Page 347 - Elements of Distribution Theory

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052184472Xc11 CUNY148/Severini May 24, 2005 17:56

11.2 Basic Properties of Convergence in Distribution 333

It follows that from Theorem 11.2
D
X n + Y → X + Y 0 as n →∞,
where Y 0 denotes a random variable that is independent of X and has the same marginal
distribution as Y.

Example 11.8 (Normal approximation to the Poisson distribution). Let Y 1 , Y 2 ,... denote
asequenceofreal-valuedrandomvariablessuchthat,foreachn = 1, 2,...,Y n hasaPoisson
distribution with mean n and let
Y n − n
X n = √ , n = 1, 2,....
n
Since the characteristic function of a Poisson distribution with mean λ is given by

exp{λ[exp(it) − 1]}, −∞ < t < ∞,
it follows that the characteristic function of X n is
√ √
ϕ n (t) = exp{n exp(it/ n) − n − nit}.
By Lemma A2.1 in Appendix 2,
n j
(it)
exp{it}= + R n (t)
j!
j=0
where
n+1 n
|R n (t)|≤ min{|t| /(n + 1)!, 2|t| /n!}.
Hence,
√ √ 1 2
exp(it/ n) = 1 + it/ n − t /n + R 2 (t)
2
where
1 3 3
|R 2 (t)|≤ t /n 2 .
6
It follows that
2
ϕ n (t) = exp{−t /2 + nR 2 (t)}
and that

lim nR 2 (t) = 0, −∞ < t < ∞.
n→∞
Hence,
2
lim ϕ n (t) = exp(−t /2), −∞ < t < ∞,
n→∞
the characteristic function of the standard normal distribution.
Let Z denote a random variable with a standard normal distribution. Then, by Theo-
D
rem 11.2, X n → Z as n →∞.
Thus, probabilities of the form Pr(X n ≤ z) can be approximated by Pr(Z ≤ z); these
approximations have the property that the approximation error approaches 0 as n →∞.