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I have been told there is a multiplicative CLT. It says that - no matter the shape of returns distributions - if you multiply consecutive iid RVs (centered at 1.1, for instance), then a lognormal is the limiting distribution for prices/wealth. The only criteria I know of is that, for this to work, both the mean and stdev must be finite (no much of a restriction).

First, is my statement of the problem sensible? Second, if so, where can I find a proof of what is essentially the multiplicative Central Limit Theorem?

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    $\begingroup$ Just take logs and apply CLT. $\endgroup$
    – fes
    Commented Oct 27, 2022 at 6:41

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Suppose $X_i$ are positive and iid. Then the multiplicative random variable $$Y_n =X_1 X_2 …….X_n$$ Hence $$log Y_n= logX_1 + logX_2 +….logX_n$$ which is the sum of iid RVs and therefore has a normal distribution in the limit, by the regular CLT. This means that $Y_n$ has a lognormal distribution in the limit.

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  • $\begingroup$ Easier than I thought. I'm surprised this doesn't get explained anywhere I have looked. $\endgroup$
    – eSurfsnake
    Commented Oct 27, 2022 at 16:50

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