5.2. Convergence in Distribution 331Remark 5.2.2(Stirling’s Formula).In advanced calculus the following approxi-
mation is derived:
Γ(k+1)≈
√
2 πkk+1/^2 e−k. (5.2.2)
This is known asStirling’s formulaand it is an excellent approximation whenkis
large. Because Γ(k+1) =k!, forkan integer, this formula gives an idea of how fast
k! grows. As Exercise 5.2.21 shows, this approximation can be used to show that
the first limit in Example 5.2.3 is 1.
Example 5.2.4(Maximum of a Sample from a Uniform Distribution, Continued).
Recall Example 5.1.2, whereX 1 ,...,Xnis a random sample from a uniform(0,θ)
distribution. Again, letYn=max{X 1 ,...,Xn}, but now consider the random
variableZn=n(θ−Yn). Lett∈(0,nθ). Then, using the cdf ofYn, (5.1.1), the cdf
ofZnis
P[Zn≤t]=P[Yn≥θ−(t/n)]=1−(
θ−(t/n)
θ)n=1−(
1 −t/θ
n)n→ 1 −e−t/θ.Note that the last quantity is the cdf of an exponential random variable with mean
θ, (3.3.6), i.e., Γ(1,θ). So we say thatZn
D
→Z,whereZis distributed Γ(1,θ).
Remark 5.2.3.To simplify several of the proofs of this section, we make use of
the limandlim of a sequence. For readers who are unfamiliar with these concepts,
we discuss them in Appendix A. In this brief remark, we highlight the properties
needed for understanding the proofs. Let{an}be a sequence of real numbers and
define the two subsequences
bn =sup{an,an+1,...},n=1, 2 , 3 ..., (5.2.3)
cn =inf{an,an+1,...},n=1, 2 , 3 .... (5.2.4)The sequences{bn}and{cn}are nonincreasing and nondecreasing, respectively.
Hence their limits always exist (may be±∞) and are denoted respectively by
limn→∞anand limn→∞an.Further,cn≤an≤bn, for alln. Hence, by the Sand-
wich Theorem (see Theorem A.2.1 of Appendix A), if limn→∞an=limn→∞an,
then limn→∞anexists and is given by limn→∞an=limn→∞an.
As discussed in Appendix A, several other properties of these concepts are useful.
For example, suppose{pn}is a sequence of probabilities andlimn→∞pn= 0. Then,
by the Sandwich Theorem, since 0≤pn≤sup{pn,pn+1,...}for alln,wehave
limn→∞pn= 0. Also, for any two sequences{an}and{bn}, it easily follows that
limn→∞(an+bn)≤limn→∞an+limn→∞bn.
As the following theorem shows, convergence in distribution is weaker than con-
vergence in probability. Thus convergence in distribution is often called weak con-
vergence.