#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 $EX_t=0$, $E|X_t|^2<infty$, 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.


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