Page 276 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 253

5.3 Convergence in Distribution
Definition 5.3.1 Consider a sequence of real valued random variables {U ; n
n
≥ 1} and another real valued random variable U with the respective distribu-
tion functions F (u) = P(U ≤ u), F(u) = P(U ≤ u), u ∈ ℜ. Then, U is said to
n n n
converge in distribution to U as n → ∞, denoted by , if and only if
F (u) → F(u) pointwise at all continuity points u of F(.). The distribution of
n
U is referred to as the limiting or asymptotic (as n → ∞) distribution of U
n
In other words, means this: with every fixed u, once we compute
P(U ≤ u), it turns out to be a real number, say, a . For the convergence in
n
n
distribution of U to U, all we ask for is that the sequence of non-negative real
n
numbers a → a as n → ∞ where a = P(U ≤ u), u being any continuity point
n
of F(.).
It is known that the totality of all the discontinuity points of any
df F can be at most countably infinite. Review Theorem 1.6.1
and the examples from (1.6.5)-(1.6.9).

Example 5.3.1 Let X , ..., X be iid Uniform (0, θ) with θ > 0. From
1
n
Example 4.2.7, recall that the largest order statistic T = X has the pdf given
n
n:n
by g(t) = nt θ I(0 < t < θ). The df of T is given by
n–1
–n
n
Now let U = n(θ – T )/θ. Obviously, P(U > 0) = 1 and from (5.3.1) we can
n
n
n
write:

so that

since m(1 – u/n) = e . See (1.6.13) for some of the results on limits.
n
–u
Now consider a random variable U with its pdf given by f(u) = e I(u > 0), so
–u
that U is the standard exponential random variable, while its distribution func-
tion is given by