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I know that the pdf of a power law distribution is $$ p(x) = \frac{\alpha-1}{x_{\text{min}}} \left(\frac{x}{x_{\text{min}}} \right)^{-\alpha}$$

But what does it intuitively mean if, for example, stock prices follow a power law distribution? Does this mean that losses can be very high but infrequently?

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2 Answers 2

This means that a stylized fact about stock returns distributions, high kurtosis or "fat tails", is better taken into account with a "Pareto-type distribution" than with a normal distribution. The latter has a finite variance, whereas the former is such that: "all moments $m \geq \alpha - 1$ diverge: when α < 2, the average and all higher-order moments are infinite; when 2 < α < 3, the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge - as more data is accumulated, they continue to grow."(Source: Wikipedia)

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A power law distribution just means that y = ax^n for some a,n. In practice, and for large values of x, a polynomial can be approximated as a power law where n = the degree of the polynomial. (Ie ignore terms x^(n-1), x^(n-2) etc as these will be much smaller than ax^n for large x).

So y = 5x^1000 is a power law. y = 2x^67 + x^2 can be approximated as a power law y = 2x^67 for large x. y = e^x and y = sin(x)^10 are not power series.

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Surely you realized that the question is about probability distribution functions. –  Did Feb 16 '13 at 7:49

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