Asymptotic Expansion for Functionals of a Marked Point Process

Abstract

We consider a functional of a marked point process and derive asymptotic expansion of the distribution. Also, we apply this result to obtain expansion for the M-estimator. A moving average process sampled by a point process and a point process marked by a diffusion process are discussed.

Keywords:

• Asymptotic expansion
• Malliavin calculus
• Marked point process
• Mixing
• Point process
• Sampling

Mathematics Subject Classification:

• Primary 62M05, 62M09, 60G55, 60F99, 60H07
• Secondary 60F05

Acknowledgments

This work was supported in part by JST Basic Research Programs PRESTO, Grants-in-Aid for Scientific Research No. 19340021 and No. 18500219; the Global COE program “The Research and Training Center for New Development in Mathematics” of the Graduate School of Mathematical Sciences, University of Tokyo; and by a Cooperative Research Program of the Institute of Statistical Mathematics.

Notes

¹ is the space of smooth functions with bounded derivatives of positive order.

²Veretennikov (1987, 1997) and Kusuoka and Yoshida (2000) provided sufficient conditions for (ii).