W4
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·
Стр.1
W4
AR(1)
Стр.2
X(k) = f(ξk), k = 1, 2, . . . ,
f(x)
nlim
x → ∞
k=1,...,n
℄
f
℄
X(k)
℄
ξk
X(k) = f(ξk)
ξk
α
f
f
ξk
α > 0.
r(n) lnn→0, n→∞.
→∞P( max ξk < a−1
an = √2 ln n−
f(ξ)
n x+an) = e−e−x
ln lnn+ln 4π
2√2 ln n
α
x0
.
,
x x0
r(k)
X(k) = f(ξk) k = 1, 2, . . .
℄
X = f(ξ)
X = f(ξ)
ξ
ξ
f(x), x ∈ R
(1)
(2)
Стр.3
α
L(x)
x
n→∞
n→∞
℄
f′(x) > 0
dn
nlim
Xk, k = 1, 2, . . .
y > 0.
f(ξk)
nlim
n→∞
α
an
h(x) = ln f(x),
nlim
→∞P(d−1
k=1,...,n
→∞P(d−1
n max f(ξk) < d−1
k=1,...,n
F(x)
n f(a−1
1−F(dn) ∼
→∞P(d−1
n max f(ξk) < d−1
y
x = αln y
f(αa−1
α
n f(αa−1
f(αa−1
y
ln f(αa−1
h(an +αa−1
h′(an)+ α2 ln y
2a2
n
y = 1
α
an
h′(an) = 1+o(1),
an+1/an →1
an
αln y
an
h′(an)+ α2 ln2 y
2a2
θn = θn(y) ∈ [0, 1].
h′′
n
n ln y)−h(an) = ln y +o(1).
h′′
an + αθn ln y
an
an + αθn lny
an
y →1
= 1+o(1)
a2 h′′
1
n
h′(z) = z
α +zg(z),
an + αθn ln y
an
an + αθn ln y/an →1
1
n ln y +an)
dn
→y
y = 1
f
n ln y +an)
f(an) →y
1−F(x) = x−αL(x),
n x+an)) = e−e−x
dn
1
n, n→∞,
n max Xk < y) = e−y−a
k=1,...,n
n lny +an)) = e−y−a
n ln y +an)−ln f(an)→ln y.
= ln y +o(1)
= o(1)
n→∞.
z2h′′ (z) = o(1)
n→∞
n→∞.
, y > 0.
dn = f(an)
.
Xk = f(ξk)
(3)
(4)
(5)
(6)
(7)
y > 0.
(8)
z →∞,
(9)
Стр.4