1、 1.1 1. f lim x!1 f(x)= 1 P n=1 b n lim x!1 a n (x)= 1 P n=1 b n 0=0 lim x!1 f(x)= 1 P n=1 b n lim x!1 a n (x)= 1 P n=1 b n 1= 1 P n=1 b n 2. f(x)=x 1 P n=1 n (x)0;8x2R f x BA “ fa j g n j=1 “ a 1 a n ;y “nBA 1 f 8n 1 1 n f A = fxjf(x+)f(x) 0g 1 n A n =fxjf(x+)f(x) 1 n g A = 1 n=1 A n f f n = 8 : f
2、;jxjn f(n) ;xn f(n) ;x0 0 X x“a j x F(a j )F(a j )F(x)F(x“) “#0 P x“a j x F(a j )F(a j )= P x“a j x F(a j )F(a j )+F(x) F(x) P x“a j x F(a j )F(a j ) x F(x) F(x) 2 f c 8x2R;8“0 F c (x)F c (x“) = F(x)F(x“)F d (x)F d (x“) = F(x)F(x“) X j b j a j (x) X j b j a j (x“) = F(x)F(x“) X x“a j x b j “#0 f c 3
3、 1.2 6. x F(x+) F(x) 0 8“ 0 F(x+“)F(x+);F(x“)F(x) F(x+“)F(x“) F(x+)F(x)0 x x 0 8y 2 U (x;);9“ y 0;F(y + “ y ) = F(y“ y ) y F (x;x+) F(x+) (x ;x) F(x) x F(x+) F(x) x fb n g 1 n=1 0 1 P n=1 b n = 1 F(x) = 1 P n=1 b n a n (x) fa n g 1 n=1 1.3 1. F 0 F =F s , F ac (x)= Z x 1 F 0 (t)dt=0;8x2R , F 0 (x)=0
4、;a:e: , Fsingular 1F =F ac , F(x)= Z x 1 F 0 (t)dt;8x2R , F(x 0 )F(x)= Z x 0 x F 0 (t)dt;8x 0; 1 P n=1 b n = 1 F 0 =0;a:e: 8x = 2fa n g 1 n=1 0 a j (x)=0;8j F F 0 (x)= 1 P n=1 b n 0 a n (x)=0 2.1 3. T k C k n k = 1; ;n T n P k=0 C k n = 22 n A 1 ; ;A n 1. A j n1 A i ;i6=j 2. fA j g B j =( i2I j A i
5、)( i2NnI j A c i ) N =f1; ;ng;I j N 2 n B j (fA j g)= (fB j g) 9 6. =N F 1 = (f1g);F j+1 = (F j ;fj +1g);j 1;F = 1 j=1 F j F j F j+1 A n =f2ng2F 2n F;n 1 A = 1 n=1 A n = 2F F 1 _ j=1 F j = ( 1 j=1 F j ) F j 9. G =fAjA= n2K n ;K g GF 8A = n2K n 2 G A c = n2K c n = n2K i6=n i = n2K c n 2 G G A j = n2K
6、 j n 2 G;j 1 1 j=1 A j 2 G G G n 2 G;8n 1 F G 3 2.2 6. E = T F 1 = T F 2 ;F 1 ;F 2 2F (F 1 S F c 2 ) T = (F 1 T ) S (F c 2 T ) = (F 2 T ) S (F c 2 T ) = F 1 S F c 2 F 1 S F c 2 2F P(F 1 S F c 2 )= 1 P(F 1 )+P(F c 2 ) P(F 1 ) P(F 2 ) P(F 1 ) P(F 2 ) P P P ()=P()=1 fE n = T F n g n1 F n 2F F 0 1 =F 1
