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A central limit theorem for jump diffusion processes in the presence of round-off errors

Part of: Limit theorems

Published online by Cambridge University Press:  25 March 2025

Diop Assane
Diop Assane*
Affiliation:
Université Cheikh Anta Diop
*
*Postal address: Département de Mathématiques et Informatique, Université Cheikh Anta Diop, Dakar-Fann, Sénégal. Email address: [email protected]
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Abstract

Let X be the sum of a diffusion process and a Lévy jump process, and for any integer $n\ge 1$ let $\phi_n$ be a function defined on $\mathbb{R}^2$ and taking values in $\mathbb{R}$, with adequate properties. We study the convergence of functionals of the type

\begin{align*} \Delta_n\sum_{i=1}^{[t/\Delta_n]} \phi_n\!\left(\alpha_n\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right], \:\frac{\alpha_n}{\sqrt{\Delta_n}}\left(\left[\frac{X_{i\Delta_n}}{\alpha_n}\right]-\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right]\right)\right),\end{align*}
where [x] is the integer part of the real number x and the sequences $(\Delta_n)$ and $(\alpha_n)$ tend to 0 as $n\to +\infty$. We then prove the law of large numbers and establish, in the case where $\frac{\alpha_n}{\sqrt{\Delta_n}}$ converges to a real number in $[0,+\infty)$], a new central limit theorem which generalizes that in the case where X is a continuous Itô’s semimartingale.

Keywords

Limit theoremdiscontinuous semimartingalesround-off errors

MSC classification

Primary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 39
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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