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How big are the lag increments of a Gaussian process? (English) Zbl 0962.60012

Since 1983 several results on the limit behaviour of increments of Wiener processes have been investigated which may be called “LIL (law of the iterated logarithm) generalizations”. The authors of this paper have chosen the following result of G. Chen, F. Kong and Z. Lin [Ann. Probab. 14, 1252-1261 (1986; Zbl 0613.60027)] to generalize further: Let \(\{W(t), 0\leq t\leq \infty\}\) be a Wiener process and \(W(s, t)= W(s)- W(s- t)\), \(s\geq t\geq 0\). Then almost surely \[ \limsup_{T\to\infty} \sup_{0\leq t\leq T} D^{-1}|W(T,t)|= 1, \]
\[ \lim_{T\to\infty} \sup_{0\leq t\leq T} \sup_{0\leq s\leq t} D^{-1}|W(T, s)|= 1,\quad \lim_{T\to \infty} \sup_{0\leq t\leq T} \sup_{t\leq s\leq T} D^{-1}|W(s, t)|= 1, \] where \(D= D(T, t)= \{2t(\log(T/t)+ \log\log t)\}^{1/2}\), \(\log t= \ln(t\vee 1)\).
There are two main results in the paper which extend this statement to the general form of an almost surely continuous, centered Gaussian process \(X(t)\) with stationary increments \(E(X(t)- X(s))^2= \sigma(|t-s|)\), where \(\sigma(\cdot)\) is some special nondecreasing function, generalizing the case of a Wiener process (when \(\sigma(t)= \sqrt t\)). The first one is very similar to that mentioned above, because one has only to substitute \(W\) by \(X\) and \(t\) by \(\sigma^2(t)\) (we mean \(D\) by \(d= d(T, t)= \{2\sigma^2(t)(\log(T/t)+ \log\log t)\}^{1/2}\)). These two results are established in some conditions on \(\sigma^2(t)\) (concavity, twice-differentiability, and so on) and use the technique of estimating upper bounds of large deviation probabilities on suprema of the Gaussian process.

MSC:

60F10 Large deviations
60F15 Strong limit theorems
60G15 Gaussian processes

Citations:

Zbl 0613.60027
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References:

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