ProcessusMarkoviens

Zd

X(Ω,F,P)
(E, E)
BEX−1(B) = {X∈B} F X
PXX(P) loi(X)mE
m(B):=P(X∈B)BEm(B)
m(E, E) = (R,B(R))
B=] − ∞, x]FX:R→[0,1]
FX(x) := P(X6x)E
E=P(E)px:= P(X=x)
x E B E
P(X∈B) = X
x∈B
px.
X Y ±1
P(X=Y= +1) = P(X=Y=−1) = x/2,
P(X= +1, Y =−1) = P(X=−1, Y = +1) = (1 −x)/2,
x0 1 X Y
{±1}P(X=Y) = x0 1
x= 0 X=−Y x = 1 X=Y x =1
2X Y
BkEk
P(Xk∈Bk,16k6n) =
n
Y
k=1
P(Xk∈Bk).
E=R
L1

ξ I
ξI:= (ξi, i ∈I). ξi:j:= (ξk)i6k6j
i6j
(Xn)n>0
m X0pn
Xn−1Xnn>0x0:nEn+1
Pm(X0:n=x0:n) = m(x0)
n
Y
k=1
pk(xk−1, xk).
m p(x, ·)E
pn(xn−1, xn) = Pm(Xn=xn|X0:n−1=x0:n−1).
m X0n>1pn(x, ·)
x E Xn
(Xn)n>0
pn(·, y)y E
Xn:= X0+Y1+··· +Yn(Yn)n>1
X0Yn=±1
ZYn=±ei16i6dZd
Yn
(Xn)n
pnpn(x, y) =
rn(y−x)rnYn

w(e)>0e
x w(x) := X
y
w(x, y)
p(x, y):=w(x, y)/w(x).
o w(e) = xne
n n + 1 o
x > 1x < 1
o d + 1
x > 1/d x 61/d
Zn+1 :=
Zn
X
k=1
ξn,k,
Zn+1 := 0 Zn:= 0
Np(0,0) = 1 i>1
p(i, j) := P(ξ1+··· +ξi=j),
(ξi)i>1ξn,k
D:= {Zn→0}
D={∃n, Zn= 0}Z0= 1
P(D|Z0=k) = P(D|Z0= 1)k
qn:= P(Zn= 0)
{Zn= 0}qn→q
qnn q =P(D)
q=f(q)
m:= E(ξ)