Definition two distribution functions u x and v x are

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Definition. Two distribution functions U (x) and V (x) are of the same type if there exist constants A > 0 and B E I. such that V(x) = U(Ax +B). In terms of random variables, if X has distribution U and Y has distribution V, then yg,X-B. A For example, we may speak of the normal type. If X o, 1 has N (0, 1, x) as its distribution and XJl,u has N(J.L, a 2 ) as its distribution, then XJl,u 4: aXo,I + J.L. Now we state the theorem developed by Gnedenko and Khintchin. Theorem 8.7.1 (Convergence to Types Theorem) We suppose U(x) and V(x) are two proper distributions, neither of which is concentrated at a point. Sup- pose for n 0 that X n are random variables with distribution function Fn and the U, V are random variables with distribution functions U (x), V (x ). We have constants On > 0, an > 0, bn E JR, f3n E R (a)If or equivalently then there exist constants A > 0, and B E I. such that as n oo and an 0, On f3n- bn B On V(x) = U(Ax +B), dU-B V=--. A (8.20) (8.21) (8.22) (b) Conversely, if (8.21) holds, then either of the relations in (8.19) implies the other and (8.22) holds. Proof. (b) Suppose
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276 8. Convergence in Distribution and Then ( an f3n- bn } Fn(CXnX + f3n) = Gn -X+ ( ) . On On Pickx such thatx E C(U(A ·+B)). Suppose x > 0. A similar argument works if x 0. Given f > 0 for large n, we have CXn f3n- bn (A- )X + B- f -X+ ( ) (A+ )X + (B +f), On On so Therefore, for any z E C(U(·)) with z > (A+ f)x + (B +f) . we have limsupFn(anX + f3n) limsupGn(Z) = U(z). Thus Since f > 0 is arbitrary, limsupFn(anX + f3n) inf U(z) = U(Ax +B) n->00 z>Ax+B by right continuity of U(-). Likewise, = U(z) n->00 for any z < (A- f)x + B- f and z E C(U(·)). Since this is true for all f > 0, liminfFn(anx + f3n) sup U(z) = U(Ax +B), n->oo z<Ax+B zeC(U(·)) since Ax+ B E C(U(·)). We now focus on the proof of part (a). Suppose
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8. 7 The Convergence to Types Theorem 277 Recall from Lemma 8.3.1 that if G n G, then also G:;- G +-. Thus we have Since U (x) and V (x) do not concentrate at one point, we can find y 1 < Y2 with y; e C(U+-) n C(V+-), fori = 1, 2, such that and Therefore, for i = 1, 2 we have (8.23) In (8.23) subtract the expressions with i = 1 from the ones with i = 2 to get Now divide the second convergence in the previous line into the first convergence. The result is Also from (8.23) Fn+-(yl)- bn -+U+-(yJ), an Fn-(yl)- f3n Fn-(Yt)- f3n ctn v-( )A _.:.:..__;;, _ __;;,_ · - --+ Yt , an Cln an so subtracting yields as desired. So (8.21) holds. By part (b) we get (8.22). 0
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278 8. Convergence in Distribution Remarks. (1) The theorem shows that when and U is non-constant, we can always center by choosing bn = Fn .... (yi) and we can always scale by choosing an = Fn+-(yz)- Fn .... (yJ). Thus quan- tiles can always be used to construct the centering and scaling necessary to produce convergence in distribution. (2) Consider the following example which shows the importance of assuming limits are non-degenerate in the convergence to types theorem . Let Then U(x) = 1°• 1, if t < C, if t ::::c. u-(t) = inf{y: U(y):::: t} = c, 1 -00, 00, if t = 0, if 0 < t :::: 1, ift>l. 8. 7.1 Application of Convergence to Types: Limit Distributions for Extremes A beautiful example of the use of the convergence to types theorem is the deriva- tion of the extreme value distributions. These are the possible limit distributions of centered and scaled maxima of iid random variables.
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