If Xn P 0 and sup n E[|Xn| 1+ ] < for some >

If Xn P

→ 0 and sup n

E[|Xn|

1+δ

] < ∞ for some δ > 0 , (11.90)

then show E[|Xn|] → 0. More generally, if the Xn are uniformly integrable in the sense supn E[|Xn|I{|Xn| > t}] → 0 as t → ∞, then (11.90) holds. [A converse is given in Dudley (1989), p.279.]