6 Q :q ¤ úÃv g - University of the Ryukyus

1 6 .. (1) Z ( ) X = Re Z, .. Y = Im Z ( ) Z = X + iY , i = 1 .. (2) Z E[| Re Z|] < E[| Im Z|] < Z . E[Z] = E[Re Z] + iE[Im Z].. Z |E[Z]| E[|Z|] . : E[|Z|] < | Re Z| |Z|, | Im Z| |Z| E[Z] = E[Z], Z. Z = 1{Z =0} . |Z|. [ ] [ ( ) ( ). ] [ ] [ ]. |E[Z]| = | | = =E Z = E |Z| Z = E |Z| Re Z + iE |Z| Im Z. | | | | | | | | | |. ( ) ( )= 0 Re | | Z | |.. Z 1 ( . ) E[|Z|] . X X (t) X (characteristic function) . X (t) = E[eitX ] = E[cos(tX)] + i E[sin(tX)], t R. (i) X . (ii) t | X (t)| 1 . (iii) X (0) = 1 X (t) = X ( t) . (iv) t X (t).

2 : (i), (ii) |eitX |2 = | cos tX + i sin tX|2 = cos2 tX + sin2 tX = 1 . (iii) X (0) = E[e0 ] = E[1] = 1, X ( t) = E[e itX ] = E[e itX ] = E[eitX ] = X (t). (iv) 0 {hn } sups R | X (s + hn ) X (s)| 0 ( . ) sups E[|eisX (eihn X 1)|] = E[|eihn X 1|]. |eihn X 1| |eihn X | + 1 = 2 E[2] = 2 < . Lebesgue E[|eihn X 1|] E[|e0 1|] = 0 . X a, b aX+b (t) = eitb X (at). : aX+b (t) = E[eiatX eitb ] = eitb E[eiatX ] = eitb X (at).. (1) X B(n, p) , q = 1 p , ( ) n ( ). n k n k n n itX itk X (t) = E[e ]= e p q = (peit )k q n k = (eit p + q)n.

3 K k k=0 k=0. 13. (2) X Poisson P ( ) .. k (eit )k . e = ee e = e (e 1) . it it X (t) = E[eitX ] = eitk e =. k! k! k=0 k=0. X N (0, 1) X (t) = e 2 t . 1 2. : Cauchy .. 1 1. eitx e 2 x dx = e 2 t e 2 (x it) dx 1 2 1 2 1 2. X (t) = E[eitX ] = ( ). 2 2 . itx 21 x2 = 12 (x it)2 12 t2 R > 0 . 4 CR ( ). CR,1 : R R, CR,2 : R R it, CR,4 : R it R. CR,3 : R it R it, . e 2 z dz = 0 . 1 2. e 2 z 1 2. C Cauchy . CR.. 4 .. e 2 z dz = e 2 z dz 1 2 1 2. CR n=1 CR,n R . R . e 2 z dz = e 2 x dx e 2 x dx =. 1 2 1 2 1 2. 2 . CR,1 R.

4 CR,2 z = R + iy dz = i dy . t |t| 1 2 2 2 (R y )+iRy e 2 z dz = e 2 (R+iy) i dy . 1 2 1 2. e i dy CR,2 0 0. |t|. e 2 (R y 2 ). dy |t|e 2 (R t2 ). 1 2 1 2. = 0. 0. CR,4 z = R + iy . 0 e 2 z dz = e 2 ( R+iy) i dy |t|e 2 (R t ) 0. 1 2 1 2 1 2 2. CR,4 t CR,3 z = x it dz = i dx . R R . 21 z 2 12 (x it)2 21 (x it)2. e 2 (x it) dx. 1 2. e dz = e dx = e dx . CR,3 R R .. 1 . e 2 (x it) dx = 0 ( ) X (t) = e 2 t 2 = e 2 t . 1 2 1 2 1 2. 2 . 2 . ( 17 ) . 1 2 2. X N (m, 2 ) X (t) = eimt 2 t . X m : Z = Z X = Z + m . X (t) = Z+m (t) = eimt Z ( t) = eimt e 2 ( t) = eimt 2.

5 1 2 1 2 2. t . 14. 1 1. X Cauchy f (x) = ( < x < ) . 1 + x2. X (t) = e |t| . : .. 1 1. X (t) = E[eitX ] = eitx dx ( ). 1 + x2.. 1st step t > 0 R > 1 2 CR ( ). CR,1 : R R, CR,2 : |z| = R, Im z 0 R R. eitz g(z) = z = i . z+i . 1 eitz 1 g(z) e t dz = dz = g(i) = . ( ). 2 i CR 1 + z 2 2 i CR z i 2i . 2 . eitz eitz dz = dz ( ). CR 1 + z2 n=1 CR,n 1 + z2. R . R . eitz eitx eitx dz = dx dx CR,1 1 + z2 R 1 + x2 1 + x2. CR,2 z = Rei , 0 , dz = Riei d . eitz eitRei eitR(cos +i sin ) dz = Rie d . i . R d . CR,2 1 + z2 0 1 + R 2 e2i.

