On central limit and iterated logarithm supplements to the martingale convergence theorem

Let { S n , n ≧ 1} be a zero, mean square integrable martingale for which so that S n → S ∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for B n ( S n – S ∞ ) where the multipliers B n ↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.