We introduce the filtered *-bialgebra which is a multivariate generalization of the unital *-bialgebra C〈X,X′,P〉 of polynomials in noncommuting variables X=X*, X′*=X′ and a projection P=P*=P2, endowed with the coproduct Δ(X)=X⊗1+1⊗X, Δ(X′)=X′⊗P+P⊗X′, with P being group-like. We study the associated convolutions, random walks and filtered 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 Γ(ℋ) over H=L2(R+,G), where 𝒢 is a separable Hilbert 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.