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Math/Stat 394, Winter 2011 F.W. Scholz
Central Limit Theorems and Proofs
The following gives a self-contained treatment of the central limit theorem (CLT). It is based on Lindeberg’s (1922) method. To state the CLT which we shall prove, we introduce the following notation. We assume that Xn1 , . . . , Xnn are independent random variables with means 0 and 2 2 respective variances σn1 , . . . , σnn with
2 2 2 σn1 + . . . + σnn = τn > 0 for all n. 2 Denote the sum Xn1 + . . . + Xnn by Sn and observe that Sn has mean zero and variance τn .
Lindeberg’s Central Limit Theorem: If the Lindeberg condition is satisfied, i.e., if for every > 0 we have that Ln ( ) = 1 2 τn n 2 E Xni I{|Xni |≥ τn } −→ 0 as n → ∞ , i=1
then for every a ∈ R we have that P (Sn /τn ≤ a) − Φ(a) −→ 0 as n → ∞ . Proof: Step 1 (convergence of expectations of smooth functions): We will show in Appendix 1 that for certain functions f we have that E [f (Sn /τn )] − E [f (Z)] → 0 as n → ∞ , (1)
where Z denotes a standard normal random variable. If this convergence would hold for any function f and if we then applied it to fa (x) = I(−∞,a] (x) = 1 if x ≤ a and = 0 if x > a, then E [fa (Sn /τn )] − E [fa (Z)] = P (Sn /τn ≤ a) − Φ(a) and the statement of the CLT would follow. Unfortunately, we cannot directly demonstrate the above convergence (1) for all f , but only for smooth f . Here smooth f means that f is bounded and has three bounded, continuous derivatives as stipulated in Lemma 1 of Appendix 1. Step 2 (sandwiching a step function between smooth functions): We will approximate fa (x) = I(−∞,a] (x), which is a step function with step at x = a, by sandwiching it between two smooth functions. In fact, for δ > 0 one easily finds (see Appendix 2 for an explicit example) smooth functions f (x) with f (x) = 1 for x ≤ a, f (x) monotone decreasing from 1 to 0 on [a, a + δ] and f (x) = 0 for x ≥ a. Hence we would have fa (x) ≤ f (x) ≤ fa+δ (x) for all x ∈ R . 1
Central Limit Theorems and Proofs
The following gives a self-contained treatment of the central limit theorem (CLT). It is based on Lindeberg’s (1922) method. To state the CLT which we shall prove, we introduce the following notation. We assume that Xn1 , . . . , Xnn are independent random variables with means 0 and 2 2 respective variances σn1 , . . . , σnn with
2 2 2 σn1 + . . . + σnn = τn > 0 for all n. 2 Denote the sum Xn1 + . . . + Xnn by Sn and observe that Sn has mean zero and variance τn .
Lindeberg’s Central Limit Theorem: If the Lindeberg condition is satisfied, i.e., if for every > 0 we have that Ln ( ) = 1 2 τn n 2 E Xni I{|Xni |≥ τn } −→ 0 as n → ∞ , i=1
then for every a ∈ R we have that P (Sn /τn ≤ a) − Φ(a) −→ 0 as n → ∞ . Proof: Step 1 (convergence of expectations of smooth functions): We will show in Appendix 1 that for certain functions f we have that E [f (Sn /τn )] − E [f (Z)] → 0 as n → ∞ , (1)
where Z denotes a standard normal random variable. If this convergence would hold for any function f and if we then applied it to fa (x) = I(−∞,a] (x) = 1 if x ≤ a and = 0 if x > a, then E [fa (Sn /τn )] − E [fa (Z)] = P (Sn /τn ≤ a) − Φ(a) and the statement of the CLT would follow. Unfortunately, we cannot directly demonstrate the above convergence (1) for all f , but only for smooth f . Here smooth f means that f is bounded and has three bounded, continuous derivatives as stipulated in Lemma 1 of Appendix 1. Step 2 (sandwiching a step function between smooth functions): We will approximate fa (x) = I(−∞,a] (x), which is a step function with step at x = a, by sandwiching it between two smooth functions. In fact, for δ > 0 one easily finds (see Appendix 2 for an explicit example) smooth functions f (x) with f (x) = 1 for x ≤ a, f (x) monotone decreasing from 1 to 0 on [a, a + δ] and f (x) = 0 for x ≥ a. Hence we would have fa (x) ≤ f (x) ≤ fa+δ (x) for all x ∈ R . 1
References: Lindeberg, J.W. (1922). “Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung.” Mathemat. Z., 15, 211-225. Thomasian, A. (1969). The Structure of Probability Theory with Applications. Mc Graw Hill, New York. (pp. 483-493) 8