The Annals of Probability

Sample Boundedness of Stochastic Processes Under Increment Conditions

Michel Talagrand

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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

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

First available in Project Euclid: 19 April 2007

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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.)

Sample boundedness moment conditions majorizing measures


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

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