§1.5 ~�5 23

§1.5 ~�5311.3 !¥, ·�?Øli Ñu�ê¼ó´Äk�U(=, ±�VÇ)�¯G�j. �!·�òïÄ�¯K´: ê¼ó´Ä�½(=, VÇǑ1)�¯G�i? 3�^$Ä;�þ, �¯G�i o�õ�g? ¤k�¯i ��mk�o5�?e¡, �½G�i. ·��Äli Ñu�âf�£G�i �¤k�m. ÷^(1.12)½Â�ÎÒ, =Pτi = inf{n ≥ 0 : Xn = i}, σi = inf{n ≥ 1 : Xn = i}.-Ti,1 := σi ǑÄg£�(=, �£G�i)��m. ér ≥ 2, 8B½Â

Ti,r :=

inf{n ≥ Ti,r−1 + 1 : Xn = i}, e Ti,r−1 <∞;

∞, e Ti,r−1 =∞.§L«ê¼ó1r g£���m. �*þ, �âfu,��Ǒ�£G�i. �o, �âê¼5,âf��Y$ÄE,´��liÑu�ê¼ó,Ïd,§2g�£G�i¤I��m�Ti,2−Ti,1Ò´σi ���Õá�E�. -Ti,0 := 0,

σi,r :=

Ti,r − Ti,r−1 e Ti,r−1 <∞;

∞, e Ti,r−1 =∞.·K1.5.1. é?¿r ≥ 1, n ≥ 1,

Pi(σ1,1 = m1, σi,2 = m2, · · · , σi,r = mr) =r∏

s=1

Pi(σi = ms). (1.22)?�Ú, ePi(σi <∞) = 1, Kσi,1, σi,2, σi,3, · · · ÕáÓ©Ù.y: 5¿�Pi(σi,1 = m1, σi,2 = m2, · · · , σi,r = mr) = Pi(σi,1 = m1)P (σi,2 = m2, · · · , σi,r =

mr|σi,1 = m1). 3¯�{σi < ∞} þ, -Yn = Xσi+n, n ≥ 0, u´σ(Y )i,r = σ

(X)i,r+1 = σi,r+1,

r ≥ 1. �ârê¼5(íØ 1.4.4), 3σi,1 = m1 þ, {Yn} ´li Ñu�ê¼ó, ÏdP (σi,2 =

m2, · · · , σi,r = mr|σ1,1 = m1) = P (σ(Y )i,1 = m2, · · · , σ(Y )

i,r−1 = mr|σi,1 = m1) = Pi(σi,1 =

m2, · · · , σi,r = mr). Ïd, Pi(σi,1 = m1, σi,2 = m2, · · · , σi,r = mr) = Pi(σi = m1)Pi(σi,1 =

m2, · · · , σi,r = mr). ù(Ü8B{L²(ؤá.,, ·�Ù¢�±Ø^rê¼5, ��y² (1.22). ·��Ñr = 2 �y², ÏǑr ≥ 3�y²aq, �´�¡�. b�σi,1 = m, σi,2 = n, =, âf3m �ǑÄg�£G�i, ��q²{�m�n 1�g�£i. ù�duâf30,m,m+ n ùn��Ǒ�¯i, ¿�3�mã[0,m+n]¥�Ù§�Ǒ�¯Ù§G�. u´,·��±òù�¯�^ m+nÚ;��Ñ.

24 1�Ù ê¼óòm ÀǑy3, Kù�¯��±�Ǒ��L�9y3�¯�A Ú��ò5�¯�B ��8,Ù¥

A = {X0 = i;Xr 6= i, ∀1 ≤ r ≤ m− 1;Xm = i} = {σi = m},B = {Xm+r 6= i, 1 ≤ r ≤ n− 1;Xm+n = i}.u´Pi(σi,1 = m,σi,2 = n) = Pi(A ∩B) = Pi(A)Pi(B|A).dê¼5

Pi(B|A) = P (Xm+r 6= i, 1 ≤ r ≤ m+ n− 1;Xn+m = i|Xm = i)

= P (Xr 6= i, 1 ≤ r ≤ n− 1;Xn = i|X0 = i) = Pi(σi = n),Ù¥, 1���Ò´ÏǑ·K 1.1.4. u´Pi(σi,1 = m,σi,2 = n) = Pi(σi = m)Pi(σi = n).ù=´ (1.22) �r = 2 ��/.?�Ú, ePi(σi < ∞) = 1, Kσi,1, σi,2, · · · Ñ´�ÅCþ. u´, (1.22) L²§�ÕáÓ©Ù. 2P

