On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables

Since the novel work of Berkes and Philipp((3)) much effort has been focused on establishing almost sure invariance principles of the form [GRAPHICS] where {x(i), i=1, 2, 3,...} is a sequence of random vectors and {X(t), t greater than or equal to 0} is a Brownian motion. In this note, we show that if {A(k), k=1, 2, 3,...} and {b(k), k=1, 2, 3,...} are processes satisfying almost-sure bounds analogous to Eq. (1), (where {X(t), t greater than or equal to 0} could be a more general Gauss-Markov process) then {h(k), k= 1, 2, 3,...}, the solution of the stochastic approximation or adaptive filtering algorithm h(k+1)=h(k)+1/k(b(k)-A(k)h(k)) for k=1,23,... (2) also satisfies an almost sure invariance principle of the same type.