Page 444 - A First Course In Stochastic Models

A. USEFUL TOOLS IN APPLIED PROBABILITY 439

Theorem A.1 Let a nm , n, m = 0, 1, . . . be real numbers. If all the numbers a nm

∞
∞
are non-negative or if n=0 m=0 |a nm | < ∞, then
∞ ∞ ∞ ∞

a nm = a nm .
n=0 m=0 m=0 n=0
This theorem is a special case of what is known as Fubini’s theorem in analysis.

Theorem A.2 Let {p m , m = 0, 1, . . . } be a sequence of non-negative numbers.
Suppose that the numbers a nm , n, m = 0, 1, . . . are such that

lim a nm = a m
n→∞
exists for all m = 0, 1, . . . .

(a) If all numbers a nm are non-negative, then
∞ ∞

lim inf a nm p m ≥ a m p m .
n→∞
n=0 m=0
(b) If there is a finite constant M > 0 such that |a nm | ≤ M for all n, m and if

∞
m=0 p m < ∞, then
∞ ∞

lim a nm p m = a m p m .
n→∞
m=0 m=0
The first part of the theorem is a special case of Fatou’s lemma and the second
part of the theorem is a special case of the bounded convergence theorem.
The above theorems can be stated in greater generality. For example, a more
general version of the bounded convergence theorem is as follows. Let {X n } be a
sequence of random variables that converge with probability 1 to a random variable
X. Then
lim E(X n ) = E(X)
n→∞
provided that |X n | ≤ Y, n ≥ 1, for some random variable Y with E(Y) < ∞.
Recall that convergence with probability 1 means that
P {ω ∈ : lim X n (ω) = X(ω)} = 1,
n→∞
where is the common sample space of the random variables X n , n ≥ 1, and the
random variable X. Often one uses the term ‘almost sure convergence’ instead of
the term ‘convergence with probability 1’.
Finally, we mention the important concept of the Cesaro limit. A sequence

n
{a n , n ≥ 1} of real numbers is said to have a Cesaro limit if lim n→∞ (1/n) k=1 k
a
exists. A sequence {a n } may have a Cesaro limit while the ordinary limit does