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Online I found many definitions of stochastic processes, but none of them was intuitive or easy to understand. I was wondering if someone could provide me with some example and definition to deeply understand this fundamental topic in time series analysis

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The stochastic process is also called random process or random function if that helps. Technically speaking, it's a set of random variables indexed by time $t$, i.e. $\{X_t\}_{t\in T}$. You can think of it as of a function $X_t = f(t)$ that is randomized. It's closely related to time-series, though as mentioned in Is a time series the same as a stochastic process?, time-series usually refer to discrete time, where the stochastic process can happen in discrete or continuous time.

What are usually referred to as stochastic processes are the ones that are not completely “random” (in the colloquial meaning). If there are some regularities, we can study them or even use them to form predictions or forecasts.

If they are “random” they are less interesting because the index $t$ in $f(t)$ is irrelevant, so there's not that much we could study or predict. However the same as we can have a constant function, the stochastic process does not need necessarily increase, decrease, or have cycles, etc over time. It can be something like white noise, that is just “random”.

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