Emergence of power law distributions for odd and even lifetimes

Avalanche lifetime distributions have been related to first-return random walk processes. In this sense, the theory for random walks can be employed to understand, for instance, the origin of power law distributions in self-organized criticality. In this work, we study first-return probability distributions, f^{(n)}, for discrete random walks with constant one-step transition probabilities. Explicit expressions are given in terms of _{2}F_{1} hypergeometric functions, allowing us to study the different behaviors of f^{(n)} for odd and even values of n. We show that the first-return probabilities have a power law behavior with exponent -3/2 only when the random walk is unbiased. In any other case, it presents an exponential decay.