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Compact Metric Space
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Gaussian Process
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Regularity of gaussian processes
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Continuity Properties of Some Gaussian Processes
Continuity Properties of Some Gaussian Processes,10.1214/aoms/1177692721,The Annals of Mathematical Statistics,Christopher Preston
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Continuity Properties of Some Gaussian Processes
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Citations: 18
)
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Christopher Preston
Let $(S, d)$ be a compact metric space; let $(\Omega, \mathscr{F}, P)$ be a probability space, and for each $t \in S$ let $X_t: \Omega \rightarrow \mathbb{R}$ be a random variable, with $E(X_t) = 0$ and such that $\{X_t\}_{t\in S}$ forms a Gaussian process. In this paper we find sufficient conditions for the
Gaussian process
$\{X_t\}_{t\in S}$ to admit a separable and measurable model whose sample functions are continuous with probability one. The conditions involve the covariance, $E(X_s, X_t)$, of the process and also the $\varepsilon$entropy of $S$.
Journal:
The Annals of Mathematical Statistics
, vol. 43, no. 1972, pp. 285292, 1972
DOI:
10.1214/aoms/1177692721
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Citation Context
(7)
...upper bound in its original form is due to Fernique (building on earlier ideas of Preston) [4, 12,
13
], while the lower was established by Talagrand in [15]...
B. Klartag
,
et al.
Empirical processes and random projections
...Thus we can apply the continuity criterion due to Preston and Garsia [
40
, 41] as formulated in Theorem 5.1 in [42]...
R. Gielerak
,
et al.
Stochastically positive structures on Weyl algebras. The case of quasi...
...The idea of majorizing measure first appeared in
Preston (1972)
and was used by Fernique (1974) and Talagrand (1987) in their study of sample paths continuity for Gaussian processes...
Juan Romo
.
The central limit theorem for empirical processes on VČ classes: A ma...
...For some recent work on sample continuity of Gaussian processes, the reader is referred to Eaves [2], Garcia, Posner, and Rodemich I3], Nisio [6], and Preston [
10
]...
Michael F. Driscoll
.
The reproducing kernel Hilbert space structure of the sample paths of ...
...Thus (2.2) means that (Xt) is subgaussian wrt the pseudometric d on T .W e then combine the majorizing measure theorems of Preston [Pr1],[
Pr2
] (see also [Fe]) and Talagrand [T] to get 116 J. Bourgain �...
J. Bourgain
.
On the IsotropyConstant Problem for “PSI2”Bodies
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Citations
(18)
Asymptotic expansions for functions of the increments of certain Gaussian processes
Michael B. Marcus
,
Jay Rosen
Journal:
Stochastic Processes and Their Applications  STOCH PROC APPL
, vol. 120, no. 2, pp. 195222, 2010
A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes and loop soups
Michael B. Marcus
,
Jay Rosen
Published in 2010.
An almost sure limit theorem for Wick powers of Gaussian differences quotients
(
Citations: 1
)
Michael B. Marcus
,
Jay Rosen
Published in 2009.
Continuity and Boundedness of Infinitely Divisible Processes: A Poisson Point Process Approach
(
Citations: 17
)
Michael B. Marcus
,
Jan Rosiński
Journal:
Journal of Theoretical Probability  J THEOR PROBABILITY
, vol. 18, no. 1, pp. 109160, 2005
Empirical processes and random projections
(
Citations: 14
)
B. Klartag
,
S. Mendelson
Journal:
Journal of Functional Analysis  J FUNCT ANAL
, vol. 225, no. 1, pp. 229245, 2005