ρij = Pi(σj <∞). (1.23)3(1.22) ¥, ém1, · · · ,mr 3{1, 2, · · · } þ�Ú=�XeíØ. §Ì��éPi(σi <∞) < 1 ��/, ÄK�ª´²��, ÏǑÙü>Ñ´1.íØ1.5.2. Pi(Ti,r <∞) = ρrii.e¡·��±ïÄê¼ó÷Ù;��¯,G��ogê. E,b�ê¼óli Ñu,k�ħ�¯i �ogê. -Vi =

∞∑

n=0

1{Xn=i} = #{n ≥ 0 : Xn = i},§L«ê¼ó÷Ù;��¯i �ogê. k�,ǑrN¤�9�ê¼ó´{Xn},·�ǑòViPǑV (X)i . eê¼óli Ñu, KVi − 1 Ò´Tê¼ó£�i �ogê. �*þ, eρii = 1, KâfÑi�7,�£i, u´§ò� 2!2�n��£i, ÏdVi = ∞; eρii < 1, KâfzgÑiѱ�½��VÇ1− ρii Ø�£i,u´ªk�gÑi�§Ø�£,ÏdVi <∞. Ǒd,·����Ý1Á�,�¡L«£�¤õ,�¡L«£��}. �ÕáÝ1Á�ØÓ�´,3·��Á�¥,�kÝ��¡â�±?1e�gÝ1(ÏǑ�k£�¤õ,!9e�g£��

§1.5 ~�5 25�mâk¿Â); ��Ý��¡, Á�(å. 3TÁ�¥, ·�'%�£�gêÒ´Ý��¡� �¡Ñy�gê,ù�ÕáÝ1Á��(J´���. ùL², Vi ÑlëêǑρii �AÛ©Ù.T(Ø�î�y²Xe: ��£�r g(=, ���¯r+1 g)�du1r g£��mk�, l Pi(Vi ≥ r + 1) = Pi(Ti,r <∞) = ρrii, ∀ r ≥ 0.þªL²Vi ÑlëêǑρii �AÛ©Ù. �âd(Ø, ·�B��: �ëêρii = 1 �, Pi(Vi =

∞) = 1 �EiVi = ∞; �ëêρii < 1 �, Pi(Vi < ∞) = 1 �EiVi =1

1−ρii< ∞. u´, Xe��J�{K¤á:

Pi(Vi <∞) = 1⇐⇒ EiVi <∞⇐⇒ ρii < 1,

Pi(Vi <∞) = 0⇐⇒ EiVi =∞⇐⇒ ρii = 1.(1.24)d�J�{KL², “�¡g£�i”ù�¯�k0-1 Æ, =Pi(Vi = ∞) = 0 ½1. edVÇǑ1,·��Ǒli Ñu�ê¼óǑ¬²~�£i, Ïd¡i Ǒ��~��.½Â1.5.3. ePi(Vi =∞) = 1, K¡G�i ´��~��, ½ö`i ´~��(recurrent). ÄK¡i ´��6�, ½¡i ´�~��(transient).d(1.24), ·��±©OÏLρii ½EiVi �OG�i �~�5, ¤^�óä©OǑáÂVÇÚ��¼ê. 30�ùü«óä� , ·�ky²~�5´pÏa�5�. Ǒd, ·�©Û�el,~��i Ñu�;�(�. 5¿�d�Pi(σi < ∞) = 1, ¿�P (Ti,r < ∞) = 1,

∀r ≥ 1. �â·K 1.5.1, σi,1, σi,2, · · · ÕáÓ©Ù. ¯¢þ, ·���±y²�r�(Ø.½Â1.5.4. b�i ~�. PEi = {~i = (i0, i1, · · · , in) : n ≥ 1; i0 = in = i; i1, · · · , in−1 6= i}.b�~Y = (Y0, · · · , Yσ)´����uEi ��Å�k���þ, Ù©Ù�Xe: ∀(i0, i1, · · · , in) ∈

Ei,

P(

~Y = (i0, i1, · · · , in))

= pi0i1 · · · pin−1in .�o, ·�¡~Y Ǒi ���i1.{ü/`, i1��´��li Ñu�ê¼ó, Äg£�i � ²{�(k���);�.·K1.5.5. b�i ~�, X0 = i. -~X(r) := (XTi,r−1 , · · · , XTi,r

), ∀ r ≥ 1.�o, ~X(1), ~X(2), · · · Ñ´i �i1, �§�ÕáÓ©Ù.

