A NON-HOMOGENEOUS MARKOV MODEL OF A CHAIN-LETTER SCHEME

Abstract

This chapter presents a nonhomogeneous Markov model of a chain-letter scheme. The role of the model in the legal proceedings is to demonstrate that the promises made by the promoters are inherently deceptive and constitute misrepresentation because it is virtually impossible for the majority of participants to make anywhere near the large sums indicated by the promotional material. Many participants can lose all or a part of their entrance fee. As a participant's earnings depend on the time of entrance, defendant promoters can produce a few winners to testify that they did well. The probabilistic approach shows that the proportion of winners is small and most participants can lose money. The chapter discusses the basic properties of the process and its limiting diffusion approximation. The transition properties of this conditional chain present two difficulties when the state variable is appropriately normalized: the increments neither have bounded conditional expectations nor their conditional variances are bounded away from zero. These difficulties are overcome by modifying the chain beyond constant boundaries and then showing that the probability of the chain moving outside these boundaries is sufficiently small so as to make the effect of this modification negligible. A suitable probability inequality for the constant boundary crossing event is obtained for this purpose using a rebounding effect in the transition probabilities of the chain.