Summary
The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these results is the establishment of second (and in some cases higher) order asymptotic expansions for martingales.
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Research supported in part by National Science Foundation grant DMS 93-05601 and Army Research Office grant DAAH04-1-0105
This manuscript was prepared using computer facilities supported in part by the National Science Foundation grants DMS 89-05292, DMS 87-03942, and DMS 86-01732 awarded to the Department of Statistics at The University of Chicago, and by The University of Chicago Block Fund