(Ω,F,P)
(An)n1
(An)n10
(An)n1L20
(An)n10
(Xn)n1Xn
X Y X =Y
X
Y Lpp∈ {1,2}
(Ω,F,P)X, X1, X2, . . . : (Ω,F,P)R
(Xn)n1
X
1n1< n2< . . .
k1
P|XnkX|>1
k1
2k.
k1Yk=Xnk(Yk)k1
(Xn)n1(Yk)k1X
(Ω,F,P) (An)n1
X
n1
P(An)<+.
P(lim sup An)=0
lim sup An:= Tk1SnkAn={ωΩ : {n:ωAn}estinfini}
X(Ω,F,P)
X
n0
nP(nX < n + 1) <+∞ ⇔ E[X]<+.
X
n1
P(Xn)<+∞ ⇔ E[X]<+.
(Xn)n1
Xn
n
P
n→∞ 0
E[|X1|]<+Xn
n
L1
n→∞ 0
E[|X1|]<+Xn
n
p.s.
n→∞ 0
(Xn)n1
n1aR
E[(Xna)2] = (E[Xn]a)2+ Var(Xn).
(Xn)n1
a
lim
n→∞
E[Xn] = alim
n→∞ Var(Xn)=0.
(Xn)n1(Xn)n1
L2X(X2
n)n1L1
X2
(Ω,F,P)
(An)n1
X
n1
P(An) = +.
P(lim sup An)=1
x1 + xex
n, m 1mn
P n
\
k=m
Ac
k!exp
n
X
k=m
P(Ak)!.
m1P
\
k=m
Ac
k!= 0
(Xn)n1
p]0,1[ 1 (Xn)n1
1 0
(Xn)n1
E[|X1|]=+
X1+...+Xn
nn1
(xn)n1x1+...+xn
nn1
lim
n→∞
xn
n= 0.
X
n1
P(|Xn| ≥ n)=+
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