7、;F 0 n =F n T F c n1 T F c 1 ;n1, E 0 n = T F 0 n ;n1 P ( P n E n )=P ( TS n F n )=P ( TP n F 0 n )=P( P n F 0 n )= P n P(F 0 n ) T F i T F j =;8i;j1!(F i T F j ) c !P(F i T F j )=0 P n P (E n )= P n P(F n )= P n P(F n T F c n1 T T F c 1 )+P(F n T (F c n1 T T F c 1 ) c ) = P n P(F 0 n )+0. P ( P n E
8、 n )= P n P (E n ) 9. C F fE j g ( 1 j=1 E j ) c , E 0 E j fp j g 0; 1 P j=0 p j = 11 P(E j ) = p j ;j 1;P(E 0 ) = p 0 P F T : F ! 2 T(E j ) = f! j+1 g;j 0 T( j2I E j ) = j2I f! j+1 g T 11. f(x 1 n ;xg n1 (x 1 n ;x)=F(x)F(x 1 n ) n!1 (x)=F(x)F(x) x F fE n g“R E n n (A)=(A T E n ), lim n!1 n (A) = (A
9、) n (x) = F n (x) F n (x) (x)=F(x)F(x) x n 12. F(x) = P i b i a i (x) fa i g A2B(R) a i 1 (A S fa i g i1 ) = (A)+(fa i g i1 ) = (A)+1 (A)=0 2 fa i g i1 F(x) = (1;x) = (1;x fa i g i1 )+(1;x fa i g c i1 ) = (fa i ja i xg)+0 = X i (a i ) a i (x) F ,(fxg)=F(x)F(x)=0;8x2R,F 22. x 11: P(x) 0 E 2F fxg E =f
10、xg S N N 18 F E;F 2F x = 2N P(F)=P(F T E)=P(F T fxg)+ P(F T N)=P(F T fxg)=0 P(E) x 22: E 22: f(x)=P(E T (1;x) f(1) = 0;f(1) = P(E) 0 f P(E) 0 x 0 f(x 0 )= 1 2 (P(E)+0) x 0f(x)f(x) =P(fxg T E)!x2E P(fxg)=P(E) fxg E x 11: 24. S 8fx n g S;n 1;x n ! x 0 x 0 3 B(x 0 ;) n x n 2 B(x 0 ;) n 0 0 B(x n ; 0 )B
11、(x 0 ;) (B(x 0 ;)(B(x n ; 0 )0 x 0 2S S c 8x 2 S c x 0 B(x; x ) S c (B(x; x ) = 0 S c = x2S c B(x; x ) Lindelof S c = n B(x n ; x n ) (S c ) P n (B(x n ; x n )=0 O (O) = 0 8x 2 O; B(x;) O (B(x;)(O)=0 x2S c OS c B 1 8“ 0; (B(x;“) F(x + “ 2 ) F(x “ 2 ) F(x+“)F(x“)(B(x;“) 4 3.1 4. G( (!) x y x F(y) (!)
12、 F(x) (!) f!jG( (!)xgf!jF(x) (!)g (!) F(x) “ 0 (!) F(x+ “) G( (!)x+“x f!j (!)F(x)gf!jG( (!) xg f!j (!)F(x)gf!jG( (!)xgf!j (!)F(x)g P(G( )x)=P( F(x)=F(x) 11. FfXg X fX 1 (B)jB2B 1 g X B X =c = A c 3.2 1. X1 0;a:s: R X1 dP =0 X1 =0;a:s: X = 0;a:s: 12. 8“ 0 P(jX1 n j “) = R jX1 n j“ dP = R fjXj“g n dP
13、P( n ) ! 0;n!1 X1 n P !0 X lim n!1 R n XdP = R 0dP =0 P(jXjn) EjXj n !0 n =fjXjng 3. ”() = 1;”(?) = 0 fA n g 1 n=1 F 1 P n=1 ”(A n ) = 1 P n=1 1 A R X1 A n dP = 1 A R X 1 P n=1 1 A n dP = 1 A R X1 1 S n=1 A n dP =”( 1 S n=1 A n ) 6. fYX n g Fatou R lim n!1 (YX n )dP lim n!1 R (Y X n )dP Y R YdP Y Fa
