ISSN:
1572-9230
Keywords:
Small ball problem
;
Gaussian Markov processes
;
Brownian motion
;
weighted norms
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let {X(t); 0≤t≤1} be a real-valued continuous Gaussian Markov process with mean zero and covariance σ(s, t) = EX(s) X(t) ≠ 0 for 0〈s, t〈1. It is known that we can write σ(s, t) = G(min(s, t)) H(max(s, t)) with G〉0, H〉0 and G/H nondecreasing on the interval (0, 1). We show that $$\mathop {\lim }\limits_{\varepsilon \to 0} \varepsilon ^2 \log P({\text{ }}\mathop {\sup }\limits_{0 〈 t \leqslant 1} {\text{ |}}X(t)| 〈 \varepsilon ) = - (\pi ^2 /8)\int_0^1 {(G'H - H'G)dt} $$ In the critical case, i.e. this integral is infinite, we provide the correct rate (up to a constant) for log P(sup0〈t≤1 |X(t)|〈∈) as ∈→0 under regularity conditions.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1021771503265
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