What is the definition of correlation length for discrete stochastic process $\{ X_i \}$?
We define variance $\text{var}(X) := E[(X - E[X])^2]$, standard deviation $\text{std}(X) := \sqrt{\text{var}(X)}$, covariance $\text{cov}(X, Y) := E[(X - E[X])(Y - E[Y])]$, correlation $\text{cor}(X, Y) := \frac{\text{cov}(X, Y)}{\text{std}(X)\text{std}(Y)}$. We assume that $\{ X_i \}$ is stationary process. I found there are two definitions of correlation length $\xi$ in physics,
- $\langle X_i X_{i+n} \rangle = E[X_{i}X_{i+n}] \propto \exp{\left(-\frac{n}{\xi}\right)}$
- $\langle (X_i - \langle X_i \rangle)(X_{i+n} - \langle X_{i+n} \rangle) \rangle = \text{cov}(X_i, X_{i+n}) \propto \text{cor}(X_i, X_{i+n}) \propto \exp{\left(-\frac{n}{\xi}\right)}$
Here, $E[X] := E[X_i]$ and $\text{cov}(X_i, X_{i+n}) = E[X_iX_{i+n}] - (E[X])^2$, so if $E[X] = 0$ then $E[X_i X_{i+n}] = \text{cov}(X_i, X_{i+n})$ and the two definitions are same.
To my understanding, for example, if I know that $\text{cor}(X_i, X_{i+n}) = p^n, p \in [0, 1]$, then the correlation length is defined according to def. 2,
$$ p^n = \alpha \text{exp}\left( - \frac{n}{\xi} \right), \alpha \in \mathbb{R} \\ n \log p = \log \alpha - \frac{n}{\xi} \\ \xi = \frac{1}{ - \log p + \frac{\log \alpha}{n}} $$
Is it right?
And is the definition 1 always thought as $E[X] = 0$?