14、tou R + X n =n1 0; 1 n ) ;n1 lim n!1 X n =0;a:s: R X n dP =1;8n 1 sup n X n = n1 1 n+1 ; 1 n ) Esup n X n = P n n( 1 n 1 n+1 ) = 1 2 3.2 11. EjXj1 fjXjag (EX 2 ) 1 2 (E1 fjXjag ) 1 2 =P(jXja) 1 2 EjXj1 fjXjag =EjXjEjXj1 fjXj 0; ja +bj p maxf1;2 p1 g(jaj p +jbj p ) 16. Fubini Z 1 1 (F(x+a)F(x)dx = Z
15、1 1 P(xX x+a)dx = Z 1 1 Z 1 fxXx+ag dPdx = Z Z 1 1 1 fxXx+ag dxdP = Z adP = a 1 LebesgueStieltjes 17. F(0)=0 Fubini Z 1 0 (1F(x)dx = Z 1 1 P(X x)dx = Z 1 1 Z 1 fxx)dx= R 1 1 P(X x)dx 3.3 2. P(X 1 X 2 =1)= 1 2 X 1 ;X 2 ;X 1 X 2 P(X 1 = 1;X 2 = 1;X 1 X 2 = 1) = 1 4 6= 1 8 = P(X 1 = 1)P(X 2 = 1)P(X 1 X
16、 2 =1) X 1 ; ;X n1 X n = X 1 X 2 X n1 n1 P(X 1 =1;X i = 1;i = 2; ;n) = 06= P(X 1 = 21) n i=2 P(X i =1) n = 3; 1 = fX 1 = 1g; 2 = fX 2 = 1g; 3 = fX 3 = 1g 2 S 3 = 1 S 2 P( 1 T ( 2 S 3 ) = P( 1 ) 6= P( 1 )P( 2 S 3 ) 3. A fi 1 ; ;i n g F i j E c k P( j F i j )= j P(F i j ) k =0 E k 1 k (k n) F i j = E
17、c j ;j = 1; ;k P( j F i j ) = P( jk1 E c j E c k jk E j ) = P( jk1 E c j jk E j )P( jk1 E c j E k jk E j ) = jk1 P(E c j ) jk P(E j ) jk1 P(E c j )P(E k ) jk P(E j ) = jk P(E c j ) jk P(E j ) = j P(F i j ) fF g B f1; ;ng P( n i=1 ( 2A i E ) c )=P( n i=1 2A i E c )= n i=1 2A i P(E c )= n i=1 P( 2A i
18、E c )= 3n i=1 P( 2A i E ) c ) 4 3.3 5. 2 A A A ;A = 2 A f (X ; 2 A );2g f1; ;ng C = fB 2 (X ; 2 A 1 )jP(B T 2kn B k )=P(B) 2kn P(B k );8B k 2X 1 (B);2A k g;C 0 = f i C i jC i 2 (X i );i2A 1 g C C 0 CC 0 C (C 0 ) (C 0 )= (X ;2A 1 ) C = (X ;2A 1 ) (X ;2A 1 ) X 1 (B);2kn;2A k 2kn (X ;2A k );1kn 10. X;Y
19、 E(jX+Yj p )= RR jx+yj p 2 (dx;dy)= RR jx+yj p x (dx) y (dy) R jx+yj p x (dx)1; y a:e: y 0 R jx+y 0 j p x (dx) 1 C r E(jXj p )= R jxj p x (dx)maxf1;2 p1 g( R jx+y 0 j p x (dx)+ R jy 0 j p x (dx)=maxf1;2 p1 g( R jx+y 0 j p x (dx)+jy 0 j p )1 14. 8x2(0;1; 1p p 1 n x n P( n =1)=p;P( n =0)=1p 18. 0;1 A
