The picture below is Definition 1.1.1 and two Remarks in page 4 of Book Nualart: The Malliavin Calculus and Related Topics (2ed).

I was able to understand the definition, it is the "Gaussian process" indexed by a Hilbert space. But I cannot figure out the second remark, as asked in the title, how could the joint normal distribution be available from multivariate normal distribution and linearity?

Any comments or hints will be appreciated.

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By a Gaussian family, it means that, for any $h_1, \ldots, h_n$, $W(h_1), \ldots, W(h_n)$ are jointly normal. That is, for any real numbers $a_1, \ldots, a_n$, $$a_1 W(h_1) + \cdots + a_n W(h_n)$$ is normal, which is obviously true, as, from the linearity, \begin{align*} a_1 W(h_1) + \cdots + a_n W(h_n) = W(a_1h_1 + \cdots + a_n h_n), \end{align*} which is normal by definition.

  • $\begingroup$ Thank you for the answer. As you said, $W(h_1),\ldots,W(h_n)$ are joint normal if and only if $a_1 W(h_1) + \cdots + a_n W(h_n)$ is normal for any $a_1, \ldots, a_n\in\mathbb R$, but why? I know the joint normality implies the normality of linear combination, but how about the converse? Thanks again. $\endgroup$ – Dreamer Oct 13 '16 at 6:19
  • $\begingroup$ I would assume that you know this or how to find it. $\endgroup$ – Gordon Oct 13 '16 at 11:32
  • $\begingroup$ Sorry I am not able to figure it out. Do you konw any reference about this? If you know, please help, thank you. $\endgroup$ – Dreamer Oct 13 '16 at 11:47
  • 1
    $\begingroup$ See Theorem A.5 in Oksendal's book. $\endgroup$ – Gordon Oct 13 '16 at 13:00

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