Definition:Sample Path of Stochastic Process

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {E, \EE}$ be a measurable space.

Let $\family {X_i}_{i \mathop \in I}$ be an $E$-valued stochastic process.

For each $\omega \in \Omega$, define $p_\omega: I \to E$ by:

$\map {p_\omega} i = \map {X_i} \omega$

for each $i \in I$.

We call the mappings $p_\omega$ the sample paths of $\family {X_i}_{i \mathop \in I}$.