1、?5?OOK9pX4.1CS?5u“O?u“O?VI4nICS?5?5?OOK9pXA?/V11?Xna(i.i.d)C,P(Xn=1)=1/2,P(Xn=1)=1/2,-Sn=nPk=1Xkk2,?*?Sn,(JXeu“O?VI4nICS?5?5?OOK9pX1CS?Xn,n 1V1A?X,eP(:Xn()X()=1.PXna.s.X.e-A=:Xn()9 X(),oP(A)=0.l?*:w,Xna.s.X?=?3VA,?A,Xn()X().A?e?43A?e?.u“O?VI4nICS?5?5?OOK9pX1CS?Xn,n 1V1A?X,eP(:Xn()X()=1.PXna.s.X.e-A=
2、:Xn()9 X(),oP(A)=0.l?*:w,Xna.s.X?=?3VA,?A,Xn()X().A?e?43A?e?.u“O?VI4nICS?5?5?OOK9pX2?Xna(i.i.d)C,P(Xn=1)=1/2,P(Xn=1)=1/2,-Sn=1nnPk=1Xk,*?eZ?Sn,(JXeu“O?VI4nICS?5?5?OOK9pXdz*?vkw?.?e?:*?zSn?Cz,?,Xe4-3n=10,25,50Sn?w,?kw?Cz?.u“O?VI4nICS?5?5?OOK9pXdz*?vkw?.?e?:*?zSn?Cz,?,Xe4-3n=10,25,50Sn?w,?kw?Cz?.u“O?V
3、I4nICS?5?5?OOK9pX2?F(x),Fn(x),n 1 OCXCS?Xn,n 1?,eF(x)?Y:x,?n ,kFn(x)F(x),KFn(x),n 1fuF(x),PFnW F.Xn,n 1?X,PXnd X.u“O?VI4nICS?5?5?OOK9pX5P34?Y:?5.k7?.e?4?.u“O?VI4nICS?5?5?OOK9pX5P34?Y:?5.k7?.e?4?.u“O?VI4nICS?5?5?OOK9pXV?)V,?*?f3?X0,1l!?C?n90 k 2n 1,X2n+k=1Xnk/2n,(k+1)/2n.?*?Xn/Xeu“O?VI4nICS?5?5?OOK9p
4、XV?)V,?*?f3?X0,1l!?C?n90 k 2n 1,X2n+k=1Xnk/2n,(k+1)/2n.?*?Xn/Xeu“O?VI4nICS?5?5?OOK9pXu“O?VI4nICS?5?5?OOK9pXV3CS?Xn,n 1V?X,e?0,limnP|Xn X|?=0.PXnP X.=?9XnX?,?9?y?V.Ve?43A?e.u“O?VI4nICS?5?5?OOK9pXV3CS?Xn,n 1V?X,e?0,limnP|Xn X|?=0.PXnP X.=?9XnX?,?9?y?V.Ve?43A?e.u“O?VI4nICS?5?5?OOK9pXn5?*)*?y(A?CXn,n 1)
5、,?k(1)k?(?),*?y?(J(CXn(),Cz.(2)zy,4Cz,?5(Xn?),5?Cz.(3)c?.?,y?*kloNw?CzA?A?(1);A?(2);V(3)?Ny.u“O?VI4nICS?5?5?OOK9pXn5?*)*?y(A?CXn,n 1),?k(1)k?(?),*?y?(J(CXn(),Cz.(2)zy,4Cz,?5(Xn?),5?Cz.(3)c?.?,y?*kloNw?CzA?A?(1);A?(2);V(3)?Ny.u“O?VI4nICS?5?5?OOK9pXn5?*)*?y(A?CXn,n 1),?k(1)k?(?),*?y?(J(CXn(),Cz.(2)zy,
6、4Cz,?5(Xn?),5?Cz.(3)c?.?,y?*kloNw?CzA?A?(1);A?(2);V(3)?Ny.u“O?VI4nICS?5?5?OOK9pXn5?*)00 P(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+P(|Y|Xn X|/2).u“O?VI4nICS?5?5?OOK9pXyeyXnP X YnP YXnYnP XY.:UV?,I?y,?00 P(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+
7、P(|Y|Xn X|/2).u“O?VI4nICS?5?5?OOK9pXyeyXnP X YnP YXnYnP XY.:UV?,I?y,?00 P(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+P(|Y|Xn X|/2).u“O?VI4nICS?5?5?OOK9pXyeyXnP X YnP YXnYnP XY.:UV?,I?y,?00 P(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+P(|Y|Xn X|/2).u“
8、O?VI4nICS?5?5?OOK9pXyeyXnP X YnP YXnYnP XY.:UV?,I?y,?00 P(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+P(|Y|Xn X|/2).u“O?VI4nICS?5?5?OOK9pXyduX,Yk?,?0,3M 0?P(|Y|M)/4,P(|X|M 1)N1P(|Xn X|1)/8,?P(|Xn|M)P(|Xn X|1,|X|1)P(|X|M 1)=1 /4.?P(|Xn|M)/4.u“O?VI4nICS?5?5?OOK9pXyduX,Yk?,?0,3M
