- "+$,&!"'$%"'&&!"($%"(&&!"#$%"#&&!"$%"&& 2009, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 38 : 5 ( 1.1!"*$%"*&& W path, x (i ρ 1, ρ 2, ±1. n, 2 n paths. 1 ρ 1, ρ 2, {1, 1}. a n, { a, n = 0 (1.1 w(n a + ρ 1 + ρ 2 + + ρ n, n = 1, 2, (ii (0, a (n, b W, u w(1 a, w(2 w(1,, w(n w(n 1, 1 v w(1 a, w(2 w(1,, w(n w(n 1, 1. 1.1. 2, ( path W : W : (0, w(0 (1, w(1 (2, w(2 (n, w(n & (!&#'% '*$!"#$%
, : Step 1., a > 0, b > 0, u v = b a, u + v = n u = n + b a 2, v = n b + a. 2 (iii (0, a (n, b W, (ii. path N(n (1.2 n! N(n = n C u = ( n + b a! ( n b + a! 2 2 n b a. = n C (n+b a/2. W : (0, a (1, w(1 (n, b, w(t 0 0 < t < n path,. (i W, x, x t. t = min{k : w(k 0}. (ii W, path W : (2.1 W : (0, a (1, w(1 (t, 0 (t + 1, w(t + 1 (n, w(n. 2, c 0 < k n, S k > c *1. "%$'& "%$*'&!"#$!%& "($&! W path (, t W x, t W path. Step 2., 1:1. (0, a (n, b x path (0, a (n, b path, path. (0, a (n, b path (0, a (n, b x path, path. *1, {w(n}, 1 c.
!"#$!%& "($& (,. "%$'& 2.1 "%$*'& (0, a (n, b x path 2.2 (. b > 0. (0, 0 (n, b path, (2.3 w(1 > 0, w(2 > 0,, w(n = b > 0!"#$%!&#'%!*#"%!"#($% 2.2 (0, a (n, b path (2.4 b n nc (n+b/2. [ ] w(1 > 0, (2.3 path (1, 1. (1, 1 (n 1, b path (2.3. (1.2 + [ ], 2.1. (i (0, a (n, b W, W x path = (0, a (n, b W path. (i (1, 1 (n, b path = n 1 C (n+b/2 1. (ii x path = (1, 1 (n, b path = n 1 C (n+b/2. (ii, (1.2,, (2.2 ( n + b + a 2 n!! ( n b a! 2 ( (1.2 a a. = n C (n+b+a/2 = n 1 C (n+b/2 1 n 1 C (n+b/2 (n 1! (n 1! = ( ( ( ( (n + b/2 1! (n b/2! (n + b/2! (n b/2 1! = n! ( (n + b/2 ( ( (n + b/2! (n b/2! n = n C (n+b/2 b n. (n b/2 n
3 path W : w(0 = 0, w(1, w(2,, w(n ( &'(*+,-./0 y(k w(k w(k 1 = ±1, k = 1,, n. path W ( : (3.1 W : w (0 = 0, w (1 y(n, w (2 y(n + y(n 1,, w (n y(n + + y(1. &'(*+,-./0 Step1. path W path W (3.1. W, W : 1., 2., 3. &'(*(+,-./01234 W w(k > 0, k = 1, 2,, n. &'(*+ path W path W. w (n = w(n w(n 1+w(n 1 w(n 2+ w(1 w(0 = w(n., Step 2. Ŵ : (0, 0 (n, b, ŵ(k > 0 for k = 1, 2,, n. path Ŵ. (3.1, Ŵ path Ŵ (3.2.
