# Generalization of Brownian motion to $\alpha$-stable distributions

Brownian motion is constructed as a limit of the sum i.i.d. Gaussian increments. Can one use a non-Gaussian $\alpha$-stable distribution (e.g. the Cauchy distribution) instead, and still construct a process? Would the scale parameter of such process evolve according to the formula $c_t = t^{1/\alpha}$?

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An even wider generalisation are Lévy processes. Given that "The probability distributions of the increments of any Lévy process are infinitely divisible" and the family of $\alpha-$stable distributions is a well-known class of infinitely divisible distributions. –  user10525 Jun 19 '12 at 20:17