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I have a time series $\{X_t\} = Z_t + \lambda(Z_{t-1} + Z_{t-2} + ... )$, where $Z_t$ is a White Noise process.

Let $\{Y_t\} = X_t - X_{t-1}$

How does $$\{Y_t\} = X_t - X_{t-1} = (Z_t + \lambda\sum_{j=1}^{\infty}Z_{t-j}) - (Z_{t-1} + \lambda\sum_{j=1}^{\infty}Z_{t-1-j}) = Z_t + (\lambda - 1)Z_{t-1}$$

I don't see how those summations can be simplified into $Z_t + (\lambda - 1)Z_{t-1}$

jon givony
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  • The simplification cannot be done since $\sum_{j=1}^\infty Z_{t-j}$ might not exist as a limit of $\sum_{j = 1}^n Z_{t-j}$ and as such you may be in trouble when subtracting the two infinite sums from each other. – Therkel Oct 11 '17 at 09:45
  • Did you mean to write $X_t = Z_t + \lambda Z_{t-1} + \lambda^2 Z_{t-j} + \ldots$? In that case, for $\lvert \lambda\rvert <1$ then $X_t$ can be written as $X_t = Z_t + \sum_{j=1}^\infty \lambda^j Z_{t-j}$ for which the subtraction of the two sums is fine (then ${X_t}$ is a stationary time series and is an $\operatorname{AR}(1)$ process). – Therkel Oct 11 '17 at 09:49

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