26 1�Ù ê¼óy: �â½Â���y, ·�� ~X(1) ´��i1. e¡y²ÕáÓ©Ù5�. PYn =

Xσi+n, n ≥ 0. drê¼5(íØ 1.4.4), {Yn} Ǒ´li Ñu�±P Ǒ=£Ý�ê¼ó,�{Yn} � ~X(1) �pÕá. l , ·��±��e¡�ü^(Ø:(Ø (a): ( ~X(2), ~X(3), · · · ) �( ~X(1), ~X(2), · · · ) Ó©Ù,(Ø (b): ( ~X(2), ~X(3), · · · ) � ~X(1) �pÕá.�â(Ø(a), ( ~X(r+1), ~X(r+2), · · · )�( ~X(r), ~X(r+1), · · · )Ó©Ù,ÏǑ§�©O´(Ø(a)¥J��ü�S��l1r �m©��YS�. d8B{, ·���(Ø (c): ( ~X(r), ~X(r+1), · · · ) �( ~X(1), ~X(2), · · · ) Ó©Ù.l ~X(1), ~X(2), · · · Ó©Ù.5¿�(Ø(b)£ã�5�´S��1���¤k�Y��pÕá. d(Ø(c), é?¿r ≥ 1, ( ~X(r), ~X(r+1), · · · )Ñ÷vT5�,l ~X(r)�( ~X(r+1), ~X(r+2), · · · )�pÕá, ∀r ≥ 1.ù(Ü8B{L² ~X(1), · · · , ~X(r) �pÕá, ∀r ≥ 1. ù=´`, ~X(1), ~X(2), · · · �pÕá. 2·K1.5.6. b�i ~��i→ j, Kj ~��Pi(Vj =∞) = Pj(Vi =∞) = 1.y: Ø��j 6= i. ·�Äky²Pi(Vj = ∞) = 1, ,�y²Pj(Vi = ∞) = 1, ��y²j~�.�Äli Ñu�;�. �į�Ar =“G�j Ñy31r �i1¥”, =Ar := {∃ n ∈[Ti,r−1, Ti,r] �� Xn = j}. d·K 1.5.5, A1, A2, · · · �pÕá, ¯¢þ, 1A1 , 1A2 , · · · ÕáÓ©Ù.Pp = P (A1). 5¿�i→ jL²j±�VÇÑy3liÑu�;�¥,ùL²p > 0. äN/,·��^�y{?1y²: XJp = 0,�o{τj <∞} = {∃ n ≥ 0�� Xn = j} = ∪∞r=1ArL²Pi(τj <∞) ≤∑∞

r=1 Pi(Ar) = 0, ù�i→ j gñ! dup > 0, dr�ê½Æ, G�j ò¬Ñy3�¡õ�i1¥, l Ñy�¡õg. äN/, Pi(limn→∞ 1n

∑nr=1 1Ar

= p) = 1. ù(ÜVj ≥∑∞r=1 1Ar

L²Pi(Vj =∞) = 1.-Yn = Xτj+n, n ≥ 0. duPi(τj < ∞) ≥ Pi(Vj = ∞) = 1, Vi = ∞ ��=�V (Y )i = ∞.�ârê¼5(·K 1.4.2), Pi(V

(Y )i = ∞|τj < ∞) = Pj(Vi = ∞). 2di ~��, 1 = Pi(Vi =

∞) = Pi(τj <∞)Pi(V(Y )i =∞|τj <∞) = Pi(τj <∞)Pj(Vi =∞), l Pj(Vi =∞) = 1.Ón, Vj = ∞ ��=�V (Y )

j = ∞. ÏdPi(Vj = ∞) = Pi(τj < ∞)Pi(V(Y )j = ∞|τj <

∞) = Pi(τj <∞)Pj(Vj =∞). d®y(ØPi(Vj =∞) = 1 �Pj(Vj =∞) = 1, =j ~�. 2þ¡�·KL²~�´pÏa�5�. e,pÏa¥�¤kG�Ñ~�, K·�¡�Ǒ~�a, ÄK¡�Ǒ�~�a. ?�Ú, ~�a�½´48. �é{`, e��pÏaØ´48, �o§�½´�~�a. Ïd, ·��I��Ä4�pÏa. d�, ·��±òê¼ó��3ù�4�pÏaþ,����Ø��ê¼ó. Ïd, 3?Ø~�5´,·�Ø�b�ê¼óØ��. d�¤kG�½öÑ~�,½öÑ�~�,·�Ǒ�¡Tê¼ó´~��,½�~

§1.5 ~�5 27��.

