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1 that j ) - JV>a(x)dG(x)|] -> 0 as n - oo. 4), that |JV(x)dG(x)- JV a (x)dG(x)| -> 0 as a -> oo. 13) J ) - JV(x)dG(x)|] - 0. 13) and Chebishev's inequality. D. Remark: The difference of this theorem from the classical weak law of large numbers is that the finiteness of second moments is not necessary. 2. 2 be satisfied, and assume in addition that supn(l/n)J^=lE[\(p(Xj)\2 +s ]< oo for some 8 > 0. s. 1 is not generally true. Suppose for example

6) Now take B! = {XE Rk: r(x) > 0}. This is a Borel set, for r is Borel measurable. 6), JBir(x)u(dx) = 0. This implies that i^B^ = 0. Similarly, we have for B 2 = {XG Rk: r(x) < 0} that v(B2) = 0. Since Bi and B 2 are disjoint we now have u(BiUB2) = + v(B2) = 0 or equivalently: 52 Introduction to conditioning P(r(X) ^ 0) = 0. s. 2) holds. The proof for the case that E [rx(X)] = 0 and/or E [r2(X)] = 0 is left to the reader as an easy exercise. D. Exercises 1. Let v be a probability measure on (Rk,33k) and let f be a non-negative Borel measurable function on R k such that Jf(x)u(dx) = c with 0 < c < oc.

The next central limit theorem is due to Liapounov. This theorem is less general than the Lindeberg-Feller central limit theorem (see Feller [1966] or Chung [1974]), but its conditions are easier to verify. ,X nkn are independent and kn—>oo. Put E (Xn,j) = anj,an = J2)l\an,p 2 0, limn^ooEjiiEIKXnj -a nJ )/ N(0,l) in distr. 209).

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