## The Annals of Probability

### Sample Boundedness of Stochastic Processes Under Increment Conditions

Michel Talagrand

#### 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

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