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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 infrequent? |
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This is an heavy tailed distribution, since the cdf is
$$
F(x) = 1 - \left( \dfrac{x}{x_\min} \right)^{1-\alpha}
$$
So the probability to exceed $x$, $(x/x_\min)^{1-\alpha}$ can be made arbitrarily close to $1$ by the proper choice of $\alpha$. For instance, if one wants the probability to exceed $10^u x_\min$ to be at least $0.9$, one should pick $\alpha$ to be at most
$$
1-\log_{10}(0.9)/u
$$
a curve represented below, with the first axis being scaled by $u$, not by $10^u x_\min$...
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It's not a peer-reviewed source, but I like this note by CMU stats professor Cosma Shalizi. He's also an author on this article, about estimating such things from data. |
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