Multiparameter Walks

Abstract

The discussions of Chapter 3 revolved around multiparameter processes that are formed by considering systems of independent one-parameter random walks. In this chapter we consider properties of genuinely multiparameter random walks. For an example of such a process, suppose each “site” t ∈ ℕ^N corresponds to an independent particle that is negatively charged with probability p and positively charged with probability 1 — p. Let X _t = 1 if the particle at site t ∈ ℕ^N is negatively charged; otherwise, set X _t = 0 Then, the total number of negatively charged particles in the rectangle [0,t] is precisely $\sum\nolimits_{s \leqslant t} {{X_s}}$. When the X’s are general i.i.d. random variables, this defines a general N-parameter random walk. To summarize the main results of this chapter, let us first suppose that the increments of the multiparameter random walk have the same distribution as some random variable ξ.