Journal of Probability and Statistics
Volume 2011 (2011), Article ID 580292, 34 pages
http://dx.doi.org/10.1155/2011/580292
Research Article

Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Andriy Yurachkivsky

Taras Shevchenko National University, 01601 Kyiv, Ukraine

Received 24 May 2011; Accepted 3 October 2011

Academic Editor: Tomasz J. Kozubowski

Copyright © 2011 Andriy Yurachkivsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let for each 𝑛 ∈ ℕ 𝑋 𝑛 be an ℝ 𝑑 -valued locally square integrable martingale w.r.t. a filtration ( ℱ 𝑛 ( 𝑡 ) , 𝑡 ∈ ℝ + ) (probability spaces may be different for different 𝑛 ). It is assumed that the discontinuities of 𝑋 𝑛 are in a sense asymptotically small as 𝑛 → ∞ and the relation 𝖤 ( 𝑓 ( ⟨ 𝑧 𝑋 𝑛 ⟩ ( 𝑡 ) ) | ℱ 𝑛 ( 𝑠 ) ) − 𝑓 ( ⟨ 𝑧 𝑋 𝑛 ⟩ ( 𝑡 ) ) 𝖯 → 0 holds for all 𝑡 > 𝑠 > 0 , row vectors 𝑧 , and bounded uniformly continuous functions 𝑓 . Under these two principal assumptions and a number of technical ones, it is proved that the 𝑋 𝑛 's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes ( 𝑋 𝑛 ( 0 ) , ⟨ 𝑋 𝑛 ⟩ ) converge in distribution to some ( ∘ 𝑋 , 𝐻 ) , then a sequence ( 𝑋 𝑛 ) converges in distribution to a continuous local martingale 𝑋 with initial value ∘ 𝑋 and quadratic characteristic 𝐻 , whose finite-dimensional distributions are explicitly expressed via those of ( ∘ 𝑋 , 𝐻 ) .