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- W2324388103 abstract "Let {X, n ? 1} be a sequence of independent and identically distributed random variables and let Sn = (X1 + + Xn) Khintchine's form of the weak law of large numbers asserts that (1) E(XI) = 0 > (Sn/n)O0 in probability. In this note we present a rather simple proof of a stronger result, namely, (2) E(Xi)= 0 = E(I Sn 1) =o(n). The Markov inequality P(j Yi ? c)?E(j Yj)Ic shows that (2) is stronger than (1). An elementary proof of (1) can be found, for example, on pp. 246-248 of Feller [1]. Our proof uses a different 'truncation' and treats the 'tails' in a simple manner. We use the standard fact that (X1 + ... + Xn) = o(n) whenever Xn +0. Proof of (2). Let F denote the common distribution function of the X's. Define the random variables Yn and Zn as follows: Yn=Xn and Zn = O if I XnItVfl, Yn-O and Zn = Xn if I Xnf > Vn. Then Xn = Yn + Zn and Sn = Tn + Un, where Tn = In Yk and Un = Xl Zk. Therefore (2) will be proved as soon as we show that E(ITnj)=o(n) and E(IUnf)=o(n). Since E( X )<oo, it follows that" @default.
- W2324388103 created "2016-06-24" @default.
- W2324388103 creator A5074172043 @default.
- W2324388103 date "1976-06-01" @default.
- W2324388103 modified "2023-09-24" @default.
- W2324388103 title "A Simple Proof of Mean Convergence in the Law of Large Numbers" @default.
- W2324388103 cites W2751862591 @default.
- W2324388103 doi "https://doi.org/10.1080/00029890.1976.11994148" @default.
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