## Moduli of continuity for a two-parameter Gaussian process on rectangles in the unit square.(English)Zbl 0943.60023

Theory Probab. Math. Stat. 58, 171-185 (1999) and Teor. Jmovirn. Mat. Stat. 58, 158-172 (1998).
The authors consider a centered Gaussian random process $$X=\{ X(x,y),0\leq x\leq 1, 0\leq y\leq 1\}$$ such that $$X$$ is almost surely continuous, $$X(0,0)=0.$$ It is assumed that the following assumption holds true: $E\{ X(x_1,y_1)-X(x_2,y_2)\}^2= \sigma^2\bigl(\bigl((x_1-x_2)^2+(y_1-y_2)^2 \bigr)^{1/2}\bigr),$ where $$\sigma(t)$$, $$t>0$$, is a nondecreasing continuous, regular varying function with exponent $$\gamma$$, $$0<\gamma<1.$$ There exists a positive constant $$C_1$$ such that $$d\sigma^2(t)/dt \leq C_1 \sigma^2(t)/t.$$ Let $$R$$ be the rectangle $$R=R(u,s,v,r)=[u,u+s]\times[v,v+r]\subset[0,1]^2.$$ The authors define the increment $$X(R)$$ by $X(R)= X(R(u,s,v,r))= X(u+s,v+r)-X(u,v+r)-X(u+s,v)+X(u,v).$ Let $$g_{t}$$ and $$h_{t}$$ be nonincreasing continuous functions, $$t>0$$, for which (i) $$0<g_{t}\leq h_{t}<1$$, (ii) there exists $$0\leq\beta\leq 1$$ such that $$\lim_{t\to\infty} g_{t}/h_{t}=\beta$$, (iii) $$\lim_{t\to\infty} h_{t}=0.$$ The main results of this paper are as follows.
Theorem 1. We have $\limsup_{t\to\infty}\sup_{0<s\leq g_{t}} \sup_{0<r\leq h_{t}} \sup_{0\leq u\leq 1-s} \sup_{0\leq v\leq 1-r}{|X(R(u,s,v,r))|\over \sqrt{2\log(h_{t}/g_{t}^3)}H(g_{t},h_{t})}\leq 1 \quad \text{a.s.} \tag{1}$ Theorem 2. Suppose that the above conditions (i) and (iii) are satisfied and there exists $$0<\beta\leq 1$$ in (ii). Assume that there exist positive constants $$C_1$$ and $$C_2$$ such that, for $$x>0$$, $${d\over dx}\sigma^2(x)\leq C_1{\sigma^2(x)\over x}$$ and $${d^2\over dx^2}\sigma^2(x) \leq C_2{\sigma^2(x)\over x^2}.$$ Then we have $\limsup_{t\to\infty} \sup_{0\leq u\leq 1-g_{t}} \sup_{0\leq v\leq 1-h_{t}} {|X(R(u,g_{t},v,h_{t}))|\over 2\sqrt{\log(1/g_{t})}H(g_{t},h_{t})}\geq 1 \quad \text{a.s.} \tag{2}$ {}.

### MSC:

 60F15 Strong limit theorems 60G17 Sample path properties