## Asymptotic behaviour of increments of random fields.(Russian, English)Zbl 1050.60036

Teor. Jmovirn. Mat. Stat. 68, 158-171 (2003); translation in Theory Probab. Math. Stat. 68, 173-186 (2004).
Let $$N_0^{d}$$ be a $$d$$-dimensional lattice with nonnegative integer coordinates, and let $$X_{n},\;n\in N_0^{d}$$, be i.i.d. random variables with the properties: (1)common generating function $$\varphi(t)=E\exp(tX_{n})$$, $$\varphi(t)<\infty$$ for some $$t>0$$; (2)$$EX_{n}=0$$, $$DX_{n}=1$$; (3) a function $$a(t),\;t\geq1$$, is given, such that $$1\leq a(t)\leq t$$, $$a(t)$$ and $$t/a(t)$$ are non-decreasing. Let us define $$a_{N}=[a(N)]$$, where $$[a]$$ is an integer part of $$a$$. This paper deals with the following increments
$S^{*}_{N}=\max_{| j|\leq N, | j-i|=a_{N}} S_{(i,j]}, \quad S_{N}=\max_{| j|\leq N, | j-i|\leq a_{N}}S_{(i,j]}, \quad S_{(i,j]}=\sum_{i<k\leq j}X_{k}, \quad (i,j]\subset N^{d}_0.$
The author proves that if conditions (1)–(3) are fulfilled and $$a_{N}/\log N\to\infty$$, $$N\to\infty$$, then $$\limsup_{N\to\infty}S^{*}_{N}\delta_{N}= \limsup_{N\to\infty}S_{N}\delta_{N}=1$$ a.s., where $$\delta_{N}=\{2a_{N}(\varepsilon_{N}+\beta_{N})\}^{-1/2}$$, $$\varepsilon_{N}=\log_{+}(N/a_{N})$$, $$\beta_{N}=d\log\log n$$, $$\log_{+}x=\log(x\vee e)$$. The cases $$a_{N}\sim C\log N$$ and $$a_{N}=O(\log N)$$ are considered, too.

### MSC:

 60F15 Strong limit theorems 60G60 Random fields

### Keywords:

asymptotic behaviour; increments of random fields