TY - RPRT
T1 - Semicontinuity of Conditional Expectations with Respect to the
Conditioning Random Variable
AU - Knudsen, Thomas Skov
PY - 1996
Y1 - 1996
N2 - 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.
AB - 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.
M3 - Report
BT - Semicontinuity of Conditional Expectations with Respect to the
Conditioning Random Variable
ER -