1. áÂVÇ�O{.b�ê¼óØ��. �½��G�, PǑo. Äk, ·��Äli Ñu�ê¼óU��G�o�VÇPi(τo <∞). X ¤ã,ei = o,Kτo = 0,l Pi(τo <∞) = 1. ei 6= o,Kτo = σo,d�·�4ê¼ókr�Ú, �âX1 �G�$^�Vúª, ��Pi(τo <∞) = Pi(σo <∞) =

∑

j∈S

Pi(X1 = j)Pi(σo <∞|X1 = j).,�, ·��ÄYn = X1+n, ∀n ≥ 0. �âê¼5(·K 1.1.4), 3X1 = j �^�e, {Yn} ´lj Ñu�ê¼ó, σo = σ(X)o = inf{n : n ≥ 1, Xn = o} = inf{1 + n : 1 + n ≥ 1, X1+n =

o} = 1 +∞{n ≥ 0 : Yn = o} = 1 + τ(Y )o , Ù¥τ (Y )

o = inf{n ≥ 0 : Yn = o}, l σo <∞ ��=�τ (Y )o <∞. u´

Pi(σo <∞|X1 = j) = Pj(τo <∞),Ù¥, þª�>�Ä�ê¼ó´{Xn}, m>�Ä�´{Yn}.e¡, òxi = Pi(τo <∞), ∀i ∈ S ÀǑ�|��ê, K§�÷ve¡��§|:

xi =∑

j∈S

pijxj , ∀ i 6= o; xo = 1, (1.25)Ù¥, xo = 1 Ǒ�¡Ǒ>.^�.¯¢þ, é?¿S �f8A, ·���A ´�¡áÂ�«�, ǑÒ´`, âf����AÒ�áÂ. PτA = inf{n ≥ 0 : Xn ∈ A} ´âfÄg��A ��Ǒ, �oxi := Pi(τA < ∞) ÒL«li Ñu�âf�ª�A áÂ�VÇ, §�¡ǑáÂVÇ. u´, é�½�f8A, ù�|áÂVÇ÷vXe�§|:

xi =∑

j∈S

pijxj , ∀ i /∈ A; xi = 1, ∀i ∈ A. (1.26)¯¢þ, ·�kXe(Ø.·K1.5.7. áÂVÇ´�§(1.26) ����K), =, e{xi, i ∈ S} Ǒ´þã�§��K),�o, xi ≥ xi, ∀i.y: b�{xi, i ∈ S} Ǒ´þã�§��K), �o, �â>.^�, é?¿i ∈ A, xi =

xi = 1 l xi ≥ xi ¤á; �i /∈ A �,

xi =∑

j∈S

pij xj =∑

j∈A

pij xj +∑

j /∈A

pij xj .

28 1�Ù ê¼ó3m>1��¥�\>.^�xj = 1, ∀j ∈ A, ¿é1��|^�§|?1S�?n, ·�kxi =

∑

j∈A

pij +∑

j /∈A

pij

(

∑

k∈S

pjkxk

)

.UYòk ∈ S ©¤k ∈ A Úk /∈ A, ¿é ö�\>.^�, é�ö?1S�, · · · · · · . �ª, ·�íÑ, é?¿n ≥ 1,

xi =∑

i1∈A

pii1 +∑

i1 /∈A,i2∈A

pii1pi1i2 + · · ·+∑

i1,··· ,in−1 /∈A,in∈A

pii1pi1i2 · · · pin−1in

+∑

i1,··· ,in /∈A

pii1pi1i2 · · · pin−1in xin .�âb�xin ≥ 0, �þª�����K. 5¿�∑

i1∈A

pii1 = Pi(X1 ∈ A) = Pi(τA = 1),

∑

i1 /∈A,i2∈A

pii1pi1i2 = Pi(X1 /∈ A,X2 ∈ A) = Pi(τA = 2),

· · · · · ·∑

i1,··· ,in−1 /∈A,in∈A

pii1pi1i2 · · · pin−1in = Pi(X1, · · · , Xn−1 /∈ A,Xn ∈ A) = Pi(τA = n).Ïd,

xi ≥ Pi(τA = 1) + Pi(τA = 2) + · · ·+ Pi(τA = n) = Pi(τA ≤ n)é?¿n ¤á. ��, -n→∞ �xi ≥ Pi(τA <∞) = xi, ∀i /∈ A. 2aq/, ·��±�ÄÄ�G�o �²þ�mEiτo. ei = o, KEiτo = 0; ei 6= o, K�ÄYn = X1+n, ∀n. 5¿�σo = σ(X)o = 1 + τ

(Y )o . u´,

Eiτo = Eiσo =∑

j∈S

pijEi(σo|X1 = j) =∑

j∈S

pij(1 + Ejτo),Ù¥, �����Ò��>�Ä�ê¼ó´{Xn}, Ùm>�Ä�´{Yn}. u´, òyi :=

Eiτo, ∀i ∈ S ÀǑ�|��ê, K§÷ve¡��§|:

yi = 1 +∑

j∈S

pijyj , ∀ i 6= o; yo = 0. (1.27)¯¢þ, yi, i ∈ S ´(1.27) ����K), y²�·K 1.5.7�aq, 3ǑSK.e¡·�?ØXÛ^áÂVÇ�O~�5. X ¤ã, ·�Ø�b�ê¼óØ��. �½G�o. d (1.24), o ~��duρoo = 1. beo ~�, �o1 = ρoo = Po(σo <∞) =

