Some universal results on the behavior of increments of partial sums. (English) Zbl 0872.60022

Einmahl, Uwe; Mason, David M.

Ann. Probab. 24, No. 3, 1388-1407 (1996).

Let \(X,X_1,X_2,\dots\) be independent and identically distributed nondegenerate random variables with common distribution function \(F\), and for each integer \(n\geq 1\) set \(S_n=X_1+\dots+X_n\). Very general results are established on the one-sided lim-sup behavior of the increments of \(S_n\), when suitably normalized. To formulate the main statements consider the quantile function \(Q(u)=\inf\{x:F(x)\geq u\},0<u<1,\) of \(F\), and for \(0<s<1\) set \(\mu(s)=\int_0^{1-s}Q(u)du,\nu(s)=\mu(s)+sQ(1-s)\) and \(\sigma^2(s)=\int_0^{1-s}Q^2(u)du+sQ^2(1-s)-\nu^2(s).\) For each \(n\geq1,0<k\leq n\) and \(0<b<1\) set \[ M_n(b,k)=\max_{0\leq i\leq n-k}\max_{0\leq j\leq k} \{j\mu(b)-S_{i+j}+S_i\}. \] Let \(0<\kappa_n\leq n\) be any nondecreasing sequence of real numbers such that \(\kappa_n/n\) is nonincreasing and \(\kappa_n/\log n\to\infty\) as \(n\to\infty\). For \(n\geq 1\), define \(k_n=[\kappa_n]\), where \([x]\) denotes the integer part of \(x, \gamma_n=\log(n\log n/\kappa_n), b_n=\gamma_n/(\kappa_n+\gamma_n)\) and \(\beta_n=(2k_n\gamma_n)^{-1/2}\sigma(b_n)^{-1}\). If \(X\geq 0\) and \(\gamma_n/\kappa_n\downarrow 0\), then with probability one \[ \limsup_{n\to\infty} \beta_n M_n(b_n,k_n)\leq 1\qquad \text{and}\qquad \limsup_{n\to\infty}\beta_n\{k_n\mu(b_n)-(S_n-S_{n-k_n})\}\geq 0. \] The constants in these inequalities are sharp in the sense that they are attained with probability one for certain distribution functions \(F\). However, if \(F\) is in the Feller class, i.e., one can find centering constants \(\delta_n\) and norming constants \(c_n\) such that \((S_n-\delta_n)/c_n\) is tight with nondegenerate subsequential limits, then with probability one \[ \limsup_{n\to\infty}\beta_n\{k_n\mu(b_n)-(S_n-S_{n-k_n})\}\geq C_1>0 \] for some constant \(C_1\) depending on \(F\). Moreover, if \(\kappa_n\) satisfies, in addition to the above assumptions, \(\log(n/\kappa_n)/\log\log n\to\infty\) as \(n\to\infty\), then with probability one \[ \liminf_{n\to\infty}\beta_n M_n(b_n,k_n)\geq C_2>0 \] for some constant \(C_2\) depending on \(F\). These results are also valid for not necessarily nonnegative random variables \(X\) if the negative part of \(X\) has a finite moment generating function in a neighborhood of zero.

Reviewer: E.Häusler (Gießen)

MSC:

60F15	Strong limit theorems
60E07	Infinitely divisible distributions; stable distributions

Keywords:

universal law of the iterated logarithm; quantile transformation; maximal inequalities; increments of partial sums