✍ Scribed by Richard Isaac
No coin nor oath required. For personal study only.
Let X k be non-negative random variables with positive, ÿnite expectation, F k an increasing sequence of -ÿelds to which the X k are adapted, and p k = E(X k | F k-1 ). It is shown how a martingale form of Kronecker's Lemma and the theorem of Abel and Dini on series can quickly yield a number of su cient conditions ensuring that X k = p k → 1 on certain sets. In this case we say that the ratios are equitable on the sets. A number of new and old results are seen to be immediate consequences, including a martingale proof of the strong law of large numbers. The paper concludes with a few simple examples illustrating non-equitable behavior.
📜 SIMILAR VOLUMES
The effects of added manufactured viscous dampers upon shear-type structures are analytically investigated here for the class of Rayleigh damping systems. The definitions of mass proportional damping (MPD) and stiffness proportional damping (SPD) systems are briefly recalled and their physical count