Abstract We consider a random variable Y and approximations Y n , 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 Y n allow to define a Dirichlet form in the space L 2 ( P Y ) where P Y 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.