❈❍❆P❚❊❘ ✶
❈♦♥✈❡r❣❡s ✐♥ Pr♦❜❛❜✐❧✐t②
■t ✐s ♣♦ss✐❜❧❡✱ ✐♥ s♦♠❡ ❝❛s❡s✱ t♦ ✜♥❞ ❜♦✉♥❞s ♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❜❛s❡❞ ♦♥ ♠♦♠❡♥ts✳ Pr♦♣♦s✐t✐♦♥ ✶✳ ✭▼❛r❦♦✈✬s ✐♥❡q✉❛❧✐t②✮ ▲❡tX ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✱ t❤❡♥ ❢♦r ❛♥② ✈❛❧✉❡t >0✱
P(X ≥a)≤ E(X) a Pr♦♦❢✳ ❋♦ra >0✱ ❧❡t
I= (
1 , X≥a
0 , otherwise
s✐♥❝❡X≥0 t❤❡♥I≤ X a✳
❚❛❦✐♥❣ ❡①♣❡❝t❛t✐♦♥s ♦❢ t❤❡ ❛❜♦✈❡ ②✐❡❧❞s t❤❛tE(I)≤E(X)a ✳ E(I) =P(X≥a)✱
t❤❡r❡❢♦r❡
P(X ≥a)≤ E(X) a
❆s ❛ ❝♦r♦❧❧❛r②✱ Pr♦♣♦s✐t✐♦♥ ✷ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞✳
Pr♦♣♦s✐t✐♦♥ ✷✳ ✭❈❤❡❜②s❤❡✈✬s ✐♥❡q✉❛❧✐t②✮ ▲❡t X ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♠❡❛♥ µ❛♥❞ ✈❛r✐❛♥❝❡ σ2✳ ❚❤❡♥✱ ❢♦r ❛♥②t >0✱
P(|X−µ| ≥t)≤σ
2
t2
Pr♦♦❢✳ ❋♦r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡ ✭t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ✐s ❡♥t✐r❡❧② ❛♥❛❧♦❣♦✉s✮✱ ❧❡t R={x:|x−µ|> t} t❤❡♥
P(|X−µ| ≥t)≤
ˆ
R
f(x)dx
✐❢x∈R✱
|x−µ|2 t2 ≥1
❚❤✉s✱
ˆ
R
f(x)dx≤
ˆ
R
|x−µ|2
t2 f(x)dx≤
∞ ˆ
−∞
(x−µ)2
t2 f(x)dx=
σ2 t2
✶✳ ❈❖◆❱❊❘●❊❙ ■◆ P❘❖❇❆❇■▲■❚❨ ✷
❋♦r ❛♥♦t❤❡r ✐♥t❡r♣r❡t❛t✐♦♥✱ s❡tt=kσ s♦ t❤❛t t❤❡ ✐♥❡q✉❛❧✐t② ❜❡❝♦♠❡s
✭✶✳✵✳✶✮ P(|X−µ| ≥kσ)≤ 1 k2
❛♥❞ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠ ✐s
✭✶✳✵✳✷✮ P(|X−µ|< kσ)≥1−k12
❈❤❡❜②s❤❡✈✬s ✐♥❡q✉❛❧✐t② ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥s❡q✉❡♥❝❡✳ ❈♦r♦❧❧❛r② ✸✳ ■❢V ar(X) = 0✱ t❤❡♥P(X =µ) = 1
Pr♦♦❢✳ ■❢P(X =µ)<1✱ t❤❡♥ ❢♦r s♦♠❡ε >0✱P(|X−µ| ≥ε)>0✳ ❍♦✇❡✈❡r✱
❜② ❈❤❡❜②s❤❡✈✬s ✐♥❡q✉❛❧✐t②✱ ❢♦r ❛♥②ε >0✱
P(|X−µ| ≥ε) = 0
❈♦♥tr❛❞✐❝t✐♦♥✱P(X=µ) = 1
❉❡❢✐♥✐t✐♦♥ ✹✳ ✭❈♦♥✈❡r❣❡♥❝❡ ✐♥ Pr♦❜❛❜✐❧✐t②✮ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Yn ✐s s❛✐❞ t♦ ❝♦♥✈❡r❣❡ ✐♥ ♣r♦❜❛❜✐❧✐t② t♦Y ✱ ✇r✐tt❡♥ Yn →pY✱ ✐❢
lim
n✙∞P(|Yn❂Y|< ε) = 1
❚❤❡♦r❡♠ ✺✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s Y1, Y2, ... ✐s s❛✐❞ t♦ ❝♦♥✈❡r✲
❣❡♥❝❡ st♦❝❤❛st✐❝❛❧❧② t♦ ❛ ❝♦♥st❛♥t c✱ ✇r✐tt❡♥Yn →stochasticc✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r