Ŵ, Ŵ : 1., 2., 3.. &'(*(+,-./01234 5 &'(*+,-(.-/,-0,-1,2-3-4567,, ( w(n > w(k for all k = 1,, n 1 path. W W. 3.1. b > 0. (0, 0 (n, b path &'(*+,-./0 ( (3.2 w(n > w(k for all k = 1,, n 1 1 b n n C (n+b/2. &'(*+,-./0 [ ] Step 1. (0, 0 (n, b path (3.2 W : (0, 0 (n, b, w(n > w(k for k = 1, 2,, n 1. paht W (3.1 W
Step 3. Step 1, 2 W : (0, 0 (n, b w (k = w(n w(n 1 + w(k w(k 1 = w(n w(k 1 > 0 (3.2 path = (3.3 path. path 2.2 b n nc (n+b/2. W (3.3 W : (0, 0 (n, b, w(k > 0 for k = 1, 2,, n. Step 2. Ŵ : (0, 0 (n, b, ŵ(k > 0 for k = 1, 2,, n. 3.2., +1, 1 path Ŵ. (3.1, Ŵ path Ŵ (3.2.,, y(k Ŵ (k Ŵ (k 1, k = 1, 2, n n. b 1 2 n b n nc (n+b/2. n, path 1 1/2 n. bw (n = y(n + y(n 1 + + y(1 = bw(n bw(0 = bw(n, bw (k = y(n + y(k = bw(n bw(k 1 ŵ (n ŵ (k = ŵ(n Ŵ (3.2. ( ŵ(n ŵ(k 1 = ŵ(k 1 > 0.
4 4.1 (. A = j, B = k, j > k, X path (0, 0 (1, 1. j + k, A j, B k, X : (0, 0 (1, 1 (j + k, j k,, A B = (1, 1 (j + k, j k path x.., A B = j k j + k., (1, 1 (j + k, j k path x. path (j + k 1! j+k 1C j 1 j+k 1 C j = (j 1! k! (j + k 1! 1 = (j 1! (k 1! k 1 = j = (j + k! j! k! j k j + k j k j k (j + k 1! j! (k 1! (j + k 1! (j 1! (k 1! j k j k = j+kc j j k j + k. [ ], A - B, X., A B, 1 A., (0, 0 (j + k, j k path j+k C j,, A B = j+k C j j k / j+kc j = j k j + k j + k. "&'($&(% &(#!"#$#% # &'( &'(#
5, 2n x path. [ ] (i (5.1 + (5.2 path, (2n 1, 1. (5.1 W : (0, 0 (2n, 0. path 2n, 2n.!%&%'!"#$%&%' (5.1 path 2n C n. 5.1 (5.2 path x path. (5.1 + (5.2 path, W : (0, 0 (2n 1, 1 (5.2., 2 2.2 [ ]. 5.1 5.1. (i (5.1, (5.2 w(1 > 0, w(2 > 0,, w(2n 1 > 0, w(2n = 0 path 1 n 2n 2C n 1. b > 0. (0, 0 (n, b path, w(1 > 0, w(2 > 0,, w(n = b > 0 (b/n n C (n+b/2. (ii (5.1, (5.3 w(1 0, w(2 0,, w(2n 1 0, w(2n = 0 path 1 n + 1 2nC n. (5.1 + (5.2 path = = 1 2n 1 (2n 1! n! (n 1! = 1 n 1 2n 1 2n 1 C n (2n 2! (n 1! (n 1! = 1 n 2n 2 C n 1.
