Our probabilistic proof of the Lindeberg-Feller Theorem develops by first showing that the small zero bias condition implies and hence, that W"-Wn=Xn-Xn→p0 Theorem 1.2 confirms that this convergence in probability to zero, mandating that Wn have its own zero bias distribution in the limit, is sufficient to guarantee normal convergence Theorem 1. 2 IfXn, n=1, 2,.. satisfies Condition 1. 1 and the small zero bias condition(14),then Vr lim P(Wn≤x)=P(Z≤x) We return now to the number Kn() of cycles of a random permutation in Sn, with mean hn and variance o? given by(6). Since 2i_ 1/i2<oo, by upper and lower bounding the n tn harmonic number hn by integrals of 1/r we nave lin 1 and therefore lim log n (15) ∞og identically makes Xin=0 for all n. Now by the linearity relatlon %s X1=1 In view of(7) and(4) we note that in this case Wn=>i Xin, as X1=1 for all a≠0, which follows directly from( 9), by(10) we have Xin=d Ui/on, where U; has distribution u-l/i, 1-1/il, i=2,.,n In particular, Uil I with probability one for all i= 1, 2, .. and therefore xn|≤1/on→0 Dy(15). Hence the small zero bias condition is satisfied, and Theorem 1.2 may be invoked to show that the number of cycles of a random permutation is asymptotically normalOur probabilistic proof of the Lindeberg-Feller Theorem develops by first showing that the small zero bias condition implies XIn,n →p 0, and hence, that W∗ n − Wn = X ∗ In,n − XIn,n →p 0. Theorem 1.2 confirms that this convergence in probability to zero, mandating that Wn have its own zero bias distribution in the limit, is sufficient to guarantee normal convergence. Theorem 1.2 If Xn, n = 1, 2, . . . satisfies Condition 1.1 and the small zero bias condition (14), then ∀x limn→∞ P(Wn ≤ x) = P(Z ≤ x). We return now to the number Kn(π) of cycles of a random permutation in Sn, with mean hn and variance σ 2 n given by (6). Since Pn i=1 1/i2 < ∞, by upper and lower bounding the n th harmonic number hn by integrals of 1/x, we have limn→∞ hn log n = 1 and therefore limn→∞ σ 2 n log n = 1. (15) In view of (7) and (4) we note that in this case Wn = Pn i=2 Xi,n, as X1 = 1 identically makes X1,n = 0 for all n. Now by the linearity relation (aX) ∗ =d aX∗ for all a 6= 0, which follows directly from (9), by (10) we have X ∗ i,n =d Ui/σn, where Ui has distribution U[−1/i, 1 − 1/i], i = 2, . . . , n. In particular, |Ui | ≤ 1 with probability one for all i = 1, 2, . . ., and therefore |X ∗ In,n| ≤ 1/σn → 0 (16) by (15). Hence the small zero bias condition is satisfied, and Theorem 1.2 may be invoked to show that the number of cycles of a random permutation is asymptotically normal. 8