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October, 1976 An Ergodic Theorem for the Square of a Wide-sense Stationary Process
A. Larry Wright
Ann. Probab. 4(5): 829-835 (October, 1976). DOI: 10.1214/aop/1176995986


Let $\{X(t), -\infty < t < \infty\}$ be a stochastic process which is stationary in the wide sense with spectral representation $X(t) = \int^\infty_{-\infty} e^{it\lambda} d\xi(\lambda)$, where the $\xi$ process is centered and has independent increments with $E\xi(\lambda) \equiv 0, E|\xi(\lambda)|^2 < \infty$. It is shown that under weak conditions $$P - \lim_{T\rightarrow\infty} \frac{1}{2T} \int^T_{-T} |X(t)|^2 dt$$ exists and is equal to $\sigma^2 + \sum J_t^2 + \sum \xi_n^2$, where $\sigma^2$ is equal to the variance of the Gaussian component of the continuous part of the $\xi$ process, $\sum J_t^2$ is the sum of the squares of the jumps of the Gaussian component of the $\xi$ process, and $\xi_N = \xi(\lambda_N + 0) - \xi(\lambda_N - 0)$, where $\{\lambda_N\}$ are the fixed discontinuities of the $\xi$ process.


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A. Larry Wright. "An Ergodic Theorem for the Square of a Wide-sense Stationary Process." Ann. Probab. 4 (5) 829 - 835, October, 1976.


Published: October, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0347.60033
MathSciNet: MR418207
Digital Object Identifier: 10.1214/aop/1176995986

Primary: 60G10
Secondary: 60G17

Keywords: Ergodic , Stationary in the wide sense , stochastic process with independent increments

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 5 • October, 1976
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