20、= 2 B fag Afag Afag = 2 B B Afag 0;1fag R 2 Lebesgue Afag2BB 4.1 1. 8M 0;P(X n 0;P( n kn fX k 0;9n0;8kn;X k m;onsomeN c ;P(N)=0 , X n !1a:s: 2. EjX n Xj = EjX n Xj1 fjX n X“jg +EjX n Xj1 fjX n X“jg EX1 fjX n X“jg +“ 2 Lebesgue 0 X n L 1 !X 3 4.1 4. f 8 0;9“0; jxj“g +Ejf(X n )f(0)j1 fjX n j“g 2MP(fjX
21、 n j“g)+Ejf(X n )f(0)j1 fjX n j“g “g)+ n!1; !0 6. Ej(X n Y n )(XY)j p =Ej(X n X)(Y n Y)j p maxf1;2 p1 g(EjX n Xj p +EjY n Yj p )!0 EjX n Y n XYj = Ej(X n X)Y n +X(Y n Y)j (EjX n Xj p ) 1 p (EjY n j q ) 1 q +(EjXj p ) 1 p (EjY n Yj q ) 1 q 1 Y n L q !Y (EjY n j q ) 1 q M M(EjX n Xj p ) 1 p +(EjXj p )
22、 1 p (EjY n Yj q ) 1 q !0 11. ) X P(jXj 0;P(jXjM)1 “0;P(jXjM) 0; 8m 0; P(jXj m) 0;8k 0;P(jf(X n )f(X)j“)P(jf(X n )f(X)j“;jXj k)+P(jXjk) f k;k 0; 8x;y2k;k; jf(x) f(y)j“ fjf(X n )f(X)j“;jXjkgfjX n Xj;jXj kgfjX n Xjg P(jf(X n )f(X)j“)P(jX n Xj)+P(jXjk) 8“ 0 0 k P(jXjk)“ 0 , n!1 “ 0 f(X n ) P !f(X) r:v:
23、 Markov 2 2 4.2 2. m(B n ;i:o:)=m(lim n!1 B n ) lim n!1 m(B n ) lim n!1 B n 4. r:v:X n b n P(jX n j b n ) 1 2 n ;8n 1 A n =nb n ; P(j X n A n j 1 n ) 1 2 n ;8n1 8“ 0; P n P(j X n A n j“) “;i:o:)=0 X n A n !0;a:s: 10. U =flim n!1 X n b;i:o:g U = a;b2Q;a bg P(X nb;i:o:) = 0 1 P(X n b;i:o:)=0 1 10 P(li
24、m n!1 X n lim n!1 X n )=0 1 3 4.3 4. n v ! 4.3.2 8 0;“ 0; n 0 (;“) n n 0 (a;b) (a+;b)“ n (a;b)(a;b+)+“ F 0 (a ;b + ) (a;b)“ j n (a;b)(a;b)j : 0; xn n1 F n n F n (x) ! x;n ! 1;8x 2 R, Lebesgue n (R) 8. 8B 2B(R) 8“ 0 U C U B C;(U) “ 2 (B)(C)+ “ 2 n (U)! (U); n (C)! (C);n!1 n (U) n (U) “ 2 ;(C) n (C)+
25、“ 2 n (U)“(B) n (C)+“ 1 n (B)“0;P(jX n Xj“)= R “ “ f X n X (x)dx = R “ “ R 1 n e x n 1 (0;1) (x)e yx 1 (1;x) (y)dydx n!1; R “ “ R e x 1 (0;1) (x)e yx 1 (1;x) (y)dydx= 1e “ 1 X n P 9X X n d !a;a f(x)= jxaj 1+jxaj f Ef(X n )! 0;n!1 4.1.5 X n a P !0 2 4.5 1. X n ! X;a:s: EjXj p 6 lim n!1 EjX n j p 6 Esup n jX n j p0 Z 1 1 jf A (x)x p jdF n (x) 6 Z jxjA jxj p dF n (x) = Z jX n jA jX n j p dP 6 Z YA Y p dP n 0 A!1 R 1 1 f A (x)dF n (x) n R 1 1 x p dF n (x) 4.5.2 1 4.5 6. fX t +Y t g 4.5.3
