Page 346 - Elements of Distribution Theory

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

332 Approximation of Probability Distributions

Hence,
∞ ∞

E[g(X n + Z/t)] = g(x + z/t)φ(z) dz dF n (x)
−∞ −∞
1 ∞ ∞
= g(x + z/t)ϕ(z) dz dF n (x)
1
(2π) 2 −∞ −∞
1 ∞ ∞
= g(x + z/t)E[exp{izZ}] dz dF n (x)
1
(2π) 2 −∞ −∞
1 ∞ ∞ ∞
= g(x + z/t)exp{izy}φ(y) dy dz dF n (x).
1
(2π) 2 −∞ −∞ −∞
Consider the change-of-variable u = x + z/t. Then
1 ∞ ∞ ∞
E[g(X n + Z/t)] = g(u)exp{iyt(u − x)}φ(y) dy du dF n (x)
1
t(2π) 2 −∞ −∞ −∞
1 ∞ ∞ ∞
= g(u)exp{−iytu}φ(y) exp{−iytx}dF n (x) dy du
1
t(2π) 2 −∞ −∞ −∞
1 ∞ ∞
= g(u)exp{−ityu}ϕ(y)ϕ n (−ty) dy du.
1
t(2π) 2 −∞ −∞
Similarly,
1 ∞ ∞
E[g(X n + Z/t)] = g(u)exp{−ityu}ϕ(y)ϕ(−ty) dy du.
1
t(2π) 2 −∞ −∞
By assumption,
lim ϕ n (−ty) = ϕ(−ty) for all −∞ < y < ∞.
n→∞
Since ϕ n (−ty)is bounded, it follows from the dominated convergence theorem (considering
the real and imaginary parts separately) that
lim E[g(X n + Z/t)] = E[g(X + Z/t)] for all t > 0.
n→∞
Hence, for sufficiently large t and n,

|E[g(X n ) − g(X)]|≤ 5 .

Since is arbitrary, the result follows.

Example 11.7. Let X 1 , X 2 ,... denote a sequence of real-valued random variables such
D
that X n → X as n →∞ for some random variable X. Let Y denote a real-valued random
variable such that, for each n = 1, 2,..., X n and Y are independent.
Let ϕ n denote the characteristic function of X n , n = 1, 2,..., let ϕ X denote the char-
acteristic function of X, and let ϕ Y denote the characteristic function of Y. Then X n + Y
D
has characteristic function ϕ n (t)ϕ Y (t). Since X n → X as n →∞,it follows from Theorem
11.2 that, for each t ∈ R, ϕ n (t) → ϕ X (t)as n →∞. Hence,

lim ϕ n (t)ϕ Y (t) = ϕ X (t)ϕ Y (t), t ∈ R.
n→∞