lecture10 - Lecture 10 Setup for the Central Limit Theorem...

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Lecture 10 : Setup for the Central Limit Theorem STAT205 Lecturer: Jim Pitman Scribe: David S. Rosenberg <[email protected]> See Durrett 2nd ed pages 116-118 for an equivalent formulation and a proof using characteristic functions. That proof leans on the continuity theorem for characteristic functions, (3.4) on page 99, which in turn relies on the Helly selection theorem (2.5) on page 88. The present approach, due to Lindeberg, is more elementary in that it does not require these tools. But note that the basic idea in both arguments is to estimate the expected value of a smooth function of a sum of independent variables using a Taylor expansion with error bound. 10.1 Triangular Arrays Roughly speaking, a sum of many small independent random variables will be nearly normally distributed. To formulate a limit theorem of this kind, we must consider sums of more and more smaller and smaller random variables. Therefore, throughout this section we shall study the sequence of sums S i = X j X ij , obtained by summing the rows of a triangular array of random variables X 11 , X 12 , . . . , X 1 n 1 X 21 , X 22 , . . . . . . , X 2 n 2 X 31 , X 32 , . . . . . . . . . , X 3 n 3 . . . . . . . . . . . . It will be assumed throughout that triangular arrays satisfy 3 Triangular Array Conditions 1 : 1. for each i , the n i random variables X i 1 , X i 2 , . . . , X in i in the i th row are mutually independent,n 2. E ( X ij ) = 0 for all i, j , and 3. j E X 2 ij = 1 for all i . Here the row index i should always be taken to range over 1 , 2 , 3 , . . . , while the column index j ranges from 1 to n i . It is not assumed that the r.vs in each row are identically distributed. And it is not assumed that different rows are independent. (Different rows could even be defined on different probability spaces.) For motivation, see section ********** below for how such a triangular array is set up in the most important application to partial sums X 1 + X 2 + · · · + X n obtained from a sequence of independent r.v.s X 1 , X 2 , . . . It will usually be the case that n 1 < n 2 < · · · , whence the term triangular. It is not necessary to assume this however. 1 This is not standard terminology, but is used here as a simple referent for these conditions. 10-1
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Lecture 10: Setup for the Central Limit Theorem 10-2 10.2 The Lindeberg Condition and Some Consequences We will write L ( X ) to denote the law or distribution of a random variable X . N (0 , σ 2 ) is the normal distribution with mean 0 and variance σ 2 . Theorem 10.1 (Lindebergs Theorem) Suppose that in addition to the Triangular Array Con- ditions, the triangular array satisfies Lindebergs Condition : > 0 , lim i →∞ n i X j =1 E [ X 2 ij 1 ( | X ij | > )] = 0 (10.1) Then, as i → ∞ , L ( S i ) → N (0 , 1) . This theorem will be proved in Section ********* below. For an alternative proof using characteristic functions, see Billingsley Sec. 27. The Lindeberg condition makes precise the sense in which the r.v.s must be small. It says that for arbitrarily small > 0, the contribution to the total row variance from the terms with absolute value greater than becomes negligible as you go down the rows. The Lindeberg condition implies
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