(ii 5.1, (i path, W : (1, 1 (2n 1, 1 (5.3. (, ρ 1, ρ 2,.,, { ρk = 1, 1/2 k ρ k = 1, 1/2 (5.4 (1, 1 (0, 0, (2n 1, 0 (2n, 0, (5.1+(5.3 path. (5.4, n n + 1,. a = 0, w(n = { 0 n = 0 n k=1 ρ k n 1 (5.1 + (5.3 path = 1 n + 1 2n C n. n. W : (0, 0 (n, w(n path 2 n, path 1/2 n. 6,. ρ 1, ρ 2, {1, 1}. a n, { a, n = 0 w(n a + ρ 1 + ρ 2 + + ρ n, n = 1, 2,., 2, ( path W : ( W : (0, 0 (1, w(1 (n, w(n path,. (i n w(n = 0, (ii n 0 w(1 0,, w(n 1 0, w(n = 0, (iii n r > 0 w(1 < r,, w(n 1 < r, w(n = r.,. W : (0, w(0 (1, w(1 (2, w(2 (n, w(n
6.1 6.1. w(0 = 0. u 2n 1 2 2n 2n C n, n = 1, 2,, u 2n. (i P[w(2n = 0], (ii P[w(1 0, w(2 0,, w(2n 0], (iii P[w(1 0, w(2 0,, w(2n 0]. [ ] (i (0, 0 (2n, 0 path 2n C n., 2n path 2 2n, (i. (ii Step 1. 5.1. P[w(0 = 0, w(1 0, w(2 0,, w(2k 1 0, w(2k = 0] = 1 2 (6.1 2 2k k 2k 2 C k 1 = 1 2k u 2k 2., { 2n } = { } { 2 }, { 4 } { 2n } P[w(0 = 0, w(1 0, w(2 0,, w(2n 0] = 1 P[w(1 0, w(2 = 0] P[w(1 0, w(2 0, w(4 = 0] P[w(1 0,, w(2n 0] (6.2 = 1 1 2 u 0 1 4 u 2 1 2n u 2n 2. 5.1 (i W : (0, 0 (2n, 0, w(1 > 0. w(2 > 0,, w(2n 1 > 0, w(2n = 0 path 1 n 2n 2C n 1. w(1 > 0, w(2 > 0,, w(2k 1 > 0, w(2k = 0 path 1 2k 1 2k 1 C k == 1 2k 1 (2k 1! k! (k 1! = 1 k 2k 2 C k 1. w(0 = 0, w(1 0, w(2 0,, w(2k 1 0, w(2k = 0 2 path, 2, k 2k 2 C k 1. Step 2. u 2k 2 u 2k = = 1 2 2(k 1 2k 2 C k 1 1 1 (2k 2! 2 2(k 1 (k 1! (k 1! ( 1 1 4, (6.2, (ii. 2 2k 2k C k (2k 1 2k k k = 1 2k u 2k 2 P[w(0 = 0, w(1 0, w(2 0,, w(2n 0] = 1 1 2 u 0 1 4 u 2 1 2n u 2n 2 = 1 (u 0 u 2 (u 2 u 4 (u 2n 2 u 2n = 1 u 0 + u 2n = u 2n.
(iii Step 1. 5.1. 5.1 (ii W : (0, 0 (2n, 0, w(1 0, w(2 0,, w(2n 1 0, w(2n = 0 path 1 n + 1 2nC n. {w(0 = 0, w(1 1, w(2 1,, w(2k 2 0, w(2k 1 < 0} = {w(0 = 0, w(1 1, w(2 1,, w(2k 2 = 0, w(2k 1 = 1} Step 2.., { 1, 2,, 2n } = { } { 1 }, (6.3 { 3 } { 2n 1 } P[w(0 = 0, w(1 0, w(2 0,, w(2n 0] = 1 P[w(1 = 1] P[w(1 0, w(2 0, w(3 = 1] P[w(1 0,, w(2n 2 0, w(2n 1 = 1] = 1 1 2 u 0 1 4 u 2 1 2n u 2n 2. (6.3 = (6.2 (ii, (iii.!&%&' 6.1!"#$(%$(' 2k 1 path!"#$"%&' P[w(0 = 0, w(1 1,, w(2k 2 0, w(2k 1 < 0] = P[w(0 = 0, w(1 1,, w(2k 2 = 0, w(2k 1 = 1] = 1 1 2 2k 1 k 2k 2 C k 1 = 1 2k u 2k 2. & $( ( RW,,. 6.2. n 1. u 2n 1 2 2n 2n C n,. (6.4 u 2n = [ ] Step 1. (6.1 n k=1 1 2k u 2k 2 u 2n 2, u 0 1. w(0 = 0 RW. P[w(0 = 0, w(1 0, w(2 0,, w(2k 1 0, w(2k = 0] = 1 2 2 2k k 2k 2 C k 1 = 1 2k u 2k 2.
