| Abstract: | Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers  holds if the class  has an envelope function that is μ-integrable, or if  is bounded in L p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class  in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date. |
