Page 352 - Elements of Distribution Theory

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

338 Approximation of Probability Distributions

D 2 2
number α, then X n → N(0, 1 + α )as n →∞. Furthermore, since for any (t 1 , t 2 ) ∈ R ,
t 1 X n + t 2 Y n = t 1 Z 1 + (t 2 + α n t 1 )Z 2 has characteristic function
2 2 2
exp{−[t + (t 2 + α n t 1 ) ]s /2}, s ∈ R,
1
which converges to
2
2
2
exp{−[t + (t 2 + αt 1 ) ]s /2}, s ∈ R,
1
it follows that t 1 X n + t 2 Y n converges in distribution to a random variable with a normal
2
2
distribution with mean 0 and variance t + (t 2 + αt 1 ) . Hence, by Theorem 11.6,
1

X n D
→ W,
Y n
where W has a bivariate normal distribution with mean vector 0 and covariance matrix
2
1 + α α
.
α 1
D
n
Now suppose that α n = (−1) , n = 1, 2,.... Clearly, Y n → Z 2 as n →∞ still holds.
n
Furthermore, since X n = Z 1 + (−1) Z 2 ,it follows that, for each n = 1, 2,..., X n has a
D
normal distribution with mean 0 and variance 2; hence, X n → N(0, 2) as n →∞.How-
n
ever, consider the distribution of X n + Y n = Z 1 + (1 + (−1) )Z 2 . This distribution has
characteristic function
2
exp(−s /2) for n = 1, 3, 5,...
n 2 2
exp{−[1 + (1 + (−1) ) ]s /2}= 2 , s ∈ R.
exp(−5s /2) for n = 2, 4, 6,...
Hence, by Theorem 11.2, X n + Y n does not converge in distribution so that, by
Theorem 11.6, the random vector (X n , Y n ) does not converge in distribution.
11.3 Convergence in Probability
A sequence of real-valued random variables X 1 , X 2 ,... converges in distribution to a ran-
dom variable X if the distribution functions of X 1 , X 2 ,... converge to that of X.Itis
important to note that this type of convergence says nothing about the relationship between
the random variables X n and X.
Suppose that the sequence X 1 , X 2 ,... is such that |X n − X| becomes small with high
probability as n →∞;in this case, we say that X n converges in probability to X. More
precisely, a sequence of real-valued random variables X 1 , X 2 ,... converges in probability
to a real-valued random variable X if, for any > 0,

lim Pr(|X n − X|≥ ) = 0.
n→∞
p
We will denote this convergence by X n → X as n →∞. Note that for this definition to
make sense, for each n, X and X n must be defined on the same underlying sample space, a
requirement that did not arise in the definition of convergence in distribution.

Example 11.11 (Sequence of Bernoulli random variables). Let X 1 , X 2 ,... denote a
sequence of real-valued random variables such that
Pr(X n = 1) = 1 − Pr(X n = 0) = θ n , n = 1, 2,...