, RW path 0 2T. 2T = 2k path 2 k 2k 2 C k 1 = 2 k u 2k 2. w(2k = 0 w(2n = 0 path 2(n kc n k = u 2n 2k. Step 2. w(0 = 0 RW. 6.1 (i w(0 = 0, P[w(0 = 0, w(2n = 0] = u 2n., RW 2n 0., (6.5 (6.5 w(0 = 0, 2k 0, w(2n = 0 path 2 k 2k 2 C k 1 2(n k C n k = 2 k u 2k 2 2 2k 2 u 2n 2k 2 2n 2k = 22n 2k u 2k 2 u 2n 2k. u 2n = P[w(0 = 0, w(2n = 0] nx = P[w(0 = 0, 2k 0, w(2n = 0] = k=1 nx k=1 2 2n 2k u 2k 2 u 2n 2k 1 2 = X n 1 2n 2k u 2k 2 u 2n 2k. k=1 %&'((*+, 7 path (excursion -./012,.,, $!"!# k { ρk = 1, 1/2 ρ k = 1, 1/2. a = 0, +,-./.012 w(n = { 0 n = 0 n k=1 ρ k n 1 n. $ %&'(*!"!# W : (0, 0 (n, w(n path 2 n, path 1/2 n.
( 3 (7.1 b > 0 path!"#$%&'(*+,-./012345 667+,- 9 (.. 0 1 path,. k 1 k path, k.. : 8! '($%& '$%& "#$%& '*$%& n b path., 3.1[ ] (7.1. b 3.1 b > 0. (0, 0 (n, b path b = w(n > w(k for all k = 1,, n 1 b n n C (n+b/2. 7.2. b., 2n, b b 1 = 2n b 2n b C n 2 2n b.,. 7.1. b., 2n b b path b 2n b 2n b C n, n b > 0,
[ 7.2 Part 1.] b > 0, (7.2, 2n b b path w &,. * +, +& &'( -& * &'( -, (, 7.1 path w * -, -& &'( &'+,( path 1, 1 L 0 b. (i, (ii 1, (iii 2,. (iv b 1 *,&,-,-.+-,&.+& &'( (& &'+&(!!+,-. -. -&!!+,-. * &'( (, -./012,,3* -./+. -&/+& &'+,( * &'(!!+,-.
path w. [ ] 7.1 path w, :! ""#$%& ""#$%& ""#$%& '(! 1 T 1, 2 T 2,. b 1 T b 1 w. w!!+,-.!!+,-. (7.3 (7.4 w (0 = 0, w (2n = 0, w (k 0 for k = 0, 1,, 2n, k = 0, 1,, 2n, w (k = 0 k b + 1. * /0 /&!!+,-. &'( 7.3. (7.3, (7.4,. 1, 1 L 0. path w, path w. 0. (i L 0 w (1 = 1, (ii w, 1 T 1. (i L 0 1, w T 1 T 1 + 1. w (T 1 + 1 = 0. T 1 + 1 w 1 0 =. ""#$%& *$ +* '$ +'!! '( ""#$%& ""#$%&
! $$%&'( *,-./ *,01/ $$%&'( $$%&'( *& +* "& +" 2",-./ 2",01/ 23,-./ 23,01/ (iii w T 1 L 0 w(t 1 + 2 = 1, (iv w, 2 T 2. (i, (ii L 0 2, w T 2 T 2 + 2. w (T 2 + 2 = 0. T 2 + 2 w 2 0 =2. "#! [ 7.2, Part 2] (7.3, (7.4 2n b 0 path w (7.3 w (0 = 0, w (2n = 0, w (k 0 for k = 0, 1,, 2n, (7.4 k = 0, 1,, 2n, w (k = 0 k b + 1. path w. (7.5 w, (7.2, 2n b b path ŵ. (v w T 1 L 0 w(t 1 + 2 = 1,. (vi w, b 1 T b 1. (i,, L 0 b 1, w T b 1 T b 1 + b 1. w (T b 1 + b 1 = 0. T b 1 + b 1 w b 1 0 =b 1. (vii w T b 1 L 0 w(t b 1 + b = 1, (viii w, 2n b b. (i,, (vii L 0 b, w 2n b 2n. w (2n = 0. 2n w b 0 =b. [ 7.2, Part 3] Step 1., Part 1 Part 2, 2n b 0 path w (7.3 w(0 = 0, w(2n = 0, w(k 0 for k = 0, 1,, 2n, (7.4 k = 0, 1,, 2n, w(k = 0 k b + 1. 2n b b path (7.2, 2n b b path w 1 : 1.
