 0$$, $$\sigma^2X_i= 1$$ and $$E\exp\{tX_i\}\leq \exp\{t^2 \sigma^2/2\}$$ for some constant $$\sigma>0$$ and all $$t>0$$. The properties of $$R^*_n(k)= \max_{0\leq i\leq n}[X_{i+1}+\cdots +X_{i+ k}]$$ and $$R_n(k)= R^*_{n- k}(k)$$ are investigated. Among other things, it is shown that under the conditions $$\ln n\ll k\ll n$$ and $u^2_n \alpha^{m- 1}_n\to\infty,\qquad u^2_n\alpha^m_n\to 0,\tag{A$$_m$$}$ where $u_n= \sqrt{2\ln\left({n\over k} \sqrt {2\ln{n\over k}}\right)},\quad \alpha_n= a_n/\sqrt k,\qquad 0< a_n< n,$ for all $$x\in\mathbb{R}$$ we have $P\Biggl[{u_n\over \sqrt k} (R^*_n(k)- u_n\sqrt k(1+ y_n))< x\Biggr]\to \exp \{-e^{-x}/ \sqrt{2\pi}\}\tag{$$*$$}$ with $$y_n= \sum^{m- 1}_{i= 1} c_i\alpha^i_n$$ and suitably chosen constants $$c_1,\dots, c_{m-1}$$; similarly, if $$(\text{A}_m)$$ is replaced by $u^2_n \alpha^m_n\to c>0,\tag{A$$^*_m$$}$ then $$(*)$$ still holds true with $$y_n= \sum^m_{i= 1} c_i\alpha^i_n$$. Moreover, exact asymptotic rates of convergence for $$P[R_n(k)< x]$$ in the Bernoulli case, $$P[X_i= 0]= P[X_i= 1]={1\over 2}$$, and $$k= [d\log_2n]$$, $$d>1$$, are found. Also a generalization of a theorem proved by P. Révész [Ann. Probab. 10, 613-622 (1982; Zbl 0493.60038)] for increments $$\xi_t= \sup_{0\leq s\leq t} (W_{s+ a_t}- W_s)$$ of the standard Wiener process $$\{W_t, t\geq 0\}$$ is given.
##### MSC:
 60F05 Central limit and other weak theorems 60F10 Large deviations 60G50 Sums of independent random variables; random walks 60E15 Inequalities; stochastic orderings
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