On lag increments of a Gaussian process
Let \(X(t)\), \(0\leq t<\infty\), be a centered Gaussian process with stationary increments \(E(X(s)- X(t))^2= C^2_0|s-t|^{2\alpha}\), \(0< \alpha< 1\), and \(X(0)= 0\). It is proved that NEWLINE\[NEWLINE\limsup_{T\to\infty} \sup_{0< t\leq T}|X(T)- X(T- t)|/\sqrt 2 C_0t^\alpha(\log(T/t)+ \log\log t)^{1/2}= 1\quad\text{a.s.}NEWLINE\]NEWLINE Other related a.s. limits are also obtained thus giving a direct extension of \textit{G. Chen}, \textit{F. Kong} and \textit{Z. Lin}'s result [Ann. Probab. 14, 1252-1261 (1986; Zbl 0613.60027)] for lag increments of a Wiener process. The proof is based on upper bounds of large deviation probabilities on suprema of a Gaussian process.
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