homework (1)

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1、8¹111.8!12.22117!115.23117!145.41§111.8!12.yP(Z0=i0,···,Zn=in)=µi0pi0i1···pin−1in,∀n≥1,i0,···,in.Ù¥µi=P(Y0=i).Uìτ?1y©,P(Z0=i0,···,Zn=in)Xn=P(Z0=i0,···,Zn=in,τ=m)+P(Z0=i0,···,Zn=in,τ>n)m=0Xn=P(Y0=i0,···,Ym=im;X06=i0,···,Xm−16=im−1,Xm=im,···,Xn=in)m=0+P(Y0=i0,···,Yn=in;X06=i0,···,X

2、n6=in)Xn=P(Y0=i0,···,Ym=im)P(X06=i0,···,Xm−16=im−1,Xm=im,···,Xn=in)m=0+P(Y0=i0,···,Yn=in)P(X06=i0,···,Xn6=in)Xn=µi0pi0i1···pim−1imP(X06=i0,···,Xm−16=im−1,Xm=im)pimim+1···pin−1inm=0+µi0pi0i1···pin−1inP(X06=i0,···,Xn6=in)!Xn=µi0pi0i1···pin−1inP(X06=i0,···,Xm−16=im−1,Xm=im)+P(X06=i0,··

3、·,Xn6=in)m=0=µi0pi0i1···pin−1in.Ù¥,Ò´ÏǑ3ê11¥)l¥,Ò´mUσ:=min{0≤m≤n:Xm=im}(min∅:=n+1)?1y©,l)lu1.§2117!115.bp16=1(Ä.´²).-η=ξ−1,ε=1η=−1,ζ=η+ε,5:ζÒ´òη3η=−1«þ?UǑ0ÅCþ.)XnǑǑnè(ÿhýèÎÝ).o,eXn6=0,Xn+1=Xn+ηn+1;eXn=0,Xn+1=Xn+ζn+1=Xn+ηn+1+εn+1.=Xn+1=Xn+ηn+1+εn+11Xn=0.Ù¥,d

4、d4íª,XnûuX0,ξ1,···,ξn,Ïdεn+1pÕá.{Xn}=£VÇgC.~5:liÑuU0VÇ.duXngUa,drê¼5,Pi(τ0<∞)=i}P1(τ0<∞),∀i≥1.Ïd{Xn~=ρ=P1(τ0<∞)=1.Yn=η1+···+ηn.ÏǑXY0X={Xn}Y={Yn}kÓ=£VÇ,¤±P1(τ0<∞)=P1(τ0<∞).eEη<0,Yn→−∞,ρ=1;eEη=0,d15!S35(½dYÙ!©

5、§(Ø),Y~,ρ=0;YeEη>0,Yn→∞,Y~,ùþE0(τ1<∞)=1²ρ<1.Ïd,Tê¼ó~=Eη

6、=µ−1≤0.?Ú,~1.2.6aq:Eη=0vkCØ©Ù,Ïd"~;Eη<0kØC©Ùπ,Ïd~.²þè:2X¯1Pn(1)limn→∞n,Ù¥X¯n=n+1m=0Xm.PP(i)~/,Eη<0.-f(i)=i,Eπf=iiπieiiπi<∞,=Eπf<∞,dH{X¯1PnPPPVi(n)X¯½nn=n+10f(Xm)→Eπf=iiπi.eiiπi=∞,dX¯n=iin+1liminfnn≥PVi(n)PPiiliminfnn+1=iiπi=∞.lokX¯n→iiπi,a.s..PVi(n)PVi(n)PM−1Vi(n)(ii)"~/,Eη=0

7、.X¯n=iin+1≥i≥Min+1≥M(1−i=0n+1),∀M≥0.u´,X¯PM−1Vi(n)liminfnn≥M(1−limni=0n+1)=M.-M→∞X¯n→∞.(2)EXn.Äk,dEXn+1=EXn+Eη+p0P(Xn=0):eEX0=∞,EXn=∞,∀n.eEX0<∞(i)~/,Eη<0.Äk,·El0Ñu§Ú±µǑ©Ù§ÍÜ.ÄE,XÚ,ThÕküè,1èY©vk,1èZ©kZ0∼µý.1èk`kh(é{`,k1è¥vký,â¬1èh),#\èÑ\1è.ä/,(Yn+1,Zn+1)=

8、(Yn+ηn+1,Zn)1Yn≥1,orξn+1≥1+(0,(Zn−1)∨0)1Yn=0,ξn+1=0.o,{Yn}´l0Ñu§,{Xn=Yn+Zn}´lZ0Ñu§.(5¿,{Zn}Ø´ê¼ó.)3ùÍÜ¥,zgYn=0,Zn(if>0)ѱVÇp0/p1(>0)CǑZn−1.du{Yn}~,§¡õg¯0,Ïd,Zn½¬CǑ0.=Zn→0a.s..ÆÝw,ùÍÜÒ´m©30ÚX0=Z0üâfYÚX.zÚ,)ÅCþη,4ùüâf©a(Y+η)∨0Ú(X+η)∨0;u´âfX[جuâfY>,=Y0=0≤X0²Yn≤Xn.ÏǑX~,XY7,(X7,

9、0,d,