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Let $X_{1}$,..., $X_{n}$ be random variables, and define $$Y_{k} := \sum_{i=1}^{k} X_{i}, k = 1,...,n.$$ Suppose that $Y_{1}$,..., $Y_{n}$ are jointly Gaussian. Determine whether or not $X_{1}$,...,$X_{n}$ are jointly Gaussian.

I do not know how to solve it!

givan
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  • Hint : use a linear transformation to relate the Y and X variables, then relate their laws. Dedcue that one of them being jointly gaussian should imply the other one being jointly gaussian. – Sarvesh Ravichandran Iyer Jan 22 '20 at 14:07

1 Answers1

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$(X_1,X_2,...,X_n)=(Y_1,Y_2-Y_1,...,Y_n-Y_{n-1})$. This implies that $X_i$'s are jointly Gaussian.

Kavi Rama Murthy
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  • can you give me a more detailed explanation? what does jointly Gaussian exactly mean? – givan Jan 22 '20 at 15:53
  • @givan Jointly Gaussian random variables are those whose densities have as specific form but it can be shown that $(X_1,X_2,...,X_n)$ is jointly Gaussian iff every linear combination of $X_i$" has a normal distribution on the real line. – Kavi Rama Murthy Jan 22 '20 at 23:25
  • How can I understand "every linear combination of $X_{i}$'' has a nornal distribution on the real line", especially the real line. How do $X_{i}$ correlate to the $Y_{i} - Y_{i-1}$? Because I cannot find the definition of jointly Gaussian. – givan Jan 23 '20 at 00:25
  • If you do not know what jointly Gaussian means you should search the net for either 'jointly Gaussian' or 'multi-dimensional Gaussian /normal distribution'. @givan – Kavi Rama Murthy Jan 23 '20 at 00:27