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Suppose that we generate many pairs time series that follows a bivariate normal distribution with a known correlation rho. What is the distribution of the correlation once the x and y are smoothed with a simple moving average of window size n?

I ask this question because often I see pairs of times series whose correlation raise sharply once it is smoothed, and I am trying to see if there is a rigorous way to calculate the significance of this rise?

Obviously I can simulate and get a distribution, but an analytical form would be nice.

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