Ch5 1
x4 a0a2a1a2a3a2a4a2a5a2a6a2a7a2a8a2a9a2a3a10a4a2a5a10a6
a0a2a1a2a3a2a4a2a5a2a6
a11 0 < p < 1
a12 q = 1 pa12a14a13 xk = k nppnpq a12a14a15a17a16a17a18a17a19 jxkj6 A a20 k a21a17a22
a20a2a23
Cknpkqn k 1p2 npqe x
2k
2 (n !1):
a24 a25a2a26
k np; n k nq:
a27a29a28a2a30a2a31 (Stirling)
a32a2a33
n! nne np2 n:
a34a2a35
Cknpkqn k = n!k!(n k)!pkqn k
= n
ne np2 npkqn k
kke kp2 k(n k)n ke (n k)p2 (n k)
= 1p2
r n
k(n k)
np
k
k nq
n k
n k:
a36a2a37a2a38a2a39a2a40
a33a2a20a2a41a2a42a44a43
r n
k(n k) =
k(n k)
n
1
2
= (np + xk
pnpq)(nq x
k
pnpq)
n
1
2
= 1pnpq 1 + xk q ppnpq + O 1n 12
1pnpq:
1
Ch5 2
a45 ln(1 + x) = x x
2
2 + (x
3)
a12 j (x3)j6jx3j ca12a47a46
ln npk k nqn k n k = kln npk (n k)ln n knq
= kln 1 +
pnpqx
k
np
(n k)ln 1 pnpq
nq
= (np + xkpnpq)ln xk
r q
np
1
2x
2
k
q
np + o
1
n
(nq xkpnpq)ln xk
r p
nq
1
2x
2
k
p
nq + o
1
n
12x2k:
a48
a16 jxkj6 A a20 ka12a47a21a2a22a2a20a2a23
np
k
k nq
n k
n k e x2k
2,
a49a2a50a2a51 a26a2a52a2a53a2a54a2a55a2a56
a43
a27a29a57a2a58a2a59a2a60a2a61a2a62a2a63a2a64 a25a2a65a67a66a29a68a10a69
a20a2a70a10a71
a59a10a60a10a61a2a62a73a72
a11 X
n B(n; p)a12a47a15a2a16a2a74a2a75a2a20 1 < a < b < 1a76a2a23
lim
n!1
P a < Xn nppnpq 6 b = 1p2
Z b
a
e x2x dx:
a77
a21
a61a17a62a17a63a17a78a17a79a17a80a17a81a17a82 n
a83a85a84a17a86a17a20a17a87a17a88a17a20a17a89a17a90a17a71a17a91a2a20a93a92
a54
a12a95a94a2a96
a35 n
a97
a98a100a99
a71a100a91a100a20a17a101a17a20a17a71a17a91a102a92
a54
a43a103a16
a34
a21a17a104a2a20a17a71a17a91a44a12
a37 a68
a21a2a105a102a106a85a107a17a108
a66a29a109a17a110
a20a17a111a2a112a44a43
x5 a113a29a114
a3a2a4a2a5a2a6
a115a2a116 a30a2a117a2a118a2a119
a12
a51a2a120a2a121a2a122
a83a123a70a2a71
a59a2a60a10a61a2a62a10a124a2a125
a21a2a104a10a20
a59a10a60a2a61a10a62
a12a47a96
a35 a68
a69
a20a93a106a29a126
a59a2a60a2a61a2a62a127a72
a5a2a6 (Lindeberg Levy) a11 X
1; X2; ; Xn;
a35a2a128 a56a93a129
a71a2a91a2a20a10a130a10a131
a132a2a133a2a134
a12 EX1 = a12 VarX1 =
2 > 0
a12 Sn = X1 + + Xna12a135a15a2a16a2a74a2a75a2a20 x a76
a23
lim
n!1
P Sn n pn 2 6 x = (x):
2
Ch5 3
a136
a106a137a12 (x) a35a17a138a17a139a17a140a17a141 a71a10a91a2a20a10a71a2a91 a118a2a119 a43a47a94a2a96 a35a10a142 a12 Sn n pn 2 a143 a71a17a91a17a144a17a145
a122
a138a2a139a2a140a2a141
a71a2a91a44a43
a24
a13
Zn = Sn n pn 2 ;
a15
a52a2a53a2a54
a96
a35a2a146a10a147a149a148 Z
n
a143
a71a2a91a2a144a2a145
a122 a140a10a141
a71a10a91a73a12a150a94a10a96
a35a10a146a151a147a67a148 Z
n a20
a30a2a117a2a118a2a119
a144a2a145
a122 a138a2a139a2a140a10a141
a71a2a91a10a20
a30a10a117a10a118a2a119
a43
a152
Y = X1 ;
a13a2a153a2a20
a30a2a117a2a118a2a119a2a125 f(t)
a43
a27 X
1; ; Xn
a128 a56a93a129
a71a2a91
a26 Z
n a20
a30a2a117a2a118a2a119a2a125
fn(t) = f tpn n:
a27 a34 EX
1 = a12 VarX1 = 2a12a47a46 EY = 0a12 VarY = 1a43
a34a2a35
a96a2a23
f0(0) = iEY = 0; f"(0) = i2EY 2 = 1:
a27 Taylor
a32a2a33
f(t) = 1 t
2
2 + o(t
2);
a34a2a35
a96a2a23
lnfn(t) = nln 1 t
2
2n + o
t2
n
! t
2
2 (n !1);
a48
fn(t) ! e t
2
2,
a106a154a126
a59a151a60a151a61a151a62
a106
a128 a56a149a129
a71a155a91a151a20a155a156a155a157
a35 a63a155a158
a20
a146a155a159
a12
a68a155a69
a108
a66a154a160
a21a155a97
a61
a62
a43
a5a2a6 (Lindeberg Feller) a11 X
1; X2;
a35a2a161a2a162a2a128 a56
a20a2a130a2a131
a132a2a133
a12a135a13
ak = EXk; b2k = VarXk; B2n =
nX
k=1
b2k;
3
Ch5 4
a163a2a164 Lindeberg
a156a2a157
a55a2a56
a12
a48
a16a2a74a10a75a10a20 " > 0a12a47a76a2a23
lim
n!1
1
B2n
nX
k=1
Z
jx akj>"Bn
(x ak)2dFk(x) = 0;
a15a2a16a2a74a2a75a2a20 xa12a47a23
limn!1P 1B
n
nX
k=1
(Xk ak) 6 x = (x):
a37 X
n
a125 a161a2a162a2a128 a56
a20a2a130a10a131
a132a151a133a10a134
a20a10a165a10a166
a68
a12 Lindeberg a156a2a157a2a167a2a168a2a168
a35
a106
a126
a59a2a60a2a61a2a62 a55a2a56
a20a2a169a10a71a10a156a10a157a73a12a170a94a10a171a10a167a10a172
a35a10a173a10a146
a156a10a157a73a43a175a174a10a176a10a20
a142
a12
a146a10a177 a40 a68a2a69
a98
a97a2a156a2a157
a72
limn!1Bn = 1; limn! bnB
n
= 0:
4