Statistics and Probability Letters 80 (2010) 899–902 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro On the Kolmogorov–Feller law for exchangeable random variables George Stoica a,∗ , Deli Li b a Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada b Department of Mathematical Sciences, Lakehead University, Thunder Bay, Canada article info Article history: Received 22 December 2009 Received in revised form 25 January 2010 Accepted 27 January 2010 Available online 2 February 2010 MSC: 60F05 60G09 Keywords: Exchangeable sequences Kolmogorov–Feller weak law of large numbers abstract We prove a Kolmogorov–Feller weak law of large numbers for exchangeable sequences, under a second order hypothesis on the truncated mixands. © 2010 Elsevier B.V. All rights reserved. Hong and Lee (1998) proved a Kolmogorov–Feller weak law of large numbers for sequences of exchangeable random variables without finite mean; some of their hypotheses are rather restrictive, and the purpose of this note is to improve that result by using a second order hypothesis on the truncated mixands, inspired by Klass and Teicher (1987) and Jiang and Hahn (2003). A sequence of random variables {X n } n≥1 on the probability space (Ω, F , P ) is said to be exchangeable if for each n, P [X 1 ≤ x 1 ,..., X n ≤ x n ]= P [X π(1) ≤ x 1 ,..., X π(n) ≤ x n ] for any permutation π of {1, 2,..., n} and any x i ∈ R, i = 1,..., n. Among examples of exchangeable sequences, we mention any weighted average of i.i.d. sequences, {X + ε n } n≥1 and {Y · ε n } n≥1 , where {ε n } n≥1 are i.i.d. and independent of X or Y , respectively. By de Finetti’s theorem, an infinite sequence of exchangeable random variables is conditionally i.i.d. given either the tail σ -field of {X n } n≥1 or the σ -field G of permutable events; cf. Chow and Teicher (1978, Theorem 7.3.3). Furthermore, there exists a regular conditional distribution P ω given G such that for each ω ∈ Ω the mixands (i.e., coordinate random variables) {ξ n ≡ ξ ω n } n≥1 of the Borel probability space (R ∞ , B(R ∞ ), P ω ) are i.i.d.; cf. Chow and Teicher (1978, Corollary 7.3.5). Namely, for all n ∈ N, any Borel function f : R n → R, and Borel set B on R, P [f (X 1 ,..., X n ) ∈ B]=  Ω P ω [f (ξ 1 ,...,ξ n ) ∈ B]dP . (1) In what follows, we denote convergence in P -probability by → P , expectation under P ω by E ω , and S n = X 1 +···+ X n . Our main result reads as follows: ∗ Corresponding author. E-mail addresses: stoica@unb.ca (G. Stoica), dli@lakeheadu.ca (D. Li). 0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.01.025