On the Continuity of the mth Root of a Continuous Nonnegative Definite Matrix-valued Function

In this paper we prove that the mth root of a nonnegative definite matrix-valued function with continuous entries is continuous. In particular, if all the entries are Ca for a g 0, 2x, the mth root is Carm, which generalizes the result on the square root with a s2 by M. Friedlin Probab. Theory Appl. 13, 1968, 375]378.