On the Interrelation of Almost Sure Invariance Principles for Certain Stochastic Adaptive Algorithms and for Partial Sums of Random Variables I

where {xi, i = 1, 2, 3,..} is a sequence of random vectors and {Xt, t>_.0} is a Brownian motion. In this note, we show that if {Ak, k = 1, 2, 3,...} and {bk, k--1, 2, 3,...} are processes satisfying almost-sure bounds analogous to Eq. (1), (where { X , t />0} could be a more general Gauss -Markov process) then {hk, k = I, 2, 3,...}, the solution of the stochastic approximation or adaptive filtering algorithm