The Multiplicative Central Limit Theorem via Log Returns

Summary

The document gives a short route from repeated multiplication to the ordinary central limit theorem. For positive, independent, identically distributed random variables, the product of successive factors becomes a sum after taking logarithms. If the logarithms satisfy the usual central limit theorem conditions, their normalized sum approaches a normal distribution; exponentiating that result motivates a lognormal approximation for the product. This is the mathematical connection behind common models of compounded returns and wealth.

The explanation is only a sketch, not a full proof, and it tightens the original question’s assumptions: finite mean and standard deviation of the original factors alone do not guarantee the required conditions for their logarithms. The variables must be positive for the logarithm to be defined, and the usual CLT needs suitable finite moments for log factors. The limiting statement concerns suitably centered and scaled log products; without that normalization, products may grow or shrink rather than converge to a fixed lognormal distribution.

Key ideas

• Taking logarithms turns a product of positive random factors into a sum.
• The ordinary central limit theorem applies to the sum when log factors meet its assumptions.
• Exponentiating an approximately normal log product yields a lognormal approximation.
• Finite mean and variance of the original factors alone may not ensure finite moments for their logarithms.
• The limit requires normalization of the log product and does not imply convergence of unscaled wealth to a fixed distribution.

Tags

# Does anyone know of a proof of the multiplicative central limit theorem?


# Does anyone know of a proof of the multiplicative central limit theorem?












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?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/73610

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.