❡✈❡r② ε >0✱
lim
n✙∞P(|Yn❂c|< ε) = 1
❚❤❡ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s t❤❛t s❛t✐s✜❡s ❚❤❡♦r❡♠ ✺ ✐s ❛❧s♦ s❛✐❞ t♦ ❝♦♥✲ ✈❡r❣❡ ✐♥ ♣r♦❜❛❜✐❧✐t② t♦ ❛ ❝♦♥st❛♥tc✱ ✇r✐tt❡♥ Yn→pc✳
❊①❛♠♣❧❡ ✻✳ ▲❡tX1, X2, ..., Xn ❜❡ ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉✲
t✐♦♥✱Xi ∼BIN(1, π)❛♥❞ ❧❡tYn=
n
P
i=1
Xi
n ✳ ❙❤♦✇ t❤❛tYn→pπ✳
❙♦❧✉t✐♦♥✳ ❯s✐♥❣ Pr♦♣♦s✐t✐♦♥ ✷✱
P(|Yn−π|< k r
π(1−π)
n )≥1−
1
k2
❈❤♦♦s❡ε=k
q π(1−π)
n )✱ t❤❡♥ lim
n✙∞P(|Yn−π|< ε)≥nlim✙∞
1−π(1−π) ε2n
❚❤❡r❡❢♦r❡✱
lim
n✙∞P(|Yn−π|< ε) = 1
❚❤❡♦r❡♠ ✼✳ ✭❙tr♦♥❣ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs✮ ■❢X1, X2, ..., Xn✐s ❛ r❛♥❞♦♠
s❛♠♣❧❡ ❢r♦♠ ❛ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ✜♥✐t❡ ♠❡❛♥µ❛♥❞ ✈❛r✐❛♥❝❡σ2✱ t❤❡♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ s❛♠♣❧❡ ♠❡❛♥s ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐❧✐t② t♦µ ♦r lim
n✙∞P(|
¯
Xn❂µ|< ε) = 1✱ ✇r✐tt❡♥ ¯
✶✳✶✳ P❘❖❇▲❊▼❙ ✹
✭✷✮ XnYn→pcd
✭✸✮ Xn
c →p1, c6= 0
✭✹✮ 1 Xn →p
1
c,∀n P[Xn 6= 0] = 1, c6= 0
✭✺✮ √Xn→p√c,∀n P[Xn≥0] = 1
Pr♦♦❢✳ ✳✳✳
❚❤❡♦r❡♠ ✶✸✳ ✭❙❧✉ts❦②✬s ❚❤❡♦r❡♠✮ ■❢Xn ❛♥❞Yn ❛r❡ t✇♦ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠
✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t Xn→pc ❛♥❞Yn→dY t❤❡♥✱
✭✶✮ Xn+Yn →dc+Y
✭✷✮ XnYn→dcY
✭✸✮ Yn
Xn →d
Y c, c6= 0
Pr♦♦❢✳ ✭❆s ❛ s♣❡❝✐❛❧ ❝❛s❡Xn ❝♦✉❧❞ ❜❡ ❛♥ ♦r❞✐♥❛r② ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡ s✉❝❤
❛sXn= (nn−1))✳✳✳
❚❤❡♦r❡♠ ✶✹✳ ■❢Yn→dY t❤❡♥ ❢♦r ❛♥② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥g(y)✱g(Yn)→d
g(Y)
Pr♦♦❢✳ ✭❆ss✉♠❡g(y)✐s ♥♦t t♦ ❞❡♣❡♥❞ ♦♥ ♥✮ ✳✳✳
❚❤❡♦r❡♠ ✶✺✳ ■❢√n(Yn−m)
c →dZ∼N(0,1)❛♥❞ ✐❢g(y)❤❛s ♥♦♥③❡r♦ ❞❡r✐✈❛t✐✈❡
❛ty=m✱g′(m)6= 0✱ t❤❡♥
√
n[g(Yn)−g(m)]
|cg′(m)| →dZ ∼N(0,1)
Pr♦♦❢✳ ✳✳✳
✶✳✶✳ Pr♦❜❧❡♠s
✭✶✮ ▲❡tX1, X2, ..., Xn❜❡ ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✱Xi∼
BIN(1, π)❛♥❞ ❧❡tYn =
n
P
i=1
(Xi−π)
