A CONVERSE TO SOME INEQUALITIES AND APPROXI¬MATIONS IN THE THEORY OF STIELTJES AND STOCHASTIC INTEGRALS, AND FOR n-th DERIVATIVES

摘要

The object of this report is to establish by counter-examples the best possible character of theorems recently obtained about stochastic integrals and Stieltjes integrals, and about n-th derivatives and finite differences. The hypotheses involve a pair of estimate functions subject to the convergence of a corresponding integral or Y-series, and it is shown that the divergence of this integral or series render in each case the conclusion false.