### 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.

Original language | English |
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Number of pages | 13 |
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Publication status | Published - 1996 |

## Cite this

Knudsen, T. S. (1996).

*Semicontinuity of Conditional Expectations with Respect to the Conditioning Random Variable*.