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A random vector $X$ is defined to be multivariate normal, if any linear combination of its components is a 1-dimensional normal distributed.

I wonder when partitioning $X$ into $(X_1, X_2)$, if $X_1$ conditional on $X_2$ equal any value is multivariate normal and $X_2$ is multivariate normal, is $X$ multivariate normal?

Thanks!

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No, not in general. Consider the example wehere $X_1$ and $X_2$ are scalars, $X_1|X_2$ is normal with zero means and variance $\exp(X_2)$.

However, if you add the additional condition that $X_1|X_2$ is multivariate normal with the mean a linear transformation of $X_2$ and a constant (i.e. non depending on $X_2$) covariance matrix, then the joint of $X_1$ and $X_2$ is multivariate normal. To prove that, you need to complete the squares of two quadratic forms in the exponential and use the properties of the determinant of block matrices. (The proof is straightforward but tedious. Maybe there is a simpler proof!)

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    *simpler proof*... Use characteristic functions.2012-11-21
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    - did is right !2012-11-21
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    Thanks! How is it done by characteristic functions?2012-11-21
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    I think the steps are as follows: 1) Compute the characteristic function of $X_1,X_2$. 2) Apply Fubini by first integrating over $X_1|X_2$ and then $X_2$. This will result in exponential of a quadratic form plus a linear form multiplied by the imaginary number (both in the argument of the characteristic function). Look at http://en.wikipedia.org/wiki/Multivariate_normal#Definition for how to manipulate the characteristic function of a multivariate normal.2012-11-21
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    You still have to complete squares in the exponential function but no need to handle determinants.2012-11-21
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    (1) What is the relation between characteristic function of X and characteristic functions of $X_1 | X_2=x_2$ and of $X_2$? (2) Is your step 2 "Apply Fubini by first integrating over X1|X2 and then X2" trying to determine the characteristic function of X?2012-11-21
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    Use $E[\exp(i u^T X)]=E[\exp(i u_1^T X_1+ i u_2^T X_2)]$. Under the integral sign, write the joint density $f(x_1,x_2)=f(x_1|x_2)f(x_2)$. Integrate first over $X_1|X_2$ by applying the formula for the characteristic function of that multivariate normal, etc...2012-11-21