When and how an error yields a Dirichlet form

We consider a random variable Y and approximations Yn, n ∈ N, defined on the same probability space with values in the same measurable space as Y .We are interested in situations where the approximations Yn allow to define a Dirichlet form in the space L2(PY ) where PY is the law of Y . Our approach consists in studying both biases and variances. The article attempts to propose a general theoretical framework. It is illustrated by several examples. © 2006 Elsevier Inc. All rights reserved