On the maximum of randomly weighted sums with regularly varying tails

Abstract

Consider the randomly weighted sums S n ( θ ) = ∑ k = 1 n θ k X k , n = 1 , 2 , … , where { X k , k = 1 , 2 , … } is a sequence of independent real-valued random variables with common distribution F, whose right tail is regularly varying with exponent - α < 0 , and { θ k , k = 1 , 2 , … } is a sequence of positive random variables, independent of { X k , k = 1 , 2 , … } . Under a suitable summability condition on the upper endpoints of θ k , k = 1 , 2 , … , we prove that Pr ( max 1 ⩽ n < ∞ S n ( θ ) > x ) ∼ F ¯ ( x ) ∑ k = 1 ∞ E θ k α .