### 2015 UC Berkeley / UC Davis Joint Statistics Colloquium

**This year, the 2015 UC Berkeley / UC Davis Joint Statistics Colloquium will be hosted by UC Berkeley on Tuesday, April 28th. Please see** **http://statistics.berkeley.edu/** **for details**

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**Seminar Date: ** Tuesday, April 28th 2015 (Time: 4:00pm-5:00pm)

**Location: ** 60 Evans Hall, Department of Statistics, **UC Berkeley**

**Speaker: Debashis Paul,** UC Davis

**Title: ** **Spectral analysis of linear time series in high dimensions**

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**Abstract: ** We study the spectral behavior of a class of $p$-dimensional stationary linear processes in two different high-dimensional regimes,

(A) $p/n \to c \in (0,\infty)$; and (B) $p/n \to 0$, as $p,n \to \infty$. The key structural assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the coefficient matrices in the linear process representation are Hermitian and simultaneously diagonalizable.

The object of our study is the symmetrized sample autocovariance matrix $S_\tau$ for any given lag order $\tau=0,1,…$.

In setting (A), we show that the empirical distribution of the eigenvalues of $S_\tau$ converges to a nonrandom limiting distribution characterized by a system of nonlinear equations uniquely describing their Stieltjes transforms. In setting (B), we show that the empirical distribution of the eigenvalues of $\sqrt{n/p}(S_\tau - \Sigma_\tau)$ converges to a nonrandom limiting distribution. Various potential applications and extensions of these results and possible relaxations of the assumptions will be discussed.