## Bernoulli

### A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors

#### Abstract

Let $X$ be a one-dimensional diffusion process. For each $n ≥1$ we have a round-off level $α n >0$ and we consider the rounded-off value $X t ( α) =α[X t/α]$. We are interested in the asymptotic behaviour of the processes $U (n,φ) t=1 n ∑ 1 ≤i≤[nt]φ(X ( i-1)/n ( α),n (X i /n ( α)-X ( i-1)/n ( α)))$ as $n$ goes to $+ ∞$: under suitable assumptions on $φ$, and when the sequence $α n$ goes to a limit $β ∈[0,∞)$, we prove the convergence of $U (n,φ)$ to a limiting process in probability (for the local uniform topology), and an associated 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 times $i /n$ and is subject to rounding off at some level $α n$ which is 'small' but not 'very small'.

#### Article information

Source
Bernoulli, Volume 3, Number 1 (1997), 1-28.

Dates
First available in Project Euclid: 4 May 2007