Power-Law Tails and Martingale Limit Behavior

Summary

The document raises a theoretical question about the long-run behavior of observations from a martingale when the sampling distribution has power-law tails. It contrasts the familiar setting in which observations have finite variance and central limit arguments can lead to Brownian motion with the possibility of heavy-tailed returns. It asks whether that change would instead produce fractional Brownian motion.

The motivation is financial: the question notes that Mandelbrot used fractional Brownian motion to model stock returns associated with power-law distributions, but the author could find no justification beyond visual resemblance. The document offers no answer, derivation, references, or empirical evidence, so it does not establish which limiting process applies. It is useful as a framing of the issue, while leaving key assumptions unspecified, including the tail exponent and dependence structure. Heavy tails alone do not settle the limit; the question itself highlights the need to distinguish distributional tails from temporal dependence.

Key ideas

• The question compares martingale central limit behavior under finite variance with behavior under power-law tails.
• It asks whether heavy-tailed sampling leads to fractional Brownian motion as a limit.
• The document distinguishes Mandelbrot’s visual motivation from a formal justification.
• It supplies no result or proof, and the tail and dependence assumptions remain unspecified.

Tags

# Martingales with power-law tails and CLT


# Martingales with power-law tails and CLT












I'm writing a course paper on stable distributions and I couldn't find any source discussing limits of Martingales with power-law tails. Suppose we have a Martingale that produces IID observations at constant intervals out of a distribution with finite variance, let's say once every moment t. If we collect observations of the Martingale on long enough intervals, eg. every 30t, following the Martingale CLT (and Classical CLT), our observations have the limit of Brownian motion.

What happens if the sampling distribution has power-law tails? Do we get fractional Brownian motion as the limit?

I know Mandelbrot refers often to fractional Brownian motion and uses it to model stock returns which he assumes come from a distribution with power-law tails. However, I couldn't find any source where he gives any reasons other than visual for doing so.

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