Compléments1

N Z
(Xi)i>1p>1 (X1,...Xp)
(X2,...Xp+1)
(Xi)i∈Zp>1 (X−p,...Xp)
(X−p+1,...Xp+1)
N
(Xk,...Xp)∼(Xk+l,...Xp+l)k6p l > 0
l= 1 (X1,...Xp)∼(X2,...Xp+1)
(Xk,...Xp−1)∼(Xk+1,...Xp)l= 1
RNN
RNB∞C
{(x1,...xn)∈A}A∈B(Rn)
(Xi)i>1∼(Xi+1)i>1:
P(RN,B∞)
L1(P)
C∞
C
B∞
L1C
B∞C∞
L1(Rd, P )P
(Xi)i>1ϕ Yk=ϕ(Xk, Xk+1, . . . )
(Xi)i∈ZYk=ψ(...,Xk−1, Xk, Xk+1, . . . )

A∈B∞Yk= 1(Xk,Xk+1,... )∈A
C
ϕ ϕ
ϕ[−n, n] 1/n
ψ
ϕ
limi→∞ XiPi>12−iXi
(Xn)n>0α
<1
Yn=
∞
X
j=0
αjXn−j=αYn−1+Xn
Y
Yn=
p
X
k=1
akYn−k+εn+
q
X
k=1
bkεn−k,
εnN(0, σ2)ak, bkσ
Zn=
Yn
Yn−p+1
, ηn=
εn
εn−q
, A =
a1a2. . . ap
1 0 . . . 0
0. . . 1 0
, B =
1b1. . . bq
0 0 . . . 0
0. . . 0 0
Zn=AZn−1+Bηn
p= 1, q = 0
Zn=Bηn+ABηn−1+A2Bηn−2+. . . ,
e
Zn= (Zn, εn,...εn−q+1)e Zn=e Ae Zn−1+e Bεn.
X= (Xi)i>1Y X
Y=f(X)
f(X) = X3+ 2 cos(X7)
T T Y X
T f(X) = X4+ 2 cos(X8).
T Y Y T
T L1

xiL1
Y= lim Xi
Y= 1Xi
Y= lim 1
n
n
X
i=1
Xi.
T Y =Y
Xn
ξΩ = [0,1] 1
log 2
dx
1+x
ξ=1
X1+1
X2+· · ·
.
X ξ ξ
T ξ =1
X2+1
X3+· · ·
=ξ−1−[ξ−1],
ξ ξ−1−[ξ−1]
x v x v
V(x) cos(x)
W(x, v) = 1
2mkvk2+V(x)y= (x, v)
m˙yt=md
dt xt
vt=mvt −∇V(xt).
W(yt) = W(y0)a∈RE={y:W(y)6a}
a yt(y)y0=y t
Tt:y07→ ytE f
d
dt ZE
f(yt(y)) dy = 0,d
dt ZE
f(xt(x, v), vt(x, v)) dxdv = 0.
yny0E E[f(yn)] = E[f(y1)]
yiy1ϕ(y1. . . yp)
y1y0
f E t = 0
t06= 0 g(y) = f(yt0(y))
d
dt ZE
f(yt(y)) dy |t=t0=d
ds ZE
g(ys(y)) dy |s=0
g f

XiY=f(X)Y=T Y
PB∞
T1A= 1AA
A A
T1A
f P (T f =f)=1 {ω:a < Y < b}
Y
Y
Xkf
E[|f(X1)]|<∞
1
n
n
X
k=1
f(Xk)−→ E[f(X1)]
L1
p{(Xk+1,...Xk+p)}k>0
f
E[|f(X1,...Xp)]|<∞
1
n
n
X
k=1
f(Xk+1,...Xk+p)L1 −→ E[f(X1,...Xp)].
XkRdf∈C∞(Rpd)
1
n
n
X
k=1
f(Xk+1,...Xk+p)L1 −→ E[f(X1,...Xp)]

Y=ϕ(X1, X2, . . . )
1
n
n
X
k=1
TkYL1 −→ E[Y]
T L1(P)Y7→ 1
nPn
k=1 TkY−E[Y]
62YB∞
L1(P)f∈C∞(Rpd)L1(P)
λ ξ [0,1] Xn=ξ+nλ
X1,...Xn
ξ f ξ L2([0,1])
f(x) = X
k
cke2iπkx, ck=Z1
0
f(x)e−2iπkxdx.
e2iπkT ξ =e2iπk(ξ+λ)T f
T f(ξ) = X
k
cke2iπkλe2iπkξ.
f=T f
ck= 0 k6= 0 f
(Xn)n>1Yn=ϕ(Xn, Xn+1, . . . )n>1
(Xn)n∈ZYn=ψ(...,Xn−1, Xn, Xn+1, . . . )
Xn
E1 E1, E2 E2
Xnπ
1
n
n
X
k=1
f(Xk)L1 −→ π(f)
f p (Xk+1,...Xk+p)