7.1 Step 2. = = b 2n b 2n b C n. (7.3, (7.4 path w. w, 2n path 2 2n,, 2n, b b 2n b 2n b C n 2 b 1 2 2n = b 1 2n b 2n b C n 2 2n b. 0 0 }{{} 2 0 }{{} 2 b 1 0 b 0 }{{} b b. (7.3 path w x. x, 2 b path. "#$%& '($%& '$%& '*$%& 8 Arc sine - k 1 k path w x, RW 1.!, 7.2 path w, {w(k 1 0, w(k > 0} {w(k 1 > 0, w(k 0},. RW., 2n, b path, b 2n b 2n b C n 2 b.
Arc sin 0.035 8.1. path w, 0.15 0.10 0.030 0.025 0.020 0.015 R(2k, 2n RW 2k, 0.05 0.010 0.005 2n 2k 2 4 6 8 10 10 20 30 40 50. R(2k, 2n = 1 2 2k 2kC k 1 2 2n 2k 2n 2kC n k. 0.15 0.10 0.05 8.2. RW n, n = R(n, 2n 1 2. 8.1 2 4 6 8 10 n = 10, n = 54. n = 10 2k 0 2 4 6 8 10 20 18 16 14 12 R(2k, 20 0.176 0.092 0.073 0.065 0.061 0.06, (= : n = 10 R(0, 20 / R(10, 20 3. n., n = 54 R(0, 54 + + R(4, 54. R(25, 54 + + R(29, 54 3.5 (, n = 49. 81% 80% 61% 60% 40% 19% 39% 20% 29% 14% 13% 0.25 0.20 0.15 0.10 0.05 0.00 1 2 3 4 5 8.2 81%, 80-61%, 60-40%, 39-20%, 19%
[ 8.1 ] u 2n 1 2 n 2n C n.. Step 1. k = 0, 2n. 6 6.1(iii w(0 = 0. $ %&'(* +*,-!.-/0!"!# P[w(1 0, w(2 0,, w(2n 0] = u 2n, RW 0 2n u 2n., R(2n, 2n = u 2n. RW, R(0, 2n = u 2n. Step 2., 1 k n 1. 2k > 0, 2(n k > 0 RW. RW x, 2T RW x Step 3. (8.1 path. 5 5.1 (i W : (0, 0 (2n, 0, (8.3 w(1 > 0, w(2 > 0,, w(2n 1 > 0, w(2n = 0 path 1 n 2n 2C n 1., 2k, 2(n k RW : w(k > 0 for k = 0, 1,, 2T (8.1 2T 2n, 2k 2T. w(k < 0 for k = 0, 1,, 2T (8.2 2T 2n, 2k. 0 = w(0 w(1 > 0 w(2t 1 > 0 w(2t = 0 path 1 T 2T 2 C T 1. 1 T k, (8.1 path = 1 T 2T 2 C T 1 2 2(n T R(2(k T, 2(n T. {z } R, 1 T n k, $ %&'(* (*+,!-.!",,/0!"!# (8.2 path = 1 T 2T 2 C T 1 2 2(n T R(2k, 2(n T. {z } R.
