For many typical instances where Monte Carlo methods are applied attem
pts were made to find unbiased estimators, since for them the Monte Ca
rlo error reduces to the statistical error. These problems usually tak
e values in the scalar field. If we study vector valued Monte Carlo me
thods, then we are confronted with the question of whether there can e
xist unbiased estimators. This problem is apparently new. Below it is
settled precisely. Partial answers are given, indicating relations to
several classes of linear operators in Banach spaces. (C) 1996 Academi
c Press, Inc.