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I am wondering what is auto correlation? From the definition of Markov Chain, the current state should only depend on previous state, why there exist auto correlation?

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  • $\begingroup$ Today is related to yesterday and yesterday is related to the day before ... If I only consider a value at time t and the value two days prior, they'll (typically) be related as well. $\endgroup$
    – Glen_b
    Jul 8, 2018 at 4:25

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Auto-correlation arises in the context of a stochastic time-series $\{ X_t | t \in \mathbb{Z} \}$ where we have one or more variables that vary over time. Within this context, auto-correlation is just the correlation of a variable at two points in time:

$$\mathbb{Corr}(X_t, X_{t+k}).$$

A Markov chain is defined by the property that the current state is conditionally independent of all states before the previous state, when you condition on the previous state. This implies that the conditional auto-correlation is zero when we condition on any intervening state:

$$\mathbb{Corr}(X_t, X_{t+k} | X_{v}) = 0 \quad \quad \quad \text{for all } 0< v < k.$$

This does not imply that the unconditional auto-correlation between separated times will be zero. Since the Markov chain depends on the previous state, and the previous state depends on the one before, and so on, it will generally be the case that states will be auto-correlated over time, beyond just a lag of one.

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  • $\begingroup$ thanks! but i am a little confused about the meaning of conditional auto-correlation and unconditional auto-correlation? $\endgroup$
    – Z宇飛
    Jul 8, 2018 at 9:07
  • $\begingroup$ In that case I would suggest going back and learning a little more about probability theory. In particular, you will need to learn about the difference between marginal and conditional probability distributions, and then marginal and conditional moments. The difference between conditional and unconditional auto-correlation is just a matter of whether or not you condition on some random variable in your analysis. $\endgroup$
    – Ben
    Jul 8, 2018 at 10:00

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