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sample function (Definition)

Let $ \lbrace X(t)\mid t\in T \rbrace$ be a stochastic process, where $ X(t)$ is a random variable on the probability space $ (\Omega,\mathcal{F},\textbf{P})$. Writing $ X(t)$ as $ X(t,\omega)$, where $ t\in T$ and $ \omega\in\Omega$, we see that if we fix the sample point $ \omega$, we have a function in $ t$: $ X_{\omega}(t) \colon t\mapsto X(t)$. This function $ X_{\omega}(t)$ of $ t$ is called a sample function, or sample path of the stochastic process.



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Also defines:  sample path
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Cross-references: function, point, fix, probability space, random variable, stochastic process
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This is version 1 of sample function, born on 2005-06-20.
Object id is 7176, canonical name is SampleFunction.
Accessed 1953 times total.

Classification:
AMS MSC: 60G05 (Probability theory and stochastic processes :: Stochastic processes :: Foundations of stochastic processes)
 60G17 (Probability theory and stochastic processes :: Stochastic processes :: Sample path properties)
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