Asymptotics Outside the Central Region

Until now we have focused on “typical” outcomes for a random walk in the “central region” of the probability distribution, which contains all of its weight as the number of steps N tends to ∞. For any finite N , now matter how large, however, there is always some chance of finding the random walk outside the central region. Such “extreme events” are not controlled by the Central Limit Theorem, and intead their probabilities depend sensitively on the “tails” of the step distribution. There is a special class of distributions for which the probabilities of extreme events follow universal law – those with “fat” power-law tails, discussed in the previous lecture. In the first part of this lecture, we explain intuitively why, in a random walk with fat-tailed IID steps, the power-law tail is preserved in the final distribution. We also sketch a proof of the additivity of the power-law tail amplitude, analogous to a (divergent) cumulant, by investigating the singularity in the characteristic function. In the second part, we move on to give a brief exposition of Laplace’s method and its generalization to the complex plane (the steepest-descent or saddle-point method). We show how the latter can be used to obtain uniformly valid asymptotic approximations for the PDF of a random walk, with an analytic characteristic function. The approximation is valid even far outside the central region, provided that N is sufficiently large (although, as we will see in the next lecture, it is not always a good approximation for small N).