A strong invariance principle for reverse martingales

Abstract

In a previous work (Scott and Huggins [5]) an embedding of reverse martingales in Brownian motion was obtained and used to give a law of the iterated logarithm for reverse martingales using properties of Brownian motion near the origin. This iterated logarithm law complements the iterated logarithm law for martingales in the same manner as results concerning the behaviour of Brownian motion as t-*0 complement those concerning the behaviour of Brownian motion as t-* ~. This becomes clearer from an examination of the embedding of doubly infinite martingales in Brownian motion given as Theorem 3 of Scott and Huggins [5]. Here we sharpen the underlying invariance principle of Scott and Huggins [5] and this result complements the corresponding martingale result of Jain, Jogdeo and Stout [3]. We then give, as an application of this invariance principle, integral tests for upper and lower functions of reverse martingales which again complement the result of Jain, Jogdeo and Stout. As we quite often follow the proofs of Jain, Jogdeo and Stout fairly closely with the appropriate changes necessary for the reverse martingale case in some places only a sketch of the proof is needed.