A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let g n : R ∞ → [ 0 , ∞ ] , n ⩾ 1 be a sequence of measurable functions such that, for every n ⩾ 1 , g n ( x + y 2 ) ⩽ C ( g n ( x ) + g n ( y ) ) and g n ( x − y 2 ) ⩽ C ( g n ( x ) + g n ( y ) ) for all x , y ∈ R ∞ , where 0 < C < ∞ is a constant which is independent of n. Let { X , X n ; n ⩾ 1 } be a sequence of i.i.d. random variables. Assume that there exist r ⩾ 1 and a function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) with lim t → ∞ ϕ ( t ) = ∞ , depending only on the sequence { g n ; n ⩾ 1 } such that lim sup n → ∞ g n ( X 1 , X 2 , … ) = ϕ ( E | X | r ) a.s. whenever E | X | r < ∞ and E X = 0 . We prove the converse result, namely that lim sup n → ∞ g n ( X 1 , X 2 , … ) < ∞ a.s. implies E | X | r < ∞ (and E X = 0 if, in addition, lim sup n → ∞ g n ( c , c , … ) = ∞ for all c ≠ 0 ). Some applications are provided to illustrate this result.