∑

i∈S

poixi, (1.28)

§1.5 ~�5 29Ù¥xi ´áÂVÇPi(τo < ∞). u´, éulo Ñu�ÚU���G�i (=, poi > 0), Ñkxi ≡ 1. ?�Ú,5¿�xi÷v (1.25),ddaí,liÑu�ÚU���G�j Ǒ÷vxj = 1.ǑÒ´`,¤kloÑuüÚU���G�j Ñ÷vxj = 1. d8B{,¤ko���G�iÑ÷vxi = 1. dê¼óØ��, ù=´`, é¤kG�i Ñkxi = 1. �L5, e¤kG�ÑiÑ÷vxi = 1,Kd (1.28), o~�. duxi ´�§ (1.25)����K),Ïd§�ðǑ1�duT�§vkÙ§([0,1]¥�)), ǑÒ´`, o ~���=��§(1.25) (3[0,1]¥)=kðǑ1�).~1.5.8. (~ 1.1.8Ú~ 1.2.3 Y) )«ó�~�5.b�{Xn} ǑØ��)«ó, Ñ)VÇǑbi, k�VÇǑdi. Pxi = Pi(τ0 < ∞), K�â(1.25), xi = bixi+1 + dixi−1, ∀i ≥ 1, l xi − xi+1 = (xi−1 − xi)

dibi

= · · · = (1 − x1)d1 · · · dib1 · · · bi

.�ªü>éi �Ú�1− xi+1 = Ri(1− x1),Ù¥Ri = 1 +

∑ik=1

d1···dk

b1···bk . -R = 1 +∑∞

k=1d1···dk

b1···bk . eR < ∞, �x1 = (R − 1)/R ∈ (0, 1),K0 < Ri(1− x1) = Ri/R < 1,u´xi+1 := 1−Ri/RÒ´��ØðǑ1�),l Tê¼ó�~�. eR =∞, K1− x1 = (1− xi+1)/Ri ≤ 1/Ri → 0.u´x1 = 1, l xi ðǑ1, ê¼ó~�. Ïd, Tê¼ó~���=�R =∞. 2~1.5.9. (~ 1.1.16 Ú~1.2.5 Y) 5KäTd þ�λ-biased �ÅiÄ.b�{Xn} ǑTd þ�λ-biased �ÅiÄ, Yn = |Xn|, K{Yn} ´)«ó, ÙÑ)ÇǑbi =

d/(λ + d), Ïddi/bi = λ/d. Xn ���:��=�Yn ��0. Ïd, {Xn} �~�5=zǑ{Yn} �~�5. �â~ 1.5.8, R = 1+∑∞

k=1(λ/d)k, ÏdR <∞ ��=�λ < d. =, {Xn}~���=�λ ≥ d. 2~1.5.10. �Ä��u{0, 1, 2, · · · } �ê¼ó. �½c ≥ 0, α ≥ 0, �p(0, 1) = 1; é?¿i ≥ 1,

pi,i+1 =1

2, pi,i−1 =

exp(−ci−α)

2, pii =

1− exp(−ci−α)

2.ù�ê¼ó´Ø���, Ù~�5�duxi := Pi(τ0 <∞) ´Ä�Ǒ1. é?¿i ≥ 1,

xi = pi,i+1xi+1 + pi,i−1xi−1 + piixi,

30 1�Ù ê¼ó�ªü>Ó�~�xi, ·���xi − xi+1 = (xi−1 − xi)pi,i−1/pi,i+1. òpi,i+1 ÀǑbi, pi,i−1 ÀǑdi. �o, aqu~ 1.5.8,Tê¼ó~���=�R =∞, Ù¥R = 1 +

∞∑

k=1

exp(−ck∑

j=1

j−α).ec = 0,KR =∞,dê¼ó~�,§Ò´���9�{ü�ÅiÄ. e�c > 0. �α > 1�,∑k

j=1 j−α <

∑∞j=1 j

−α < ∞, exp(−c∑kj=1 j

−α) ≥ ε > 0, u´R = ∞, dê¼ó~�; �α < 1 �,∑k

j=1 j−α ≈ k1−α/(1 − α), exp(−c∑k

j=1 j−α) �exp(− c

1−αk1−α) Ó�. u´R <∞,Tê¼ó�~�; �α = 1�,

∑kj=1 j

−1 ≈ log k, exp(−c∑kj=1 j

−α)�k−c Ó�. u´, R =∞ ��=�c ≤ 1, d�, Tê¼ó~�. o�, ~�5�6uëê�'Xã�ã 1.5.Ù¥, ¢�L«~�, J�L«�~�. 2

c

1

1 0 ã 1.5: ~�: α > 1; α = 1�c ≤ 1; α < 1�c = 0.

