# Noise correlations depending on distance

If one has noise being generated over time $t\in[a,b]$, and the correlations of the value of the noise at times $t_0$ and $t_1$ turn out to be distributed according to a density function that depends only on the distance $|t_0-t_1|$ between them, then what is the distribution of the noise? How do the correlations determine the global distribution? Does it, for example, have to be equivalent to some version of the Ising model necessarily?

Thanks a lot!

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First I think you need to distinguish between what we call the marginal and joint distributions. Let N(t) denote the noise at time t. The marginal distirbution would be the distribution of the noise at a fixed time t. We might assume that this distribution doesn't change with t. Another distribution is a bivariate distribution which represents the distribution of the pair (N(t$_0$), N(t$_1$)). All that you are telling me is that this joint distribution depend on the time only through the different ∆t=t$_1$-t$_0$. That is not enough to determine the joint distribution of (N(t$_0$), N(t$_1$)) and it tells me nothing about the marginal distribution of N(t).