Uniform convergence in probability (${\sup }_{\theta \in \Theta }|Q_n(\theta )-Q(\theta )|\stackrel{P}{\to }0$)

Uniform convergence in probability means that the supremum of the absolute discrepancy on the entire index set tends to zero in probability. It controls evaluation at random indices in that set, unlike pointwise convergence in probability. On a fixed compact index set it is a special case of uniform convergence on compacts in probability.