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

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.

### MSC:

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