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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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    $\begingroup$ 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. $\endgroup$
    – user10525
    Jun 19, 2012 at 20:17

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My quick answer would be yes, but I am not sure about the scale parameter. You can view a Gaussian random walk as a subset of random walks with stable distributions. All stable distributions have the property that a linear combination of two i.i.d. stable distributions is also stable. (All this is related to a generalized central limit theore and functional analysis, but that's too much to deal with here.)

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