# #StackBounty: #time-series #probability #mathematical-statistics #expected-value #covariance-matrix Upperbound for Norm of Lagged Autoc…

### Bounty: 100

Suppose $$X_t$$ is a vector valued time series in $$mathbb{R}^d$$. Assume for the moment that $$X_t$$ is stationary with $$EX_t=0$$, $$E|X_t|^2, and let

$$C_h = E[X_0 X_h^top]$$
denote the autocovariance matrix at lag $$h$$. What I wish to show is that

$$|C_h|2 le |C_0|_2,$$
or find a counterexample to this statement. Here $$|cdot|_2$$ is the Hilbert-Schmidt Norm:
$$|A|_2^2 = sum$$
{i,j=1}^d a_{i,j}^2. \$\$

So far all I can show is the weaker statement that
$$|C_h|_2 le trace(C_0).$$

To see this, we have by the Cauchy-Schwarz inequality for expectation and stationarity that
$$|C_h|2^2 = sum{i,j=1}^d (E[X_{0,i}X_{h,j}])^2 le sum_{i,j=1}^d E[X_{0,i}^2]E[X_{h,j}^2] = sum_{i,j=1}^d E[X_{0,i}^2]E[X_{0,j}^2] =[trace(C_0)]^2.$$
Part of me believes that this bound must be sharp (i.e. there is a counter example), but I really have no idea! Any help/advice is much appreciated.

Get this bounty!!!

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