6 0 |R2 e2i + 1|. tR sin . e tR sin e R. = R d R d 2 0. 0 |R e R 1 R 1. 2 2i + 1| 2. 0. 2 |R2 e2i + 1| |R2 e2i | 1 = R2 1 0 . sin 0 t > 0 e tR sin 1 ( ), ( ) .. eitx1 e t 2. dx =. 1 + x2 i 2i . 1 eitx 2i ( ) X (t) = dx = e t . 1 + x2. 2nd step t = 0 X (0) = 1 (iii) . t < 0 ( ) y = x .. 1 1 1 1. X (t) = eit( y) dy = ei( t)y dy = e ( t) = e |t| . 1 + ( y)2 1 + y2. 3 t > 0 1st step . (k). X E[|X|k ] < X (t) C k - X (t) = ik E[X k eitX ].. : 16 . 15. Cauchy t = 0 Cauchy .. p X = (X1 , .. , Xp ) Rp X (t) X .. { p }. X (t) = E[eit X ] = E[exp i tj Xj ], t = (t1.)

7 , tp ) Rp j=1.. p X p A p b AX+b (t) = eit b X (A t).. : AX+b (t) = E[eit AX eit b ] = eit b E[ei(A t) X ] = eit b X (A t).. X = (X1 , .. , Xp ) p N (m, ) m = (m1 , .. , mp ) Rp , 1 . = ( ij ) X (t) = eit m 2 t t . : P = (pij ) D = ( ij ) P P = D . Y = (Y1 , .. , Yp ) = P (X m) Y N ((0, .. , 0) , D) . ( 19 ) Y1 , .. , Yp Yj . N (0, jj ) . Y (t) = E[eit1 Y1 eitp Yp ] = E[eit1 Y1 ] E[eitp Yp ]. 1 . = e 2 11 t1 e 2 pp tp = e 2 t Dt . 1 2 1 2. X = P Y + m .. t) D(P t) 1 1 . X (t) = eit m Y (P t) = eit m e 2 (P = eit m e 2 t P DP.

8 1. t = eit m 2 t t .. Dynkin . X R X : X (A) = P (X A), A B(R). ( ). B(R) R Borel X X (distribution) . X X (R, B(R)) . ( , F, P ) ( ) . X FX (x) FX (x) = P (X x) = X (( , x]) . X, Y X, Y : X = Y , X (A) = Y (A) ( A . B(R)) FX (x) = FY (x) ( x R) . - Dynkin . (1) S P . (a) S P, (b) A, B P A B P. 2 . (2) S D Dynkin . 16. (a) S D. (b) A, B D A B A\B D.. (c) An D, An An+1 ( n N ) n=1 An D. 3 . S C C Dynkin L(C) D ( ) Dynkin .. D Dynkin ( ) {D } C Dynkin .. L(C) = D D D ( ) C .. Dynkin 0 D 0 = D . (Dynkin ) P L(P) = (P) *4.)

9 : - Dynkin ( 20(2) ) L(P) (P) .. L(P) (P) L(P) - . 1st step A P GA = {B; A B L(P)} GA P Dynkin .. B P A B P L(P). B GA , P GA . (a) S P GA.. (b) B1 , B2 GA , B1 B2 A B1 , A B2 L(P) A B1 A B2 A (B1 \B2 ) =. (A B1 )\(A B2 ) L(P). B1 \B2 GA . ( ). (c) Bn GA , Bn Bn+1 ( n N ) A Bn L(P), A Bn A Bn+1 A n=1 Bn =.. n=1 A Bn L(P). n=1 Bn GA . L(P) GA A P, B L(P) A B L(P) .. 2nd step A L(P) GA = {B; A B L(P)} .. 1st step P GA (a) S P GA .. (b), (c) 1st step GA P Dynkin . A, B L(P) A B L(P) L(P) . 3rd step L(P) - . (i) S L(P) (ii) A L(P) (i) S L(P) Ac = S\A Ac L(P).

10 N (iii) An L(P) (n N ) Bn = k=1 Ak (ii) L(P) . ( n ). c c . Bn = k=1 Ak L(P) (c) k=1 Ak = k=1 Bk L(P) .. : (= ) A = ( , x] . ( =) J = {( , x]; x R} {( , )} A = {A B(R); X (A) = Y (A)} J. A Dynkin ( 20(3). ) x ( , ) X (( , x]) = FX (x) = FY (x) = Y (( , x]) x = . X (( , )) = Y (( , )) = 1. J A (J ) = L(J ) A. J - (J ) Borel B(R) B(R) A .. f (x) f 0 |f (x)| X (dx) < E[f (X)] = f (x) X (dx). R R. *4 (P) P - . 17.. : f (x) f (x) = ai 1Ai (x) .. E[f (X)] = ai P (X Ai ) = ai X (Ai ) = f (x) X (dx). i i R. , f (x) 0 {fq (x)}.))))