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Principle of the single big jump

Learned about an interesting probability principle about random walk called "principle of the single big jump"

A high overall displacement with respect to the orgin resulting from doing random walk might be essentially contributed by one single very large step, i.e. a leap.

Technically, assume the step sizes are independent random variables X_1, X_2, \cdots with heavy-tailed (technically, subexponential) distributions, then the maximum and the sum have the same asymptotic distribution. That is, as x goes to infinity,

\lim{x \rightarrow \infty} P( X_1 + X_2 + \cdots + X_n > x ) = \lim{x \rightarrow \infty} P( max(X_1, X_2, \cdots , X_n > x )




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