2. ��¼ê�O{.é?¿i, j ∈ S, -Gij = EiVj = Ei

∞∑

n=0

1{Xn=j} =

∞∑

n=0

Pi(Xn = j) =

∞∑

n=0

p(n)ij ,¿¡�Ǒê¼ó���¼ê. �â�J�{K (1.24), i ~���=�Gii =∞.~1.5.11. (~1.1.11Y) d �{ü�ÅiÄ�d = 1, 2 �´~��, �d ≥ 3 �´�~��.): òZd ��:PǑ0. ·�òy², �d = 1½2�, G00 =∞;�d ≥ 3�, G00 <∞.éud �{ü�ÅiÄ, §²LÛêÚ£��:�VÇǑ0. b�§²L2n Ú£��:, Ù¥,1r ���þ !�©Ùrnr Ú. K

p(2n)00 = P0(S2n = 0) =

∑

n1+···+nd=n

(2n)!

(n1!)2 · · · (nd!)2· 1

(2d)2n. (1.29)3þ¡��Ú±9e©¥, n1, n2, · · · , nd Ñ���K�ê.

§1.5 ~�5 31�d = 1 �, ��$^Stirlingúªm! ≈ (m/e)m√2πm, Ù¥≈ L«�mü>�û3n →

∞ �ªu1. u´,

p(n)00 =

(2n)!

(n!)2· 1

22n≈ 1√

πn,ùL²��{ü�ÅiÄ~�.�d = 2 �,

P0(S2n = 0) =∑

n1+n2=n

(2n)!

(n1!)2(n2!)2· 1

42n=

(2n)!

n!n!· 1

42n

∑

n1+n2=n

n!

n1!n2!· n!

n2!n1!

= Cn2n ·

1

42n

n∑

n1=0

Cn1n Cn−n1

n = Cn2n ·

1

42n· Cn

2n ≈1

πn,ùL²��{ü�ÅiÄ~�.�d ≥ 3 �, ·��±y²�n ¿©��,

P0(S2n = 0) ≤ Cn−d/2, (1.30)Ù¥C ´�6ud �~ê. ùL²n�±þ�{ü�ÅiÄ�~�. e¡, ·�±d = 3 Ǒ~y² (1.30). ·�òZd ¥�n���©O¡Ǒx, y, z. �Ä��kn�¡�ú²Úf,Ý�(J�UǑx, y, z, ±9n�ú²M1, ©O¡ǑM1x, M1y, M1z. e¡, ·�Õá/Ý�Úf�ùn�M1, ¿UìUXeö�$1Z3 þ�{ü�ÅiÄ. z�g, kwÝ�Úf�(J, XJÝ�(JǑw (Ù¥w �Ǒx, y, z), �o2Ý�M1w, Ý��¡Kâf3w ��þ ?�Ú, Ý��¡K3w ��þ�ò�Ú. ò©OdM1x, y, z �)���{ü�ÅiÄPǑ{Xm}, {Ym}, {Zm}, §�Ñ´l0 Ñu���{ü�ÅiÄ. b�n gö��, âf3x, y, z ��þ©O$ÄK,L,M Ú. �o, K, L, M ÑÑl��©ÙB(n, 1/3). �5¿§�Ø�pÕá, ÏǑ§�÷v���å^�K + L +M = n. ��5¿�´, ÝÚf, Ýx M1, Ýy M1, Ýz M1´�pÕá/?1�, Ïd, (K,L,M), ({Xm}, {Ym}, {Zm}) ´�pÕá�. duSn = (XK , YL, ZM ), ·�kSn = 0 ��=�XK = YL = ZM = 0. ��¡,

P0(S2n = 0) = P (XK = YL = ZM = 0)

=∑

n1+n2+n3=n

P (K = 2n1, L = 2n2,M = 2n3)P (X2n1 = Y2n2 = Z2n3)

=∑

n1+n2+n3=n

(2n)!

(2n1)!(2n2)!(2n3)!· 1

d2n× Cn1

2n1· 1

22n1× Cn2

2n2· 1

22n2× Cn3

2n3· 1

22n3,ù=´ (1.29). ,��¡, 5¿�EK = 2n/3. -

A = {K,L,M ≥ n/3} .

