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Journal Article

A Central Limit Theorem for Normalized Functions of the Increments of a Diffusion Process, in the Presence of Round-Off Errors

Sylvain Delattre and Jean Jacod
Bernoulli
Vol. 3, No. 1 (Mar., 1997), pp. 1-28
DOI: 10.2307/3318650
Stable URL: http://www.jstor.org/stable/3318650
Page Count: 28
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Central Limit Theorem for Normalized Functions of the Increments of a Diffusion Process, in the Presence of Round-Off Errors
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Abstract

Let X be a one-dimensional diffusion process. For each n ≥ 1 we have a round-off level $\alpha _{n}>0$ and we consider the rounded-off value Xt (α n)=α n[Xt/α n]. We are interested in the asymptotic behaviour of the processes $U(n,\varphi)_{t}={\textstyle\frac{1}{n}}\sum_{1\leq i\leq [nt]}\varphi (X_{(i-1)/n}^{(\alpha _{n})},\sqrt{n}(X_{i/n}^{(\alpha _{n})}-X_{(i-1)/n}^{(\alpha _{n})})$ as n goes to +∞: under suitable assumptions on φ, and when the sequence $\alpha _{n}\sqrt{n}$ goes to a limit β ϵ [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'.

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