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Definition: A $(\mathbb R,\cal B)$ valued stochastic process, i.e. a family $(X_t)_{t\in I}$ of real valued random variables on the same probability space $(\Omega,\cal F,\mathbb P)$ is called gaussian process, if every finite subsystem is multidimensional normal distributed.

I do not know exactly how to work with this definition, especially what does 'subsystem' mean. An other definition , which is more handy, is given by:

Definition: $Z=(Z_1,\dots,Z_d)\in \mathbb R^d$ is a gaussian vector if any linear combination $\sum_{i=1}^n \lambda_i Z_i$ is a gaussian random variable in $\mathbb R$.

Definition: The stochastic process $(X_t)_{t\ge 0}$ is gaussian if $(X_{t_1},\dots X_{t_n})$ is a gaussian vector for any choice of $t_i, \,i=1,\dots ,n$.

Could someone explain my definition and why this is equivalent to theirs?

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The first definition is equivalent to the third, as you could interpret finite subsystem as finite subset of a stochastic process (so $(X_{t_1},\dots X_{t_n})$ is a finite subsystem of $(X_t)_{t\in I}$, because $(X_{t_1},\dots X_{t_n})\subseteq (X_t)_{t\in I}$). A normal distribution is a Gaussian distribution, so multidimensional normal distributed is equivalent to gaussian vector.

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  • $\begingroup$ @J.D The linear combination is to create a vector (with of course all of them linear independent elements), and a multidimensional subsystem is also a vector. $\endgroup$ Jan 20 '18 at 0:30

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