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A Course in Mathematical Statistics phần 5

Tham khảo tài liệu 'a course in mathematical statistics phần 5', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 8.5 Further Limit Theorems 203 Since as 0 FX z c FXzc we have J X lim sup P X z n - V Yn 7 FX zc z eft 6 Next Xn z c Xn z c n 1 - c Xn z c n IYn - c c IYn - c IX z c n 1 - c . By choosing c we have that Yn - c is equivalent to 0 c - Yn c and hence r . . í X X z c - I Y - c X z if N Yn 7 z 0 and X z c n n - c c X z if z 0. N Y n 7 That is for every z e R Xn Jc n IYn - c s fX z N Y n 7 and hence Xn z c c y - c U fX zi z e w. N Y ri 7 Thus p x z c p Yn -c pfX zl N Yn 7 Letting n and taking into consideration the fact that P Yn - c 0 and P Xn z c Fv z c we obtain FX z c . . . _í X lim inf P X z n N Yn 7 z e w. Since as 0 FX z c FX zc we have X Fx zc limmf p X z z e R 7 Relations 6 and 7 imply that lim P Xn Yn z exists and is equal to Fx zc p x z Fx c z . 7 Thus 204 8 Basic Limit Theorems X Y - z FxJy- z n FX z z e as was to be seen. REMARK 12 Theorem 8 is known as Slutsky s theorem. Now if X j 1 . n are i.i.d. r.v. s we have seen that the sample variance n 2 n sn n z Xj - Xn n t X2 - Xn2. Next the r.v. s X2 j 1 . n are i.i.d. since the X s are and e x 2 Ơ2 Xj EXj 2 Ơ2 u2 if u e X j G G Xj which are assumed to exist . Therefore the SLLN and WLLN give the result that 1 V v2 . 2 J X j G u a.s. n and also in probability. On the other hand xn u2 a.s. and also in 2 2 n tt probability and hence Xn u a.s. and also in probability by Theorems 7 i and 7 i . Thus n A X V 2 V 2 . 2 2 2 _2 X j X n Ơ u u G a.s. nj-11 1 and also in probability by the same theorems just referred to . So we have proved the following theorem. THEOREM 9 Let Xj j 1 . n be i.i.d. r.v. s with E X u ơ2 Xy Ơ2 j 1 . n. Then S2 G2 a.s. and also in probability. REMARK 13 Of course S2 S 2 Ơ2 implies n 1 n n tt 2 n tt n 1 G since n n 1 1. n tt COROLLARY If X1 . Xn are i.i.d. r.v. s with mean u and positive variance Ơ2 then TO THEOREM 8 .--- r n 1 Xn u N 0 1 and also nX u N 0 1 . S n S n tt PROOF In fact Xn u N 0 1 G n tt 8.5 Further Limit Theorems 205 by Theorem 3 and Jn _Sn P . 1 vn-ĩ Ơ - by Remark 13. Hence the .