6.2 (8.4 2 n R(2k, 2n = (8.1 + (8.2 kx 1 = T 2T 2 C T 1 2 2(n T R(2(k T, 2(n T T =1 n k X 1 + T 2T 2 C T 1 2 2(n T R(2k, 2(n T. T =1 n 1. u 2n 1 2 2n 2n C n,.. u 2n = nx k=1 1 2k u 2k 2 u 2n 2, u 0 1. Step 4. (8.5 m = 1, 2, n 1,, R(2k, 2m = u 2k u 2(m k,,. R(2k, 2n = u 2k u 2(n k, k = 0,, m k = 0,, n kx 2 n R(2k, 2n = 2 2n 1 1 2T u 2T 2 u 2k 2T u 2n 2k T =1 {z } u 2k n k X +2 2n 1 1 2T u 2T 2 u 2(n k 2T u 2k T =1 {z } u 2(n k = 2 2n 1 u 2k u 2n 2k + u 2n 2k u 2k = 2 n u 2k u 2n 2k. k = 0, 2n Step 1. 1 k 2n 1. (8.4 2 n R(2k, 2n = kx 1 = T T =1 2T 2C T 1 {z } 2 2T 2 u 2T 2 2 2(n T R`2(k T, 2(n T {z } (8.5 u 2k 2T u 2n 2k 9 (i,. (ii,,. n k X 1 + T T =1 2T 2C T 1 2 2(n T R`2k, 2(n T {z } {z } 2 2T 2 (8.5 u 2T 2 u 2k u 2n 2k 2T = kx T =1 1 T 22n 2 u 2T 2 u 2k 2T u 2n 2k n k X 1 + T 22n 2 u 2T 2 u 2k u 2n 2k 2T. T =1 6
9.1 U : (i 1, +1, 1. (ii, U p (0 < p < 1. (iii a, 0 (=. (. (i U n, 1 n a 1. { n + 1 (ii 1. U n 1. (iii, U 2 n + 1 n 1 (=.! ' #$%& # #$% #&% (!" " (,. -. 9.1. U n (0 < n < a. U,. a Q(n (i 2 p n + 1, q = 1 p n 1. (ii a (=.. (iii, 0 (=,.. 9.1, (9.1. (9.1 Q(n = p Q(n + 1 + q Q(n 1, Q(a = 0, Q(0 = 1, p + q = 1.
9.2 Case 1. p = 1/2. p = 1/2 = q = 1 p (9.1 Q(n = 1 2 Q(n + 1 + 1 Q(n 1 2 Q(n + 1 Q(n = Q(n Q(n 1 Q(n + 1 Q(n = = Q(1 Q(0 = Q(1 1. Q(n = ( ( Q(n Q(n 1 + Q(1 Q(0 + Q(0 = (Q(1 1n + Q(0 = (Q(1 1n + 1. (9.1 2 Q(n Q(n 1 = γ n 1{ Q(1 Q(0 } = γ n 1{ Q(1 1 }. Q(n = ( ( Q(n Q(n 1 + Q(1 Q(0 + Q(0 = { Q(1 1 }{ γ n 1 + γ n 2 + + γ + 1 } + 1 = { Q(1 1 } 1 γ n 1 γ + 1 = 1 { { γ + Q(1 + 1 Q(1} γ n }. 1 γ Q(1. 0 = Q(a = 1 + (Q(1 1a Q(1 1 = 1/a. 1 Step 2. (9.1 2, (9.2 p = 1 2 Q(n = 1 n, n = 0, 1,, a. a 0 = Q(a = 1 1 γ { γ + Q(1 + ( 1 Q(1 γ }. Q(1 = γa γ γ a 1. Case 2. Step 1. p 1/2 p + q = 1, (9.1 Step 1 2 (p + q Q(n = Q(n = p Q(n + 1 + q Q(n 1 p { Q(n + 1 Q(n } = q { Q(n Q(n 1 }. γ q/p, : { Q(n + 1 Q(n } = γ { Q(n Q(n 1 }. (9.3 p 1 2 Q(n =, n = 0, 1,, a. γa γ a 1 γn γ a 1 = γa γ n γ a 1.
