# Linear process autocovariance function converges to zero as h goes to infinity

Assume that time series $$(X_t)$$ is given by: $$$$X_t = \sum_{i=0}^{\infty} c_i \varepsilon_{t - i},$$$$ where $$(\varepsilon_t)$$ is a weak white noise $$\text{WN}(0, \sigma^2)$$ and sequence $$(c_n)$$ converges in $$l^2$$.

How to prove, that the autocovariance function: $$$$\gamma_X(h) = Cov(X_{t+h}, X_t)$$$$ converges to zero, as $$h \rightarrow \infty$$?

• Commented Mar 3, 2021 at 12:34

We know that $$$$\gamma_X(h) = \sigma^2\sum_{i = 0}^\infty c_ic_{i+h}.$$$$ To test the convergence consider $$$$\frac{1}{\sigma^2}|\gamma_X(h)| \leq \sum_{i=0}^\infty|c_i|\cdot|c_{i+h}| \leq \sqrt{\sum_{i=0}^\infty|c_i|^2}\cdot\sqrt{\sum_{i=0}^\infty|c_{i+h}|^2} = ||c_i||_{\mathcal{l}^2}\cdot||c_{i+h}||_{\mathcal{l}^2}.$$$$ As $$(c_i)$$ is a member of the $$\mathcal{l}^2$$ space, so $$||c_i||<\infty$$. What remains is to check $$\lim_{h\rightarrow\infty}||c_{i+h}||_{\mathcal{l}^2}$$. $$$$||c_{i+h}||_{\mathcal{l}^2}^2 = \sum_{i=0}^\infty|c_{i+h}|^2 = \sum_{i=h}^\infty|c_i|^2 \rightarrow 0,\;\;\text{as}\;\;h\rightarrow\infty.$$$$ Which holds because the $$\mathcal{l}^2$$ space is complete and tails of convergent series converge to zero.
Hence $$$$|\gamma_X(h)| \leq \sigma^2 ||c_i||_{\mathcal{l}^2}\cdot||c_{i+h}||_{\mathcal{l}^2} \rightarrow 0,\;\;\text{as}\;\;h\rightarrow\infty.$$$$