32 1�Ù ê¼óKp(A) ≈ 1 �3A þ, þ¡L�ª¥�n1 , n2, n3 Ñ¿©�l �±^Stirling úª. ·�?1Xe©):

P0(S2n = 0) = P (Ac, S2n = 0) + P (A,S2n = 0). (1.31)éþª¥�1��, ·�y²ÙVÇǑo( 1√n3 );é1��,·��±^��{ü�ÅiÄ�(J?1�O.äN/, 'u(1.31)m>�1��, ·��±^4 ��'ÈÅØ�ª, ��

P (Ac) ≤ 3P (K < n/3) ≤ 3P (|K − EK| > n/3) ≤ 3E(K − EK)4

(n/3)4≤ 3000

n2≤ 1√

n3.Ù¥, 31���ª¥·�^�K,L,M Ó©Ù, 3�����ª¥^�K ∼ B(2n, 1/3). I�`²�´: ·��8I´y² (1.30), 4 ��'ÈÅØ�ª�Ñ�´P (Ac) = O( 1

n2 ).ù��O3d ≥ 5 ��/�´Ø�. Ǒd, ·�I��°[��O, äNO�Xe: 5¿�K ∼ B(2n, 1/3),=K d= ξ1+· · ·+ξ2n,Ù¥ξ1, · · · , ξ2nÕáÓ©Ù,�P (ξ1 = 1) = 1−P (ξ1 =

0) = 1/3. u´, é?¿a > 0, P (K < n/3) ≤ Eea(K−n/3) = e−an/3EeaK , Ù¥EeaK = Eea

∑2ni=1 ξi =

(

Eeaξ1)n

= (ea/3 + 2/3)2n.�é{`,

P (K < n/3) ≤ ϕ(a)2n, ∀a > 0, (1.32)Ù¥ϕ(a) = e−a/6(ea/3 + 2/3). AO/, �a = 2/5 ��ϕ(a) �����ρ = ϕ(2/5) < 1. u´, P (K < n/3) ≤ ρ2n = o( 1√nd

), ∀d.'u(1.31)m>�1��, �2n1, 2n2, 2n3 ≥ n/3 �,

P (K = 2n1, L = 2n2,M = 2n3, S2n = 0)

= P (K = 2n1, L = 2n2,M = 2n3, X2n1 = Y2n2 = Z2n3 = 0)

= P (K = 2n1, L = 2n2,M = 2n3)P (X2n1 = Y2n2 = Z2n3 = 0), (1.33)Ù¥�����ª^�{Xm}, {Ym}, {Zm} �(K,L,M) �pÕá. 5¿��n ¿©��,

P (X2n1 = Y2n2 = Z2n3 = 0) = P (X2n1 = 0)P (Y2n2 = 0)P (Z2n3 = 0) ≈ 1√πn1· 1√

πn2· 1√

πn3.

(1.34)l , ·�kP (X2n1 = Y2n2 = Z2n3 = 0) ≤ 3√n3. u´, P (A,Sn = 0) ≤ 3√

n3P (A) ≤ 3√

n3. nþ��, �n ¿©��, P0(Sn = 0) ≤ 4/

√n3.F�ÆöKakutaniéd�Ñ��k��5): “A drunk man can find way home, a

drunk bird can not”. 2

§1.5 ~�5 33,, ·�Ǒ�±^��¼êy²~�5´pÏa�5�(·K 1.5.6�Ü©(Ø). y²Xe:ej = i K·K¤á. e�j 6= i. Äk, ·�^�y{y²j → i. ÄK, d·K 1.3.1�(1)Ú(3),

Pi(X1 = i1, · · · , Xn−1 = in, Xn = j,Xn+m 6= i, ∀m ≥ 1) = pi0i1 · · · pin−1inPj(Xm 6= i, ∀m ≥ 1) > 0., , ÏǑi1, · · · , in−1 ÑØ´i, ¤±d�ª�>�¯��íÑXn 6= i, ∀n ≥ 1,=Ti =∞. ùL²ρii = Pi(Ti < ∞) < 1. d(1.24), i �~�, gñ! di ↔ j 9·K 1.3.1, �3n,m ≥ 1 ��p(n)ij , p(m)ji > 0. �âChapman-Kolmogorov�ª,=·K1.1.17,·�k

p(m+r+n)jj =

∑

k,l∈S

p(m)jk p

(r)kl p

(n)lj ≥ p

(m)ji p

(r)ii p

(n)ij , ∀ r ≥ 0.ü>Ó�ér�Ú, �Gjj ≥ p(m)

ji p(n)ij Gii. l i ~�Kj ~�. 2·K1.5.12. �π ´��ØC©Ù, j �~�, Kπj = 0.y: b�i 6= j, ·��Äli Ñu�ê¼ó�¯j �²þogê. Äk, Tê¼ó�U��j, Ùg, l§Ä�j �m©, �¯j �ogêÒ´lj Ñu�ê¼ó�²þ£�gê. Ïd, l�*þ·���

