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Let $(Y_t)_{t \in Z}$ be a covariance-stationary stochastic process. According to Hamilton (page 46-47), we say that the process is

  1. Ergodic for the mean if $$\overline{Y}\equiv \frac{1}{T}\sum_{t=1}^T Y_{t} \stackrel{\mathbb{P}}{\to} E[Y_t]=: \mu,\quad \text{as} \quad T \to \infty.$$
  2. Ergodic for the second moments if $$ \frac{1}{T-j}\sum_{t=j+1}^T (Y_{t} - \mu)(Y_{t-j} - \mu) \stackrel{\mathbb{P}}{\to} E\left[(Y_t - \mu)(Y_{t-j} - \mu)\right] =: \gamma_j,\quad \text{as} \quad T \to \infty.$$

Hamilton makes the following comment:

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In the same page, Hamilton says

Sufficient conditions for second-moment ergodicity will be presented in Chapter 7.

Related to the ergodicity for the mean, I managed to identify the Theorem in chapter 7 showing that [3.1.15] implies ergodicity for the mean (see page 186-188):

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I just can't identify what conditions guarantee me ergodicity for the second moments. Could you tell me where in the book he shows this? Or even a way for me to demonstrate this.

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  • $\begingroup$ In definitions 1 and 2, what are convergence modes? Convergence in probability or almost surely or in $L^2$? $\endgroup$
    – Zhanxiong
    Nov 13, 2023 at 23:47
  • $\begingroup$ I believe it is convergence in probability, as theorem 7.5 says that convergence is m.s. and this implies convergence in probability. However, I've seen some books define ergodicity using almost certain convergence. $\endgroup$
    – user346624
    Nov 14, 2023 at 0:27

1 Answer 1

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The relevant result that Hamilton is possibly hinting at is when he is discussing the sufficient conditions for which $$\frac1 T\sum_{t=1}^TY_tY_{t-k}\overset{\mathbb P}{\longrightarrow}\mathbf E(Y_t Y_{t-k});$$

he defines a new random variable $$X_{t, k}:=Y_tY_{t-k}-\mathbf E(Y_tY_{t-k}) =\sum_{u=0}^\infty\sum_{v=0}^\infty\psi_u\psi_v[\varepsilon_{t-u}\varepsilon_{t-k-v}-\mathbf E(\varepsilon_{t-u}\varepsilon_{t-k-v}) ]$$ based on taking the general linear structure of $Y_t=\sum_{j=0}^\infty\psi_j\varepsilon_{t-j},~\sum_{j=0}^\infty|\psi_j|<\infty$ and $\langle \varepsilon_t\rangle$ are iids with finite moments greater than $2.$

Let the information available at $t-m$ be $\Omega_{t-m}\equiv \{\varepsilon_{t-m},\varepsilon_{t-m-1},\ldots\}.$ He then concluded $\langle X_{t, k}\rangle$ is $\rm L^1$-mixingale by showing the expectation of the absolute value of the forecast of $X_{t, k}$ based on $\Omega_{t-m}$ is bounded. In fact, it is uniformly integrable based on proposition $7.7\rm b. $ Rest follows by applying LLN for $\rm L^1$-mixingales as stated in proposition $7.6.$

From this, one can deduce that

$$\frac1 T\sum_{t=k+1}^T(Y_t-\bar{Y}_t)(Y_{t-k}-\bar{Y}_t)\overset{\mathbb P}{\longrightarrow}\mathbf E(Y_t-\mu) (Y_{t-k}-\mu).$$

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