❈❍❆P❚❊❘ ✶
❈♦♥✈❡r❣❡s ✐♥ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❉✐str✐❜✉t✐♦♥
❉❡❢✐♥✐t✐♦♥ ✶✳ ▲❡t{Xn}✱X ❜❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❚❤❡♥{Xn} ❝♦♥✈❡r❣❡s ✐♥ ❞✐str✐❜✉t✐♦♥ t♦X ❛sn→ ∞✱ ✇r✐tt❡♥Xn →dX✱ ✐❢
lim
n✙∞
P(Xn≤X) = lim
n✙∞
FXn(x) =FX(x) ❢♦r ❡❛❝❤ ❝♦♥t✐♥✉✐t② ♣♦✐♥t ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥F(x)✳
❊①❛♠♣❧❡ ✷✳ ▲❡tX1, X2, ..., Xn ❜❡ ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ ✉♥✐❢r♦♠ ❞✐str✐❜✉✲
t✐♦♥✱Xi ∼U N IF(0,1)✱ ❛♥❞ ❧❡tYn =Xn:nt❤❡ ❧❛r❣❡st ♦r❞❡r st❛t✐st✐❝✳ ❋✐♥❞ ❧✐♠✐t✐♥❣
❞✐str✐❜✉t✐♦♥ ♦❢Yn✳
❙♦❧✉t✐♦♥✳ ❋r♦♠ ❡q✉❛t✐♦♥ ❄❄ t❤❡ ❈❉❋ ♦❢Yn ✐sGYn = (FX(y))
n❛♥❞
FX(x) =
0, x,
1,
0≥x
0< x <1
x≥1
❚❤❡r❡❢♦r❡
GYn(y) =
0, yn,
1,
0≥y
0< y <1
y≥1
❛♥❞
lim
n✙∞
GYn(y) =
lim
n✙∞
0 = 0,
lim
n✙∞ yn= 0,
lim
n✙∞
1 = 1,
0≥y
0< y <1
y≥1
❚❤✉s
GY (y) = lim
n✙∞
GYn(y) =
(
0 , y <1 1 , y≥1
❊①❛♠♣❧❡ ✸✳ ❙✉♣♣♦s❡ t❤❛t X1, X2, ..., Xn ✐s ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ P❛r❡t♦
❞✐str✐❜✉t✐♦♥✱ Xi ∼P AR(1,1) ♦r fX(x) = (1 +x)−2, x >0✱ ❛♥❞ ❧❡t Yn =nX1:n✳
❚❤❡ ❈❉❋ ♦❢Xi ✐sFX(x) = 1−1+x1 , x >0✱ ❋✐♥❞ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢Yn
❙♦❧✉t✐♦♥✳ ❋r♦♠ ❡q✉❛t✐♦♥ ❄❄✱
GYn(y) =
(
1− 1−FX ynn= 1− 1 1+yn
n
= 1− 1 +ny−n
,0< y
0 , y≤0
❛♥❞
GY (y) = lim n✙∞
GYn(y) =
(
lim
n✙∞1− 1 + y n
−n
= 1−e−y ,0< y
0 , y≤0
✶✳ ❈❖◆❱❊❘●❊❙ ■◆ P❘❖❇❆❇■▲■❚❨ ❆◆❉ ❉■❙❚❘■❇❯❚■❖◆ ✷
❊①❛♠♣❧❡ ✹✳ ❘❡✇♦r❦ ❊①❛♠♣❧❡ ✸ ❢♦rYn =Xn:n
❙♦❧✉t✐♦♥✳ ❋r♦♠ ❡q✉❛t✐♦♥ ❄❄✱
GYn(y) =
(
(FX(y))n=1−1+y1
n
,0> y
0 , y≥0
❛♥❞
GY (y) = lim n✙∞
GYn(y) =
lim
n✙∞
y 1+y
n
= 0 ,0> y
0 , y≥0
❚❤✉sYn ❞♦❡s ♥♦t ❤❛✈❡ ❧✐♠✐t ❞✐str✐❜✉t✐♦♥✳
❉❡❢✐♥✐t✐♦♥ ✺✳ ❚❤❡ ❢✉♥❝t✐♦♥G(y)✐s t❤❡ ❈❉❋ ♦❢ ❞❡❣❡♥❡r❛t❡ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡
✈❛❧✉❡y=c ✐❢
GY (y) =
(
0 , y < c
1 , y≥c
■♥ ♦t❤❡r ✇♦r❞s✱G(y)✐s t❤❡ ❈❉❋ ♦❢ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥ t❤❛t ❛ss✐❣♥s ♣r♦❜❛❜✐❧✐t②
♦♥❡ ❛t t❤❡ ✈❛❧✉❡y=c❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳
