Canji meant de &'

exam

Exo1 1 . Vain ours .

2 . Tx en in temps d'

anent

now- tout oc

~~

On a I - = L ° 6-xn ↳ e Xn he

Enulilisat Markov fat et l'

invariance

pentathlonan q

IEOC4 C T ,e+ , - tx ) I Ft . . ] = En [ 4 C t.nl ]= lEo[ YCT , 1 ]

par constant Tan- Tn 1 Ee et Tn at ,e=T# en bin

Epartial 'm T , :( µ - Tu etiivhtpadatde IT ,Tz , ... , Tn

et Amite des ( Tan - Tn) an indeutitnntdimitmie .

On endidvit fire thin . tnlny . at iii. d. .

3. A

' T,

st integrate , d'

aprio a ( mi micidesc - 1

E [ Tr ) = E IE ( Tgn - Ty ) = x tE[ In ] < ay=o

done IE [ T.it c a et IE ( Te ) = 0 ( > c) .

4. Xn et born Ee done de came

'

irtejakei.

a er adapter

.

# [ XTIFN. D= EH, + 2Xm DXN + (DXNIY Faa )

2

= Xu } t Xu. , tE[ DX n I Fn . i ) +1 = xu . , +

Il-

= O

d ' on'

iE[ Xp . n I Fn . , ]= XP - Cn- i ) .Done ( Xi - n ) stare martingale

2- -5

. D'

apws le the nine d'

and t (

XmnEn-

na En ) st

we martingale . En particular

* I ' E [ xihnen ] = LECMAE. ] tn > 0 .

Xnoffw I xr

et Xnn2En → xtFe = it ps can INET , ,< + xp . s .

dompancomayeuudomime E[ xihnag ]→x~

De plus main I In E en e it

^^ En → In p . s

dampenoweugenu dominie

ltfnnntn) → LELIN ] .

On pent dnc passer a- lh limitdans ( *T at obteniraims E[ In ] = x

?Paraillewas If an , etdmc cE[ En]=ttE[T

,]=Ofa)

done x~= 0 ( x ) cetin'

at absmde .

Ex 82. 1 .

k st l ' imbiu du prochain bus pour-

.

µ Voyageurarrival on temps m ,

done

Tenst

le temps depangede a bus It Than. m st letemps d

'attente du bus ponce voyageur .

± .2.

#T n T

kn - l Tkm knt 1

hi Xnzs , alan knn = Ian at done Xnn= Xn - 1 .

Mi Xn= 0 , ahh but, = kn +1 et done Xnn= T- 1

knts

En conditimnavt par rapport eikneten uhilisatll counter i. i. d. des CTD on ablietPC Xnn = l 1 X. = le , ... / ×n , = ln . , i Xn = 0 )= I Pt = lt 11X. = e. , ... , xn . , sent ,Xn=o ,kik) Plkyk)k ktt -E T ( Tz , . . , Tk )= PC Tk+ , = eti ) = PLT , =L +1 ) = vcl )Ceqn

' il

illaitdemurrer .

3.

On a P"

C x. o ) =1 .

de plus cmme Test man

brute ie excite y 2 se tee

que v( y )

> 0 . dncPY

' " + 'T 0 , x ) 3 Plo , ylplg ,y - i ) . . PC x.nu )=r ( y ) > 0

did 0 cmmunitne own tout le monde dnc be

Chaime st imeduchtshe .

4- y.IT#.K - i k

foil - T le temps de retain en 0 peutatde 0

.

avertµ

T= Xst 1 . dmEo[T

]=Eo[×

,+1 ]

= left . ]=n< a

done O eat returner pesififone G Chain ehrecurrent pent 've .

the hometstinuaiatessiT ( x ) = I Gehlt.IT101 r be ) txzo6 qii se want enIT C. c) = I ( 01 E V ( y )y ? K -Cequ

'on part encore riecnne en

dl Ge ) = C PCT,

> x )

Cmme E Pct . > x7 = ET Te ]=µ

KEN

On en didn't que l'

canine probeinnate st

P ( En > ' e) fee IN .donnie per IGet

-

r

5.

Pnloio ) - In Plo , H P" "

1h , ol

ny

=[ V ( k ) P

" ' - k( 0,0 )

he 0

= In.

,

Ptak ) Pnhlao )i

done pm ( o

,o|=[E I Plt , ,=ki )

ill by +. . + ki =m f

#

Pm ( o , 01 > 0 ⇐> Fk , , . . ,kiEIN

*

knt . . .tk ; = n

et PCT ,=kj ) > 0 tj

d'

on' pgcd ( lnzi . Pm 6,01>041

= pgcd ( his> 1 : PCtn=n ) > 04 )La chain est apeiiodique site support

de la

loi de Teen aponiodique .

6 . Si cdtecondition st veiifiee on a d

'

anis le

bhiaeme de convergence

him R(×n=o) = to ) = Fen

G P ( Keo ) at be probe qu 'm busarrive an

temps n .

arrivedllinsreutn .