A sufficient condition for linear growth of variances in a stationary random sequence

Abstract:

Suppose $({X_k},k = \ldots , - 1,0,1, \ldots )$ is a weakly stationary random sequence. For each positive integer $n{\text {let }}{S_n} \equiv {X_1} + \cdots + {X_n}$ and $\tau (n) = \operatorname {Sup} \{ |\operatorname {Corr} (\sum _{k = - I}^0{X_k},\sum _{k = m}^{m + I}{X_k})|:m \geqslant n,I \geqslant 0\}$. If Var ${S_n} \to \infty$ as $n \to \infty$ and $\sum _{n = 0}^\infty \tau ({2^n}) < \infty$, then ${n^{ - 1}}$ Var ${S_n}$ converges to a finite positive limit as $n \to \infty$. A bound on the rate of convergence is obtained.