∞∑

n=0

p(n)ij ≤ Gjj <∞. (1.35)Ùî�y²3ǑSK. ü>Ó��±πi ¿éi �Ú, �∑i∈S πi

∑∞n=0 p

(n)ij ≤ Gjj . ���ÚgS�, þª��>´∑∞

n=0

∑

i∈S πip(n)ij . dØC©Ù�½Â, ù=´∑∞

n=0 πj . m>Gjj k�, ÏǑj �~�. l , πj = 0. 2±�·�òy², éu��Ø��ê¼ó, ØC©Ùπ e�3, �o§3j ��πj Ò´;��¯j �ªÇ�4�(H{½n), Ò�3ÕáÝ1�.¥, Ý��¡�VÇÒ´�¡Ñy�ªÇ�4���. lù�ÆÝw, �~��G���¯�ogêk�. Ïd, πj = 0, =, ªÇǑ0, ´ég,�(Ø.öSK1. y²: é?¿i 6= j, (1) e¡n^�d: ρij > 0, i → j, Gij > 0; (2) Gij ≤ Gjj ; (3)

ρii = 1− 1/Gii.

2. �E,«�¬I�²L �ü�óS.3�¤1��óS��, 10%�\ó�¤¢¬,

20%�\ó�I��ó,�{�70%K?\1��óS.3�¤1��óS��, 5%�\ó�¤¢¬, 5% �\ó�I��£�1��óS, 10% �\ó�I��£�1��óS; �{�80% �±Ñ�. Áïá��ê¼ó�[dXÚ, ¿���)�L§�¢¬Ç.

34 1�Ù ê¼ó3. �ξ1, ξ2, · · · ´ÕáÓ©Ù�lÑ.�ÅCþS�, Eξ1 6= 0. -S0 = 0, Sn = ξ1 + ξ2 +

· · ·+ ξn. y²: �ÅiÄ{Sn}´�~��. (5: eEξ1 = 0,K{Sn}~�,ùÜ©y²I��õVÇØ�;��£, ®²�Ñ�Ö�ùÇ��.)

4. 3~1.1.9 ¥, ÁïÄPzL§�2Â�#L§�~�5.

5. 3~1.1.12 ¥, ÁïÄ{Xn} �{Yn} �~�5.

6. y²: (1) eA ´k�48, K�3~�aC ��C ⊆ A; (2) k�G��mþ�ê¼ók~��.

7. �ê¼ó�G��mǑ{0, 1, · · · , 6},=£ÝXe.Á(½=G�´~��, =´�~��, ¿�3z�~�aþ�ØC©Ù.

1/2 0 1/8 1/4 1/8 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

1 0 0 0 0 0 0

0 0 0 0 1/2 0 1/2

0 0 0 0 1/2 1/2 0

0 0 0 0 0 1/2 1/2

.

8. ^�8>/Á÷�²¡,¤·¶G.�ÄÙþü�Õá�{ü�ÅiÄ.Á¯N���© ��±�y§�[Ø��? ?�Ú,|^þKy²: XJ\��^��Ø�÷v, §�7½����g.

9. b�ê¼óli Ñu, PFij Ǒτj �1¼ê, =Fij(s) =∑∞

n=0 Pi(τj = n)sn. y²:

Gij(s) = Fij(s)Gjj(s).

10. �â (1.33)�(1.34)y²: �3~êCd ��P0(S2n = 0) ≈ Cdn−d/2, ∀d ≥ 1. (l , U? (1.30)).

11*. �ÄÄg�£�:��mσ0 . y²:

(a) ed = 3 ½4, K∑∞k=1 2kP0(σ0 = 2k) <∞;

(2) ed ≥ 5, K∑∞k=1 2kP0(σ0 = 2k) =∞. ù«�/¡Ǒr6�(stongly transient).

12. �{Xn} Ǒ�ÅiÄ, X0 = 1, Ú�©ÙǑP (ξ = 2) = P (ξ = −1) = 1/2. τ = inf{n ≥ 0 :

Xn = 0}. Áy: φ(s) = Esτ ÷vsφ3 − 2φ+ s=0.

13. �½o ∈ S. y²: eyi ≥ 0, ∀i ∈ S �§�Ǒ÷v(1.27), �oyi ≥ Eiτo, ∀i ∈ S.

§1.5 ~�5 35

14. b�S Ø��, ~�, A,B ⊆ S �A ∩B = ∅. Pxi = Pi(τA < τB)

15. y²yi = Eiσo, i ∈ S ´(1.27) ����K).