Summary
A Skorokhod embedding approach is used to give functional laws of the iterated logarithm which involve the process up to timen in the reverse martingale case and the tail of the process in the martingale case. This complements the more usual versions of the iterated logarithm laws for martingales and reverse martingales.
Article PDF
Similar content being viewed by others
References
Barbour, A.D.: Tail sums of convergent series of independent random variables. Math. Proc. Camb. Philos. Soc.75, 361–364 (1974).
Eagleson, G.K., Weber, N.C.: Limit theorems for weakly exchangeable arrays. Math. Proc. Camb. Philos. Soc.84, 123–130 (1978)
Hall, P.G., Heyde, C.C.: On a unified approach to the law of the iterated logarithm for martingales. Bull. Aust. Math. Soc.14, 435–447 (1976)
Heyde, C.C.: On central limit and iterated logarithm supplements to the martingale convergence theorem. J. Appl. Probab.14, 758–775 (1977)
Loynes, R.M.: An invariance principle for reversed martingales. Proc. Am. Math. Soc.25, 56–64 (1970)
Loynes, R.M.: On the weak convergence ofU-statistics, and of the empirical process. Math. Proc. Camb. Philos. Soc.83, 269–272 (1978)
Miller, R.G., Jr., Sen, P.K.: Weak convergence ofU-statistics and von Mise's differentiable statistical functions. Ann. Math. Statist.43, 31–41 (1972)
Scott, D.J., Huggins, R.M.: On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse martingales. Bull. Aust. Math. Soc.27, 443–459 (1983)
Scott, D.J., Huggins, R.M.: A law of the iterated logarithm for weakly exchangeable arrays. [Submitted for publication]
Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 211–226 (1964)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Huggins, R.M. On functional laws of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 243–250 (1985). https://doi.org/10.1007/BF02450282
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02450282