Weak law of large numbers for linear processes

We establish sufficient conditions for the Marcinkiewicz–Zygmund type weak law of large numbers for a linear process { X k : k ∈ Z } defined by X k = ∑ j = 0 ∞ ψ j ε k - j for k ∈ Z , where { ψ j : j ∈ Z } ⊂ R and { ε k : k ∈ Z } are independent and identically distributed random variables such that x p Pr { | ε 0 | > x } → 0 as x → ∞ with 1 < p < 2 and E ε 0 = 0 . We use an abstract norming sequence that does not grow faster than n 1 / p if ∑ | ψ j | < ∞ . If ∑ | ψ j | = ∞ , the abstract norming sequence might grow faster than n 1 / p as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz–Zygmund type weak law of large numbers for the linear process.