In Lemma 13.4.2, show that Condition (13.41) can be replaced by the assumption that, for some n

In Lemma 13.4.2, show that Condition (13.41) can be replaced by the assumption that, for some βn = o(n1/2), lim sup n→∞

EGn [|Yn,i − μ(Gn)|I{|Yn,i − μ(Gn)| ≥ βn}] = 0.

Moreover, this condition only needs to hold if βn = o(n) if it is also known that supn EGn |Yn,i − μ(Gn)| < ∞.