Necessary and sufficient conditions for the Robbins-Monro method

This paper examines the relation between convergence of the Robbins-Monro iterates Xn+1= Xn-an[latin small letter f with hook](Xn)+an[xi]n, [latin small letter f with hook]([theta])=0, and the laws of large numbers Sn=an[Sigma]n-1j=0 [xi]j-->0 as n-->+[infinity]. If an is decreasing at least as rapidly as c/n, then Xn-->[theta] w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]1) implies Sn-->0 w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]1) as n-->+[infinity]. If an is decreasing at least as slowly as c[+45 degree rule]n and limn-->+[infinity]a n=0, then Sn-->0 w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]2) implies Xn-->[theta] w.p. 1 (resp. in Lp, p[greater-or-equal, slanted]2) as n -->+[infinity]. Thus, there is equivalence in the frequently examined case an[reverse similar, equals]c[+45 degree rule]n. Counter examples show that the LLN must have the form of Sn, that the rate of decrease conditions are sharp, that the weak LLN is neither necessary nor sufficient for the convergence in probability of Xn to [theta] when an[reverse similar, equals]c[+45 degree rule]n.Robbins-Monro method differential inequalities stochastic approximation Liapunov function laws of large numbers