A CENTRAL-LIMIT-THEOREM FOR NORMALIZED FUNCTIONS OF THE INCREMENTS OFA DIFFUSION PROCESS, IN THE PRESENCE OF ROUND-OFF ERRORS

Let X be a one-dimensional diffusion process. For each n greater than or equal to 1 we have a round-off level alpha(n) > 0 and we consider t he rounded-off value X-t((alpha n)) = alpha(n)[X-t/alpha(n)]. We are i nterested in the asymptotic behaviour of the process U(n,phi)(t) = 1/n Sigma(1 less than or equal to i less than or equal to[nt])phi(X(i-1/n )((alpha n)),root n(X-i/n((alpha n)) - X(i-1/n)((alpha n)) - X(i-1/n)( (alpha n))) as n goes to +infinity: under suitable assumptions on phi, and when the sequence alpha(n) root n goes to a limit beta is an elem ent of [0, infinity), we prove the convergence of U(n,phi) to a limiti ng process in probability (for the local uniform topology), and an ass ociated central limit theorem. This is motivated mainly by statistical problems in which one wishes to estimate a parameter occurring in the diffusion coefficient, when the diffusion process is observed at time s i/n and is subject to rounding off at some level alpha(n) which is ' small' but not 'very small'.