Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics

• Published in: Journal of theoretical probability Vol. 34; no. 2; pp. 728 - 754
• Main Authors: Hashorva, Enkelejd, Mishura, Yuliya, Shevchenko, Georgiy
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2021; Springer Nature B.V
• Subjects: Asymptotic properties; Gaussian process; Hilbert space; Lower bounds; Mathematics; Mathematics and Statistics; Metric space; Normal distribution; Optimization; Probability Theory and Stochastic Processes; Statistics; Reproducing kernel Hilbert space; Gaussian process; Compact metric space; 60F10; 60G70; Large deviations; Cameron–Martin theorem; Constrained quadratic optimization problem; Boundary non-crossing probability; 60G15
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-020-01002-3

We study boundary non-crossing probabilities
P
f
,
u
:
=
P
(
∀
t
∈
T
X
t
+
f
(
t
)
≤
u
(
t
)
)
for a continuous centered Gaussian process
X
indexed by some arbitrary compact separable metric space
T
. We obtain both upper and lower bounds for
P
f
,
u
. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case
P
y
f
,
u
,
y
→
+
∞
, which are two-term approximations (up to
o
(
y
)
). The asymptotics are formulated in terms of the solution
f
~
to the constrained optimization problem
h
H
X
→
min
,
h
∈
H
X
,
h
≥
f
in the reproducing kernel Hilbert space
H
X
of
X
. Several applications of the results are further presented.