  We introduce the filtered *-bialgebra which is a noncommutative analog of the
*-bialgebra of multivariate polynomials with the canonical coproduct and
counit. We study the associated filtered convolutions, random walks and random
variables. The GNS representations of the limit states lead to filtered
fundamental operators which are the CCR fundamental operators on the multiple
symmetric Fock space, multiplied by appropriate projections. The importance of
filtered random variables and fundamental operators stems from the fact that by
addition and strong limits one obtains from them the main types of
noncommutative random variables and fundamental operators, respectively,
regardless of the type of noncommutative independence.
