Semicontinuity of Conditional Expectations with Respect to the Conditioning Random Variable - Danish National Research Database-Den Danske Forskningsdatabase

^{1} Department of Mathematics, Technical University of Denmark^{2} Department of Applied Mathematics and Computer Science, Technical University of Denmark

Abstract:

In this paper we prove that if x_i is a random variablewith values in a complete, separable metric space and Z in L_p,p>2 is a real random variable then for sequences x_i,nwhich converge 'nicely' in probability to x_i, the conditionalexpectations E[Z| x_i,n] converge in L_2 to E[Z| x_i].This kind of nice convergence includes convergence in probability ofrandom variables with values in a denumerable set. For a larger classof sequences x_i,n which converge in probability, it isshown that if the conditional expectations converge in L_2, thelimit is at least as close in L_2 to Z as E[Z| x_i]. Themotivating example is taken from nonlinear filtering and the problemof robustness and approximations of such filters.