9.3, : 9.2. U a, n (0 n a, 1 1. Q(n 1 n a, p = 1/2, Q(n = (1/p 1 n (1/p 1 a 1 (1/p 1 a, p 1/2. a = 100 a = 1000 p 0.49 0.495 0.497 0.498 0.5 Q(50 0.881 0.731 0.646 0.599 0.5 1 p 0.49 0.499 0.4995 0.4997 0.5 Q(500 1 0.881 0.731 0.646 0.5 2 ( 9.2 Q(n p, a = 100, n = 50, a = 1000, n = 500, Q(n. 1 2, : 1.0 0.8 0.6 0.4 0.2 0.475 0.480 0.485 0.490 0.495 0.500 9.1 a = 100 (, a = 1000 (
(i 1 (a = 100, n = 50, p = 0.5 Q(50 = 0.5,, p = 0.48 2% Q(50 = 0.982. (ii 2 (a = 1000, n = 500, 1 p. Q n 0.8 0.6 0.4 0.2 p = 0.5 Q(50 = 0.5, p = 0.499 0.1% Q(500 = 0.881. 9.2 10 20 30 40 p = 0.51( p = 0.52( n, p > 0.5, (. a = 100, p = 0.52, n Q(n n 10 15 20 25 30 a = 100, p = 0.51 Q(n 0.664 0.540 0.439 0.356 0.238 a = 100, p = 0.52 Q(n 0.449 0.301 0.201 0.135 0.09 3, 2 % (p = 0.52, 1/10, 50%., : 9.3. (i Q(n, p a (ii,,. a, p. ( Q(n.
9.4 9.2, 9.3. 9.5. Q(n Q (n. 9.1 1,. [ ] Q (9.4 : (1/p 1 2n (1/p 1 2a = ˆ(1/p 1 n (1/p 1 a ˆ(1/p 1 n + (1/p 1 a 9.4. 9.1, 1/2, 1/2 Q (9.4 : 1 (1/p 1 2a = ˆ1 (1/p 1 a ˆ1 + (1/p 1 a, U Q (n. [ ] Step1. [ 9.4 ],,. Q (n = [ (1/p 1 n (1/p 1 a][ (1/p 1 n + (1/p 1 a] [ 1 (1/p 1 a ][ 1 + (1/p 1 a] = Q(n (1/p 1n + (1/p 1 a 1 + (1/p 1 a. Step 2., 9.2 : n 2n, a 2a., Q (n., p 1/2 (9.4 Q (n = (1/p 12n (1/p 1 2a 1 (1/p 1 2a. p < 1 2 1 p 1 > 1 (1/p 1n + (1/p 1 a 1 + (1/p 1 a > 1. Q(n Q (n 9.6. 0 < p < 1 2 Q (n > Q(n, 1 2 < p < 1 Q (n < Q(n.
9.5 1/3, 1/4,. Q(n n, a,. 9.7. (i p 1/2,. (ii p 1/2,. 10 -.,. 10.1 (i 1, +1, 1. (ii, U p (0 < p < 1. (iii a, 0 (=. (, (. n, a. D(n * 1 + 9.8.,. 10.1. D(n.,,. Step 1. (i n + 1, (ii, n 1, (iii, 1, : *1..
