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I have a general question about stochastic processes. Is the linear combination of any number of stochastic processes, also a valid stochastic process?

What about non-linear combinations?

Any book/reference to read about it?

I know that the linear combination of two independent Gaussian Processes is also a Gaussian process and hence a valid stochastic process. But for the Gaussian case is easy to show because the linear combination of Gaussian random variables are also a Gaussian. So my question is for "any" stochastic process beyond the Gaussian one.

Thanks so much.

jdeJuan
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    Stochastic process is a collection of random variables, linear combination of random variables is a random variable, so why not? – Tim May 05 '21 at 16:01
  • It comes down to what you mean by "non-linear combination." If you are referring to collections of processes *defined on the same parameter space* and are operating on them component by component, then your question reduces to "when are combinations of random variables also random variables," which has an easy answer. Possibly an adequate definition of "stochastic process" will help you: see https://stats.stackexchange.com/questions/48911 and https://stats.stackexchange.com/questions/126791, for instance. – whuber May 05 '21 at 18:50
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    Thank you. I was thinking more in that stochastic processes are infinite collections of random variables. But I agree that they are no more than "random variables", so linear combinations of them are also valid ones. I guess that same applies to non-linear combinations. Do all non-linear combinations applied to random variables provide a random variable? – jdeJuan May 06 '21 at 16:21
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    Almost: when "non-linear combinations" are *measurable functions,* they are guaranteed to produce a random variable. Otherwise, all bets are off. – whuber May 07 '21 at 15:02

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