THE STABILITY OF LMS

New and weak conditions are given under which the LMS algorithm is exp onentially convergent with probability one in a stochastic setting. Th ese results show that LMS works under very broad conditions: a small g ain, input signal with finite fourth moments; time varying persistence of excitation; and weak assumptions on the correlation structure of t he input signal. Previous results at this level of generality linked c onvergence to peak signal amplitude rather than average amplitude. Und er the same conditions stochastic boundedness of a forced LMS system i s also established.