A limit theorem for expectations conditional on a sum

Sandy L Zabell*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Let ξ1, ξ2, ξ3,... be a sequence of independent random variables, such that μj{equals colon}E[ξj], 0<α≤Var[ξj] and E[|ξjj|2+δ] for some δ, 0<δ≤1, and every j≥1. If U and ξ0 are two random variables such that E[ξ02]<∞ and E[|U|ξ02]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξj:j≥1}, then under appropriate regularity conditions {Mathematical expression} where Sn{equals colon}ξ12+⋯+ξnj{equals colon}E[ξj], sn2{equals colon}Var[Sn], and cn=O(sn).

Original languageEnglish (US)
Pages (from-to)267-283
Number of pages17
JournalJournal of Theoretical Probability
Issue number2
StatePublished - Apr 1 1993


  • Bartlett's formula
  • Conditional expectations

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)


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