Summary
A class of non-pathological martingales is exhibited to which current martingale central limit theory does not apply. The problem consists in the convergence of the normalized sums of squares being only in distribution. The class is that of degenerate U-statistics.
Article PDF
Similar content being viewed by others
References
Alvo, M., Cabilio, P., Feigin, P.D.: Degenerate U-statistics and the problem of n rankings. Operations Research, Statistics and Economics Mimeograph Series No. 251, Technion (1980)
Anderson, T.W.: On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann. Math. Statist. 30, 676–687 (1959)
Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968
Diaconis, P., Graham, R.L.: Spearman's footrule as a measure of disarray. J. Roy. Statist. Soc. Ser. B 39, 262–268 (1977)
Dvoretsky, A.: Asymptotic normality for sums of dependent random variables. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. 2, 513–535. University of California (1972)
Eagleson, G.K.: Martingale convergence to mixtures of infinitely divisible laws. Ann. Probability 3, 557–562 (1975)
Eagleson, G.K.: Orthogonal expansions and U-Statistics. Austral. J. Statist. 21, 221–237 (1979)
Feigin, P.D.: Some comments concerning a curious singularity. J. Appl. Probability 16, 440–444 (1979)
Gregory, G.C.: Large sample theory for U-statistics and tests of fit. Ann. Statist. 5, 110–123 (1977)
Hall, P.G.: Martingale invariance principles. Ann. Probability 5, 875–887 (1977)
Hoeffding: A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325 (1948)
McLeish, D.L.: Dependent central limit theorems and invariance principles. Ann. Probability 2, 620–628 (1974)
Rootźen, H.: A note on convergence to mixtures of normal distributions. Z. Wahrscheinlichkeitsth. verw. Gebiete 38, 211–216 (1977)
Sproule, R.N.: Asymptotic properties of U-statistics. Trans. Amer. Math. Soc. 199, 55–64 (1974)
Author information
Authors and Affiliations
Additional information
This work was done while the first two authors were on Sabbatical leave at the Technion.
Work supported in part by National Research Council of Canada, Grant No. A-9068
Work supported in part by National Research Council of Canada, Grant No. A-9076
Rights and permissions
About this article
Cite this article
Alvo, M., Cabilio, P. & Feigin, P.D. A class of martingales with non-symmetric limit distributions. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 87–93 (1981). https://doi.org/10.1007/BF00536197
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00536197