Page 325 - Applied Probability

14. Poisson Approximation
314
10. Consider an infinite sequence W 1 ,W 2 ,... of independent, Bernoulli
random variables with common success probability p. Let X α be the
indicator of the event that a success run of length t or longer begins
t
W k and
at position α. Note that X 1 =

k=1
j+t−1

X j =(1 − W j−1 ) W k
k=j
for j> 1. The number of such success runs starting in the first n po-

sitions is given by S = X α , where the index set I = {1,...,n}.
α∈I
The Poisson heuristic suggests the S is approximately Poisson with
t
mean λ = p [(n−1)(1−p)+1]. Let B α = {β ∈ I : |β −α|≤ t}. Show
that X α is independent of those X β with β outside B α . In the Chen-
Stein bound (14.3), prove that the constant b 2 = 0. Finally, show
t
2
that the Chen-Stein constant b 1 ≤ λ (2t +1)/n +2λp for 1 <t ≤ n.
(Hint:
2t
2t
b 1 = p +2tp (1 − p)
2t
2
+[2nt − t + n − 3t − 1]p (1 − p) 2
exactly. Note that the pairs α and β entering into the double sum for
b 1 are drawn from the integer lattice points {(i, j): 1 ≤ i, j ≤ n}.An
upper left triangle and a lower right triangle of lattice points from
this square do not qualify for the double sum defining b 1. The term
p 2t in b 1 corresponds to the lattice point (1, 1).)
11. Let X 1 ,...,X n be n independent, exponentially distributed waiting
times with common intensity λ, and define M n = max 1≤i≤n X i . Show
that λM n − ln n converges in distribution to the extreme value sta-
tistic having density e −e −u e −u . (Hints: This assertion can be most
easily demonstrated by considering the moment generating function
)
of λM n −ln n. Since M n has density n(1−e −λx n−1 λe −λx , prove that

∞
)
E[e s(λM n −ln n) ]= e s(λx−ln n) n(1 − e −λx n−1 λe −λx dx
0

∞ 1
su
)
e
= e (1 − e −u n−1 −u du.
− ln n n
Argue that in the limit
∞ −u

su −e
lim E[e s(λM n −ln n) ]= e e e −u du
n→∞
−

∞
= w −s −w dw (14.9)
e
0
= Γ(1 − s),
where Γ(x) is the gamma function. )