第11・12回：確率過程と ブラウン運動 (Brownian Motion) シ …

1 1 11 ( brownian Motion) .. (random walk) x n=0 x=0 ( n=1) ( x=1) ( x(n+1)= x(n)+1) ( x(n+1)= x(n) -1) 1/2 n x(n) n x(n) x (n) n x(n) n (= 0,1,2,..) x(=0, 1, 2.)

2 0 n n n , n n n x x n n n .. xnnnnn , = =+ + + 2 ,2 xnnxnn =+= +n )21,(nB x (even) = =nxnnxn,,3,1,,4,2,0KK x (odd)),,2,1,0,(nnnK= +nnnxnxnnnnnnnnnnnnnBnxp +== = === + +++ +21!2!2! ) !!! ( 2121 ))(21,(),( n x(n) 2 n x x = 2n+-n 4)1())((2)())(21,(22nppnnEnpnnEnnnB= = == ===++++ () 44)2(4)2( 022)2(222nnnnEnnEnnnnE= = = == = = +++ 0 n p(x,n) B(n,p=1/2) n N( =np=n/2, = np(1-p) = n/4) n+ N(n/2, n/4) x = 2 n+-n= 2( n+-n/2) N(0, n) ),0(21!

3 2!2!),(nNxnxnnnxpn += )2log(21log21)!log( + + nnnnn n! (Stirling s formula) log Stirling's Formula-50050100150200250300350400159131 7212529333741454953576165697377818589939 7nlog(n!)(n+1/2)log n - n +1/2 log(2 )nxnxnnnxp +=21!2!2! ),( + +++ = + + +++ = + ++++ + + ++ ++ ++= + + ++ + ++ + + + + =nxxnnxxnnnnxnxnnxnxnnxnxnxnxnxnxnnxnxnn xnxnxnxnxnxnnnnxnxnnnxpn1log211log212log 2log )2log(212loglog21log21 )2log(212log212212 )log(212)log(212log21212122 2log)2log(2122log212 )2log(2122log212)2log(21log21 21log!

4 2log!2log!log ),(log + +++ nxxnnxxnnnxp1log211log212log2log ),(log L+ + =+432413121)1log( += + + = + + +++ + + = + + ++= +++<<2232323221211log211log21 21211log2121211log21 1nxnxnxnxnnxxnxxnnxxnnxnxnxxnnxxnnxnxnxx nnxxnnx + 222122log ),(lognxnxnnxp 3 + 222122log ),(lognxnxnnxp ),0(2 21exp22 ),( 2nNnxnnxp = 0 n 2 p(x,n) x=2 p(x,n) n x x p(x,n) nxnxnnnxp +=21!

5 2!2! ),( n x(n) {} ),1( ),1(21 )1,( nxpnxpnxp ++=+{}() =+= + +++=+= ++++ + ++= + ++ ++= ++=++)1,(21!21!21!1212121!21!21!21!21!21 !21!21!21!21 ),1( ),1(21 111nxpxnxnnnxnxnxnxnnxnxnnxnxnnnxpnxpnnn n444344421{} ),1( ),1(21 )1,( nxpnxpnxp ++=+ n+1 x n x+1 x :1/2 n x-1 x :1/2 ),1( nxp {} ),1( ),1(21 )1,( nxpnxpnxp ++=+ x+1xx-1 1/2 1/2),1(nxp+ )1,( +nxp , f ( ,t) {} ),1( ),1(21 )1,( nxpnxpnxp ++=+{} ),(2),1( ),1(21 ),( )1,( nx p nxpnxpnxpnxp ++= + , = =ntx = =),( ),(),(tftfnxp.

6 = =),( ),(),(tftfnxp{} ),(2),1( ),1(21 ),( )1,( nx p nxpnxpnxpnxp ++= +{} ),( ),(),( ),(21 ),( ),( ),(2),( ),(21 ),( ),( 2tftftftftftft f tftftftf + = + ++= +D= 220, 22),(),( = tfDttf D 4 = =),( ),(),(tftfnxp =DtDttf4exp41),(2 =DtDtnxnnxp4exp41 2exp21),(220, ,2 ,2 = = = Dntx22),(),( = tfDttf {} ),(2),1(),1(21 ),( )1,( nxpnxpnxpnxpnxp ++= + 7