On the rareness of generalized sub- and supermartingales in the class of all uniformly L₁ bounded stochastic processes

Abstract

In the present paper we consider stochastic processes {X_n n ∈ ℕ} over a given filtrated probability space (Ω,∑,P, {F_n : n ∈ ℕ}). Earlier papers e.g. LJUNG [7], GOODWIN et al.[3], SIN and GOODWIN [8], SOLO [9] and LANDAU [6] have enhanced the interest in the use of a martingale-type approach to the study of convergence properties, in a stochastic enviroment, of recursive identification and control schemes. Therefore the question arises how "big" the classes of generalized sub- and supermartingales in the class of all uniformly L₁ -bounded stochastic processes are. The aim of the paper is to prove that these classes are nowhere dense. In this sense we can say that generalized sub- nd supermartingales are very rare