PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
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1. DISTRIBUTION THEORY<br />
Consider the second-order Taylor expansion,<br />
M U (t) = M U (0) + M ′ U(0)t + M ′′<br />
U(0) t2 2 + r(t)<br />
M U (t) = 1 + 0 + t 2 /2 + r(t),<br />
↓<br />
(M U (0))(as M ′ U(0) = 0)<br />
r(s)<br />
where lim = 0<br />
s→0 s 2<br />
⇒<br />
= 1 + t 2 /2 + r(t).<br />
M Zn (t) = { M U (t/ √ n) } n<br />
= {1 + t 2 /2n + r(t/ √ n)} n<br />
= (1 + a n ) n ,<br />
where<br />
a n = t2<br />
2n + r(t/√ n).<br />
Next observe that lim na n = t2<br />
n→∞ 2<br />
[<br />
To check this observe that:<br />
for fixed t.<br />
nt 2<br />
lim<br />
n→∞ 2n = t2 2<br />
and lim<br />
n→∞<br />
nr(t/ √ n)<br />
= lim<br />
n→∞<br />
t 2 r(t/ √ n)<br />
(t/ √ n) 2<br />
= t 2 r(s)<br />
lim = 0 for fixed t.<br />
s→0 s 2<br />
Note: s = t/ √ n → 0 as n → ∞. ] 75