Martingales, Lévy’s continuity theorem, and the martingale central limit theorem
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منابع مشابه
A Central Limit Theorem for Local Martingales with Applications to the Analysis of Longitudinal Data
SUMMARY A functional central limit theorem for a local square integrable martingale with persistent disconti-nuities is given. By persistent discontinuities, it is meant that the martingale has jumps which do not vanish asymptotically. This central limit theorem is motivated by problems in the analysis of longitudinal and life history data.
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Let {Xi}i≥1 be a martingale difference sequence with Xi = Si − Si−1. Under some regularity conditions, we show that (X 1+· · ·+X2 Nn)SNn is asymptotically normal, where {Ni}i≥1 is a sequence of positive integer-valued random variables tending to infinity. In a similar manner, a backward (or reverse) martingale central limit theorem with random indices is provided. Keywords—central limit theorem...
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Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating marting...
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تاریخ انتشار 2015