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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
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