Wn an approximate fixed point of the zero bias transformation, and there- fore approximately normal. This explanation, when given precisely, becomes a probabilistic proof of the Lindeberg-Feller central limit theorem under a condition equivalent to( 8)which we call the small zero bias condition We first consider more precisely this special property of the zero bias transformation on independent sums. Given Xn satisfying Condition 1. 1, let X*=in: I<i<nf be a collection of random variables so that xin has the Xin zero bias distribution and is independent of Xn. Further, let In be a random index, independent of Xn and Xn, with distribution P(In= i) and write the variable selected by In, that is, the mixture, using indicator functions as 1(0=iX ,n and X* 1(In=1)Xn(12) Then Wr=Wn-Ximn+Xi has the Wn zero bias distribution. For the simple proof of this fact, see 6 From(13) we see that the Clt should hold when the random variables XIm, n and Xin are both small asymptotically, since then the distribution of Wn is close to that of W*, making Wn an approximate fixed point of the zero bias transformation. The following theorem shows that properly formalizing the notion of smallness results in a condition equivalent to Lindeberg's. R call that ay a sequence of random variables Yn converges in probability to y and write Y-Y. if limP(Yn-Y1≥∈)=0 for all e>0. Theorem 1.1 For a collection of random variables Xn, n=1, 2, .. satisfy ing Condition 1. 1, the small zero bias condition and the Lindeberg condition (8) are equivalent. 7Wn an approximate fixed point of the zero bias transformation, and there￾fore approximately normal. This explanation, when given precisely, becomes a probabilistic proof of the Lindeberg-Feller central limit theorem under a condition equivalent to (8) which we call the ‘small zero bias condition’. We first consider more precisely this special property of the zero bias transformation on independent sums. Given Xn satisfying Condition 1.1, let X∗ n = {X∗ i,n : 1 ≤ i ≤ n} be a collection of random variables so that X∗ i,n has the Xi,n zero bias distribution and is independent of Xn. Further, let In be a random index, independent of Xn and X∗ n , with distribution P(In = i) = σ 2 i,n, (11) and write the variable selected by In, that is, the mixture, using indicator functions as XIn,n = Xn i=1 1(In = i)Xi,n and X ∗ In,n = Xn i=1 1(In = i)X ∗ i,n. (12) Then W∗ n = Wn − XIn,n + X ∗ In,n (13) has the Wn zero bias distribution. For the simple proof of this fact, see [6]. From (13) we see that the CLT should hold when the random variables XIn,n and X∗ In,n are both small asymptotically, since then the distribution of Wn is close to that of W∗ n , making Wn an approximate fixed point of the zero bias transformation. The following theorem shows that properly formalizing the notion of smallness results in a condition equivalent to Lindeberg’s. Re￾call that we say a sequence of random variables Yn converges in probability to Y , and write Yn →p Y , if limn→∞ P(|Yn − Y | ≥ ) = 0 for all  > 0. Theorem 1.1 For a collection of random variables Xn, n = 1, 2, . . . satisfy￾ing Condition 1.1, the small zero bias condition X ∗ In,n →p 0 (14) and the Lindeberg condition (8) are equivalent. 7