The Annals of Probability

Sample Boundedness of Stochastic Processes Under Increment Conditions

Michel Talagrand

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Abstract

Let $(T, d)$ be a compact metric space of diameter $D$, and $\|\cdot \|_\Phi$ be an Orlicz norm. When is it true that all (separable) processes $(X_t)_{t \in T}$ that satisfy the increment condition $\|X_t - X_s\|_\Phi \leq d(t, s)$ for all $s, t$ in $T$ are sample bounded? We give optimal necessary conditions and optimal sufficient conditions in terms of the existence of a probability measure $m$ on $T$ that satisfies an integral condition $\int^D_0 f(\varepsilon, m(B(x, \varepsilon))) d\varepsilon \leq K$ for each $x$ in $T$, where $f$ is a function suitably related to $\Phi$. When $T$ is a compact group and $d$ is translation invariant, we are able to compute the necessary and sufficient condition in several cases.

Article information

Source
Ann. Probab., Volume 18, Number 1 (1990), 1-49.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990936

Digital Object Identifier
doi:10.1214/aop/1176990936

Mathematical Reviews number (MathSciNet)
MR1043935

Zentralblatt MATH identifier
0703.60033

JSTOR
links.jstor.org

Subjects
Primary: 60G17: Sample path properties
Secondary: 28A99: None of the above, but in this section 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords
Sample boundedness moment conditions majorizing measures

Citation

Talagrand, Michel. Sample Boundedness of Stochastic Processes Under Increment Conditions. Ann. Probab. 18 (1990), no. 1, 1--49. doi:10.1214/aop/1176990936. https://projecteuclid.org/euclid.aop/1176990936


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