Lecture Notes 2: Random Fields

In these notes, I discuss some properties of collections of random variables, which I call random fields. In terms of jargon, a collection of random variables indexed by scalars is known as a stochastic process. Random fields generalize the standard notion of a stochastic process by allowing for collections where the index is more complicated. To be clear, the definitions of each vary across textbooks. The random variables I work with, loosely speaking, correspond to outcomes or choices in the previous notes. Here, my focus is on probability ideas. Kindermann and Snell (1980) is a gentle and intuitive introduction to the literature. i. Extension theorems In many contexts, economic theory provides a set of conditional probability measures that relate individual behaviors to others. An equilibrium for such an environment is a joint probability measure that is compatible with the conditional measures. A joint probability measure generated from a given set of conditionals is known as an extension of the conditional measures. Extension theorems provide conditions under which extensions exist from some initial set of probability statements. The classic example of an extension theorem is the eponymous Kolmogorov Extension Theorem, which is a cornerstone of the theory of time series. Consider a collection of random variables (i.e. a stochastic process) tω, t Z ∈.