# Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on the topic returned Sven Åberg's paper on Wishart-Levy matrices, but the results and methods in that paper are in a sense insufficient. Is there any exact result available?

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you'll have to tell us how the $a_i$'s at different times are correlated; if they're uncorrelated, then the $X_i$'s will just have a normal distribution, won't they? – Carlo Beenakker Dec 20 '13 at 20:48
The $a_i$'s are uncorrelated. And I agree, the $X_i$'s have a normal distribution. What I find difficult is that the $X_i$'s are also auto-correlated as a Markov chain. Hence W does not have a Wishart distribution, which as I understand is valid only if $X_i(t)$ is independent of $X_i(t')$ for any $t\neq t'$. Please kindly correct me if I get something wrong :) – Sinbaski Dec 27 '13 at 14:57
Is $X$ a matrix or a vector? Is $X_i$ a row, column, scalar, or something else? Is $\phi_i$ a scalar or a matrix? I take it $\phi_i$ are non-stochastic. If the entries of $X$ are Gaussian and independent of one another at a fixed time, then $XX^T$ will have the Wishart distribution. Autocorrelations across time should not change that conclusion as long as entries of $X$ are independent at a fixed time. – Will Nelson Dec 29 '13 at 9:44
Hi, X is a NxT matrix, $\{\phi_i\}_{i=1}^N$ are N constant scalars. I did some calculations in the simple case where $\phi_1=\phi_2=\cdots=\phi_N=\phi$ and found that the joint PDF of the entries of $XX^T$ is $w(AA^T) = w(RMM^TR^T)$, where $w(\cdot)$ denotes the Wishart joint PDF and the M matrix has entries $M_{ij} = \delta_{ij} -\phi\delta_{i+1,j}$. The main point is to use the property: if $A = CBC^T$ has a Wishart distribution $W(\Sigma, T)$, and C is non-singular, then B also has a Wishart distribution $W(C^{-1}\Sigma (C^{T})^{-1}, T)$. I look forward to your comments! – Sinbaski Dec 31 '13 at 21:03