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

Shcherbakova, O. E.

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.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60F15	Strong limit theorems
60G60	Random fields

Keywords:

asymptotic behaviour; increments of random fields