❊①❛♠♣❧❡ ✻✳ ▲❡t X1, X2, ..., Xn ✐s ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛♥ ❊①♣♦♥❡♥s✐❛❧ ❞✐s✲
tr✐❜✉t✐♦♥✱Xi∼EXP(θ)♦r fX(x) = 1θe−
x
θ, x >0✱ ❛♥❞ ❧❡tYn=X1:n ❙♦❧✉t✐♦♥✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❈❉❋ ♦❢Yn ✐s
GYn(y) =
(
1−e−nyθ , y >0
0 , y≤0
❚❤❡♥
GY (y) = lim n✙∞
GYn(y) =
(
lim
n✙∞
1−e−nyθ = 1 ,0< y
0 , y≤0
✇❤✐❝❤ ✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ ❞❡❣❡♥❡r❛t❡ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ ✈❛❧✉❡y= 0✳
❉❡❢✐♥✐t✐♦♥ ✼✳ ❆ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Y1, Y2, ... ✐s s❛✐❞ t♦ ❝♦♥✈❡r✲
❣❡♥❝❡ st♦❝❤❛st✐❝❛❧❧② t♦ ❛ ❝♦♥st❛♥tc✱ ✇r✐tt❡♥Yn→stochasticc✱ ✐❢ ✐t ❤❛s ❛ ❧✐♠✐t✐♥❣
❞✐str✐❜✉t✐♦♥ t❤❛t ✐s ❞❡❣❡♥❡r❛t❡ ❛ty=c✳
❊①❛♠♣❧❡ ✽✳ ❋♦r ❊①❛♠♣❧❡ ✷ ❛♥❞ ❢r♦♠ ❉❡✜♥✐t✐♦♥ ✺ ❛♥❞ ❉❡✜♥✐t✐♦♥ ✼✱ ❛♥❞G(y)
✐s t❤❡ ❈❉❋ ♦❢ ❞❡❣❡♥❡r❛t❡ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ ✈❛❧✉❡ y= 1❛♥❞Yn →stochastic1✳
❚❤❡♦r❡♠ ✾✳ ✭❈♦♥t✐♥✉✐t② ❚❤❡♦r❡♠✮ ▲❡tY1, Y2, ...❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠
✈❛r✐❛❜❧❡s ✇✐t❤ r❡s♣❡❝t✐✈❡ ❈❉❋✬sGY1(y), GY2(y), ...❛♥❞ ▼●❋✬sMY1(t), MY2(t), ...✳
■❢ MY(t)✐s t❤❡ ▼●❋ ♦❢ ❛ ❈❉❋ GY (y)✱ ❛♥❞ ✐❢ lim n✙∞
MYn(t) =MY(t)❢♦r ❛❧❧ t ✐♥ ♦♣❡♥ ✐♥t❡r✈❛❧ ❝♦♥t❛✐♥✐♥❣ ③❡r♦✱ −h < t < h✱ t❤❡♥ lim
n✙∞
GYn(y) = GY (y) ❢♦r ❛❧❧ ❝♦♥t✐♥✉✐t② ♣♦✐♥ts ♦❢ GY (y)✳
■♥ ♦t❤❡r ✇♦r❞s✱
lim
n✙∞
MYn(t) =MY (t)⇒ lim
n✙∞
GYn(y) =GY (y)⇒Yn→dY
❊①❛♠♣❧❡ ✶✵✳ ▲❡tX1, X2, ..., Xn ❜❡ ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ ❇❡r♥♦✉❧❧✐ ❞✐str✐✲
❜✉t✐♦♥✱Xi∼BIN(1, p)✱ ❛♥❞ ❝♦♥s✐❞❡rYn= n
P
✶✳ ❈❖◆❱❊❘●❊❙ ■◆ P❘❖❇❆❇■▲■❚❨ ❆◆❉ ❉■❙❚❘■❇❯❚■❖◆ ✹
❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ ▼●❋ ♦❢Zi ✐sMZ(t) = e
1 2t
2
✱ t❤❡♥ ❢r♦♠ ❚❤❡♦r❡♠ ❄❄ ▼●❋ ♦❢
Zn ✐s✱
MZn(t) =E e
Znt
=E
e
n
P
i=1Zi + 1
n
√n
t
=E e
t
√n+
t
√n
n
P
i=1
Zi
!
=e√t
nE
e√t
nZ1e t
√
nZ2...e t
√
nZn
=e√t
n
MZn
t √ n
n
❚❤❡r❡❢♦r❡✱
lim
n✙∞
MZn(t) = lim
n✙∞ e√t
n
et
2 2n
n
= lim
n✙∞ e√t
net
2 2 =e
t2
2