Theory of Statistics - George Mason University

1.3 Sequences of Events and of Random Variables 93
large. It may be useful to magnify the difference Xn − bn by use of some
normalizing sequence of constants an:
Yn = an(Xn − bn). (1.190)
While the distribution of the sequence {Xn − bn} may be degenerate, the
sequence {an(Xn − bn)} may have a distribution that is nondegenerate, and
this asymptotic distribution may be useful in statistical inference. (This approach
is called “asymptotic inference”.) We may note that even though we
are using the asymptotic distribution of {an(Xn −bn)}, for a reasonable choice
of a sequence of normalizing constants {an}, we sometimes refer to it as the
asymptotic distribution of {Xn} itself, but we must remember that it is the
distribution of the normalized sequence, {an(Xn − bn)}.
The shift constants generally serve to center the distribution, especially
if the limiting distribution is symmetric. Although linear transformations are
often most useful, we could consider sequences of more general transformations
of Xn; instead of {an(Xn −bn)}, we might consider {hn(Xn)}, for some
sequence of functions {hn}.
The Asymptotic Distribution of {g(Xn)}
Applications often involve a differentiable Borel scalar function g, and we may
be interested in the convergence of {g(Xn)}. (The same general ideas apply
when g is a vector function, but the higher-order derivatives quickly become
almost unmanageable.) When we have {Xn} converging in distribution to
X + b, what we can say about the convergence of {g(Xn)} depends on the
differentiability of g at b.
Theorem 1.46
Let X and {Xn} be random variables (k-vectors) such that
an(Xn − bn) d → X, (1.191)
where b1, b2, . . . is a sequence of constants such that limn→∞ bn = b < ∞, and
a1, a2, . . . is a sequence of constant scalars such that limn→∞ an = ∞ or such
that limn→∞ an = a > 0. Now let g be a Borel function from IR k to IR that is
continuously differentiable at each bn. Then
an(g(Xn) − g(bn)) d → (∇g(b)) T X. (1.192)
Proof. This follows from a Taylor series expansion of g(Xn) and Slutsky’s
theorem.
A common application of Theorem 1.46 arises from the simple corollary for
the case when X in expression (1.191) has the multivariate normal distribution
Nk(0, Σ) and ∇g(b) = 0:
Theory of Statistics c○2000–2013 James E. Gentle