On estimation of random variables via the martingale convergence theorem

Abstract

Conditional expectation and the martingale convergence theorem are frequently used in attempts to estimate a random variable as a function of other random variables. Examples are exhibited which illustrate difficulties arising from a careless approach to these techniques. Real-valued random variables, including a Gaussian random variable, are constructed such that the martingale convergence theorem, or any scheme based on conditional expectation, can fail to be of any value whatsoever in estimating them.