应用随机过程补充讲义-马氏链

1、 5 ij , i, j 1 cd y noT¸ ¥ ( T X0 = 1, Xn n x Xn+1 = j =1 nj . d Xn; n 0 Ö W4qrsTeDn x h n © µ¶lm ¨ y no áe FvT½ ¦T 4ã à ú w 9 ¨no X © t¨lm» T 1874 C Francis Galton â Reverend H.W. Watson ¾ 0 Æ Î

2、7 S Á ¨ T e Galton-Watson Th GW ©¥2 Ç de¨f¹3©£ w ¥0½ a Î £¯ gh¨i Thj yØ Á È k ¨ 1 ¹©ðD¥4 1 ðTöp l # m § n ! S ¥¦ÌÄ© d T T4 o p a 0 T qr P£E&

3、#169; Xn = 0, Xn+k = 0 k 1. X 0 eD st ©u = minn, Xn = 0. C Üù vw P ( < ¤ x D 1. ¥4 8 ÛÜ = P ( < , f (s = k P ( = k = p k , k = 0, 1 , 2 , · · · . pk sk , m= kpk . 8. (1EXn = mn . ¤ £ s = f (s ¨ ¦ ¥ y © < 1 £

4、; × w £ m > 1. (2 ÷ø. (1 EX1 = E = m. z C T Xn EXn+1 = E j =1 l=1 l nj = E l l=1 j =1 1Xn =l nj =E ¥ l ¦ (2 = 1Xn =l nj = j =1 l P (Xn = lml = mEXn = mn+1 . vw 1 ¤ £ s = f (s ¨¥ y © f (s 0,1 (¬v a s ÿ TÔ T 2T |V y = f (x Ý

5、 A V y = x 0,1 (¬ú x = 1 f (1 = m. £ m < 1 g ©£ m = 1 2T |V y = f (x Ý A V y = x 0,1 (¬ú x = 1 g © £ m > 1, |V y = f (x Ý A V y = x 0,1 (¬¢ g Y©©' © 11 P ( < , X1 = k = pk P ( < |X0 = k = pk k . T m <

6、1, m = 1, m > 1, nen x ¤p kT k, â k. n x ¨ Ú ÕsAØ4 © Dà T Y¨ ¥i ©Ú 4n x U E ¨lmnoTeD noT D PGW ; ðDn x 4¥ ( ¨lmno GW áeTá§Ûܽ¥ (¨lmno pk UE ¨lmno©§ GW pk no 

7、¡ ¸¨ TeD GW © Di¥iT Dhi¥iT d = 0 . plâl¨nx 0 U¨lmno¥º Á 0, PGW (0=1, PGW (=0. l#$T PGW (0 = , Aa A , e» PGW ( = 1 . PGW (A = i ¨ÙªlmT à ÛÜ vw P . D X Pwe PGW GW GW ¨ ©Aa T I e4Y o D T YTA Y

8、 x, e» |x| D x Ý o e» |e| = min|e0|, |e1|. ¸Â e ¨ËÌ w(e = |e|. ¥Õ ¢ 4à e = (e0 , e1 , ˨À©ì áÎTÛÜ ½ ¨ D¥¤4 PQR T e ¨ PW R(e = |e|. d R(o, ¤ ¨ ÿ ©

9、;e» PGW (A . PGW ( e br(T D T ¨ branching number. 9. P (br(T = m = 1. ¨ nTbr(T m. TËÌ w(e e»¨qrs©¼ m, d ÷ø. w©GW gبqrs¦§¨©¾x 9 R(o, =, ÛÜ ¨ ¤ Dirichlet Principle. ¸ a0 = 0; br (T

10、= sup, R(o, < . T ö k ¨YD Xk , ¸ f (y = a|y|, d EGW u n vS al = 1, l n; ak = k s s=1 ( m , n s s=1 ( m 1 k n 1. (up(u, v (f (u f (v 2 = 2EGW k =1 n k+1 Xk (ak ak1 2 . n k k =1 ( m 2 = 2 k =1 k+1 mk (ak ak1 2 = 5 Å 4 a 0, a¤ PGW (oo = 0 = 1. ½ X ᨠGW ¨

11、 PW ¤ £ n h t #©©' © 12 © F4 ¸ ( ¨cd ¡ ¤qrs©ÀÁ ٪ ýzWD qrs Ä 2 Ã9. ±Å ²Æ n ÁÇÈ a WTu 4 È ¿Ô 4 ÈÉÊÅ 0 h 1 2TËt 10. Ê

12、mD p. X0 ÇÈ ÌÍ È ­ 9 ¨ Î © Xk ÇÈ k p ÉÊ Ët ¨ Î T Wqrs©¶ © Xk Ö FÎÙªlm¨ ÐÑ »© Á = P (X0 = 1, = 1 , q = 1 p. Ï y ¤¥¦§¨TA x, 

13、10; pxy (n pyy (n. ¶© Fz© 11. n=0 n=0 © F ³ qr © 4ò 12. §Ô cÝ A m, n 1; A x1, · · · , xn, y S, 1 2 n+m n n 1 1 n1 n1 A A n1 < n2 < · · · < nr ; x1 , · · · , xr S, r P (X0 = 1|Xn = 1 = + (p q n , 1

14、 + ( (p q n P (X = y |X = x , X = x , · · · X =x = P (Xm+n = y |Xn = xn . P (Xn1 = x1 , · · · Xnr = xr = P (X1 = x1 3 P (Xm1 = y1 , · · · , Xms = ys |Xn1 = x1 , · · · Xnr = xr , Xn = x = P (Xn1 = x1 , · · · Xnr = xr |Xn = x. 13. 14. A r, s 1, n1 < n2 < · · · < nr < n < m1 < m2 < · · · < ms , A