A martingale inequality and the law of large numbers

Abstract

The original Kolmogorov's inequality [6] has been extended to a martingale inequality by Levy [8] and Ville [12] and later to a semimartingale inequality by Doob [3]. In this note we will extend (1) to a semi-martingale inequality which contains Doob's inequality as a special case. As Kolmogorov's inequality is the key to the proof of the of large for a sequence of independent random variables, we will use our inequality to prove a law of large numbers for a martingale, which will be shown to include the extensions of Kolmogorov's of large for independent random variables [7] made by Brunk [I], Chung [2], Kawata and Udagawa [5], and Prohorov [11], and for dependent random variables made by Levy [8] and Loeve [9]. In the following (W, F, P) will be a probability space, cl, c2, . . . a nonincreasing sequence of positive numbers, xl, x2, * * * a sequence of random variables, yk=XlX2+x2 ? * +Xk and Fk the Borel field generated by xi, x2, * * *, Xk for each k, and for a random variable z we put z+=max(z, 0).