(10.1 D(n = p D(n + 1 + (1 p D(n 1 + 1 : for 1 n a 1, D(0 = 0 = D(a. Step 2. 2, (. (10.1 1 ((10.1 2 D(n = p D(n + 1 + (1 p D(n 1, Q(n (9.1. A + B( 1 p p n, p 1 2, D(n = A + B n, p = 1 2 p 1 p 10.2. n, a., D(n, (10.2 D(n = 1 1 2p (n a 1 ( (1 p/p n 1 ( (1 p/p a p 1/2, n (a n, p = 1/2.,, {b n } b n = n 1 2p, p 1 2, n 2, p = 1 2. 10.3. U n = 1, a = 1000. p, : p 0.49 0.499 0.4999 0.49999 0.5 D(1 50 462.6 932.7 992.4 999 (10.1 (. Step 3. D(n = A + B( 1 p p n + n 1 2p, p 1 2, A + B n n 2, p = 1 2, 600 500 400 300 200 100 0.492 0.494 0.496 0.498 0.500 (10.1, A, B.
U Q(n 1 n a, p = 1/2, Q(n = (1/p 1 n (1/p 1 a 1 (1/p 1 a, p 1/2. U Q(1 p 0.49 0.499 0.4999 0.49999 0.5 Q(1 1 0.99993 0.9992 0.99902 0.999.. 10.2 a. S k, k = 0, 1,, 0 a.. : u(z, n z, n 0,, 0 z a. 10.4. : I. u(z, n : 1 z + 1 z 1 p = 0.49 Q(1 1 D(1 = 50, p = 0.5 Q(1 1 D(1 = 999 (!. (10.3 u(z, n = p u(z + 1, n 1 + q u(z 1, n 1, 2 z a 2, n 2.. ( =06/23(, 13:20. A4,.,. (10.4 u(0, n = 0 = u(a, n, n 1 u(0, 0 = 1; u(z, 0 = 0, 1 z a. II. (10.3+ (10.4, generating function : (10.5 U(z, s s n u(z, n, s 1. n=0 ( u(z, n
(i U(z, 0 = u(z, 0. (ii U(z, s s U (z, s = n=1 n sn 1 u(z, n. U (z, 0 = u(z, 1. IV. (10.7. Step 1. s < 1. x = p s x 2 + q s ( (10.7 (iii U(z, s s 2 U (z, s = n=2 n(n 1 sn 2 u(z, n. U (z, 0 = 2! u(z, 2. (iv U(z, s s k 1 (n k s n k u(z, n. U (k (z, 0 = k! u(z, k. U (k (z, s = n=k n(n, U(z, s, u(z, k. (10.7 λ 1 1 + 1 4p q s, 2 p s λ 1 1 + 1 4p q s. 2 p s U(z, s = A(s (λ 1 (s z + B(s (λ 2 (s z. III. u(z, n (2.3 u(z, n = p u(z + 1, n 1 + q u(z 1, n 1. (10.3 sn. (10.5 n=1 (10.4 = U(z, s = p s U(z + 1, s + q s U(z 1, s. (10.6 U(0, s = 1, U(a, s = 0. (10.7 U(z, s = p s U(z + 1, s + q s U(z 1, s. 2 Step 2. (10.6, A(s, B(s : A(s = B(s = (λ 2 (s a (λ 1 (s a (λ 2 (s a, (λ 1 (s a (λ 1 (s a (λ 2 (s a. λ 1 (s λ 2 (s = qs/ps = q/p u(z, n, n 0 U(z, s 10.5. U(z, s (10.8 U(z, s = (λ 1(s a (λ 2 (s z (λ 2 (s a (λ 1 (s z (λ 1 (s a (λ 2 (s a = ( q z (λ 1 (s a z (λ 2 (s a z p (λ 1 (s a (λ 2 (s a. q = 1 p.
V. a. v(z, n z n a,, 0 z a., v(z, n V (z, s. s n v(z, n n=0 10.6. V (z, s. [ ] U(z, s., p q, q p, z a z, a z z µ 1 1 + 1 4p q s, 2 q s µ 1 1 + 1 4p q s. 2 q s V (z, s = ( p z a (µ 1 (s z (µ 2 (s z q (µ 1 (s a (µ 2 (s a.