Weak convergence of probability measures in spaces of smooth functions (Q1086899)

The paper deals with conditions of weak convergence of measures on the space \(C^k\) of \(k\)-times differentiable functions defined on \(\mathbb R^n\) (or on a ball of \(\mathbb R^n)\) with values in \(\mathbb R^N\) \((N\) and \(n\) are positive integers). The author has found necessary and sufficient conditions for tightness of a sequence of measures on \(C^k\) by use of the distributions of the modulus of continuity; these conditions together with the weak convergence of finite-dimensional projections provide necessary and sufficient conditions for weak convergence of measures on \(C^k\). No proofs are given. The results of the paper with proofs have been published by the author earlier in his Thesis, Univ. New South Wales (1983).