Page 356 - Elements of Distribution Theory

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

342 Approximation of Probability Distributions

Proof. Fix and consider Pr(| f (X n ) − f (0)| < ). By the continuity of f , there exists a
δ such that | f (x) − f (0)| < whenever |x| <δ. Hence,
Pr(| f (X n ) − f (0)| < ) = Pr(|X n | <δ).

The result now follows from the fact that, for any δ> 0,
lim Pr(|X n | <δ) = 1.
n→∞
According to Theorem 11.2, a necessary and sufficient condition for convergence in
distribution is convergence of the characteristic functions. Since the characteristic function
p
of the random variable 0 is 1 for all t ∈ R,it follows that X n → 0as n →∞ if and only if
ϕ 1 ,ϕ 2 ,..., the characteristic functions of X 1 , X 2 ,..., respectively, satisfy
lim ϕ n (t) = 1, t ∈ R.
n→∞
Example 11.14 (Gamma random variables). Let X 1 , X 2 ,... denote a sequence of real-
valued random variables such that, for each n = 1, 2,..., X n has a gamma distribution
with parameters α n and β n , where α n > 0 and β n > 0; see Example 3.4 for further details
regarding the gamma distribution. Assume that
lim α n = α and lim β n = β
n→∞ n→∞
for some α, β.
Let ϕ n denote the characteristic function of X n . Then, according to Example 3.4,
log ϕ n (t) = α n log β n − α n log(β n − it), t ∈ R.
p
Hence, X n → 0as n →∞ provided that α = 0, β< ∞, and
lim α n log β n = 0.
n→∞

Example 11.15 (Weak law of large numbers). Let Y n , n = 1, 2,... denote a sequence of
independent, identically distributed real-valued random variables such that E(Y 1 ) = 0. Let

1
X n = (Y 1 +· · · + Y n ), n = 1, 2,....
n
The characteristic function of X n is given by

ϕ n (t) = ϕ(t/n) n
where ϕ denotes the characteristic function of Y 1 and, hence

log ϕ n (t) = n log ϕ(t/n).
Since E(Y 1 ) = 0, by Theorem 3.5,

ϕ(t) = 1 + o(t)as t → 0
and, hence,

log ϕ(t) = o(t)as t → 0;