9、 0?P(|Y|M)/4,P(|X|M 1)N1P(|Xn X|1)/8,?P(|Xn|M)P(|Xn X|1,|X|1)P(|X|M 1)=1 /4.?P(|Xn|M)/4.u“O?VI4nICS?5?5?OOK9pXyduX,Yk?,?0,3M 0?P(|Y|M)/4,P(|X|M 1)N1P(|Xn X|1)/8,?P(|Xn|M)P(|Xn X|1,|X|1)P(|X|M 1)=1 /4.?P(|Xn|M)/4.u“O?VI4nICS?5?5?OOK9pXyduX,Yk?,?0,3M 0?P(|Y|M)/4,P(|X|M 1)N1P(|Xn X|1)/8,?P(|Xn|M)P(|Xn
10、X|1,|X|1)P(|X|M 1)=1 /4.?P(|Xn|M)/4.u“O?VI4nICS?5?5?OOK9pXy?,?0,3N N1,?n NP(|Yn Y|/2M)/4,P(|Xn X|/2M)/4.d?n NP(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+P(|Y|Xn X|/2)P(|Xn|M)+P(|Yn Y|/2M)+P(|Y|M)+P(|Xn X|/2M)/4+/4+/4+/4=.u“O?VI4nICS?5?5?OOK9pXy?,?0,3N N1,?n NP(|Yn Y|/2M)/4,
11、P(|Xn X|/2M)/4.d?n NP(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)+P(|Y|Xn X|/2)P(|Xn|M)+P(|Yn Y|/2M)+P(|Y|M)+P(|Xn X|/2M)/4+/4+/4+/4=.u“O?VI4nICS?5?5?OOK9pXy?,?0,3N N1,?n NP(|Yn Y|/2M)/4,P(|Xn X|/2M)/4.d?n NP(|XnYn XY|)=P(|XnYn XnY+XnY XY|)P(|XnYn XnY|+|XnY XY|)P(|Xn|Yn Y|/2)
12、+P(|Y|Xn X|/2)P(|Xn|M)+P(|Yn Y|/2M)+P(|Y|M)+P(|Xn X|/2M)/4+/4+/4+/4=.u“O?VI4nICS?5?5?OOK9pXSlutskys Theoremvkc?oK$5,?kXe(J:n2eXnd X,YnP c(),K(1)Xn+Ynd X+c;(2)XnYnd cX;(3)ec=0,KXnYnP 0;(4)ec,0,Yn,0,KXn/Ynd X/c.u“O?VI4nICS?5?5?OOK9pXSlutskys Theoremvkc?oK$5,?kXe(J:n2eXnd X,YnP c(),K(1)Xn+Ynd X+c;(2)Xn
13、Ynd cX;(3)ec=0,KXnYnP 0;(4)ec,0,Yn,0,KXn/Ynd X/c.u“O?VI4nICS?5?5?OOK9pXa.s.?Onn3e?Kp?d:(1)Xna.s.X.(2)?0,Pn=1m=n|Xm X|?=0.Tn=1Sm=n|Xm X|?u),X?n,3d?o3,Xm?|Xm X|?,L(1)kXn,?|Xn X|?,(2)Xn?X.u“O?VI4nICS?5?5?OOK9pXa.s.?Onn3e?Kp?d:(1)Xna.s.X.(2)?0,Pn=1m=n|Xm X|?=0.Tn=1Sm=n|Xm X|?u),X?n,3d?o3,Xm?|Xm X|?,L(1)
14、kXn,?|Xn X|?,(2)Xn?X.u“O?VI4nICS?5?5?OOK9pXBorel-Cantellinn4?An,n 1Vm(,F,P)?,PAn,i.o.=n=1m=nAmL/Anu)g0.(1)eXn=1P(An)0,kXn=1P(|Xn X|?)0,kXn=1P(|Xn c|?).u“O?VI4nICS?5?5?OOK9pXV?n7eCS?Xn,n 1vlimnEX2n=0,KXnP 0.4?X1,X2,EX1=,VarX1=2,PSn=X1+Xn,KSnnP.u“O?VI4nICS?5?5?OOK9pXV?n7eCS?Xn,n 1vlimnEX2n=0,KXnP 0.4?
15、X1,X2,EX1=,VarX1=2,PSn=X1+Xn,KSnnP.u“O?VI4nICS?5?5?OOK9pXA?%Vn8Xna.s.X%XnP X,!Xe?X0,1l!?C-X1=1X0,1X2=1X0,1/2,X3=1X1/2,1,X4=1X0,1/3,X5=1X1/3,2/3,X6=1X2/3,1,Xn(n1)/2+k=1X(k1)/n,k/n,1 k nyXna.s.9 0,?XnP 0.u“O?VI4nICS?5?5?OOK9pXA?%Vn8Xna.s.X%XnP X,!Xe?X0,1l!?C-X1=1X0,1X2=1X0,1/2,X3=1X1/2,1,X4=1X0,1/3,X5
16、=1X1/3,2/3,X6=1X2/3,1,Xn(n1)/2+k=1X(k1)/n,k/n,1 k nyXna.s.9 0,?XnP 0.u“O?VI4nICS?5?5?OOK9pXA?%Vn8Xna.s.X%XnP X,!Xe?X0,1l!?C-X1=1X0,1X2=1X0,1/2,X3=1X1/2,1,X4=1X0,1/3,X5=1X1/3,2/3,X6=1X2/3,1,Xn(n1)/2+k=1X(k1)/n,k/n,1 k nyXna.s.9 0,?XnP 0.u“O?VI4nICS?5?5?OOK9pXA?%Vn8Xna.s.X%XnP X,!Xe?X0,1l!?C-X1=1X0,1X2=1X0,1/2,X3=1X1/2,1,X4=1X0,1/3,X5=1X1/3,2/3,X6=1X2/3,1,Xn(n1)/2+k=1X(k1)/n,k/n,1 k nyXna.s.9 0,?XnP 0.u“O?VI4nICS?5?5?OOK9pXA?%Vn8Xna.s.X%XnP X,!Xe?X0,1l!?C-X1=1X0,1X2=1X0,1/2,X3=1X1/2,1,X4=1X0,1/3,X
