Lower tail probabilities for Gaussian processes.
The authors study the asymptotic behavior of lower tail probability for Gaussian random processes and their applications. Let \(X=(X_t)_{t\in S}\) be a real-valued centered Gaussian process indexed by \(S\). They first show a general estimate of the probability \(P(\sup_{t\in S}(X_t-X_{t_0})\leq x)\) as \(x\to 0\), with \(t_0\in S\) fixed, under mild conditions for the covariance. As the corollary, they obtain the asymptotics of lower tail probability for several stationary Gaussian processes. Especially, they identify its leading order for the fractional Brownian motion case. Their argument also works for the fractional Brownian motion pursuit problem. A lower bound of the main estimate is proved by combining a clever decomposition of time interval and Slepian's lemma. The LDP estimate for Gaussian process plays a crucial role in the argument. An upper bound is given by a comparison estimate with i.i.d. Gaussian random variables.
scientific article; zbMATH DE number 2100717
scientific article