  We introduce a new analysis of an adaptive mixture method that combines
outputs of two constituent filters running in parallel to model an unknown
desired signal. This adaptive mixture is shown to achieve the mean square error
(MSE) performance of the best constituent filter, and in some cases outperforms
both, in the steady-state. However, the MSE analysis of this mixture in the
steady-state and during the transient regions uses approximations and relies on
statistical models on the underlying signals and systems. Hence, such an
analysis may not be useful or valid for signals generated by various real life
systems that show high degrees of nonstationarity, limit cycles and, in many
cases, that are even chaotic. To this end, we perform the transient and the
steady-state analysis of this adaptive mixture in a "strong" deterministic
sense without any approximations in the derivations or statistical assumptions
on the underlying signals such that our results are guaranteed to hold. In
particular, we relate the time-accumulated squared estimation error of this
adaptive mixture at any time to the time-accumulated squared estimation error
of the optimal convex mixture of the constituent filters directly tuned to the
underlying signal in an individual sequence manner.
