Page 190 - Elements of Distribution Theory

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176 Stochastic Processes

knowing the limit by showing that, roughly speaking, the terms in the sequence become
closer together as one moves down the sequence. See Section A3.2 in Appendix 3 for
further details.
The same type of approach may be used to establish convergence in mean square of a
sequence of random variables. Theorem 6.4 gives such a result; as in the case of convergence
of a sequence of real numbers, the advantage of this result is that it may be shown that a
limiting random variable exists without specifying the distribution of that random variable.

Theorem 6.4. Let Y 0 , Y 1 ,... denote a sequence of real-valued random variables such that
2
E(Y ) < ∞ for all j = 0, 1,.... Suppose that for every > 0 there exists an N > 0 such
j
that
2
E[(Y m − Y n ) ] <
for all m, n > N; in this case we say that Y n ,n = 0, 1,..., is Cauchy in mean square.
If Y n ,n = 0, 1,..., is Cauchy in mean square, then there exists a random variable Y
2
with E(Y ) < ∞ such that
2
lim E[(Y n − Y) ] = 0.
n→∞
Proof. Fix j = 1, 2,.... Since Y 0 , Y 1 ,... is Cauchy in mean square, we can find m, n,
m > n, such that
1
2
E[(Y m − Y n ) ] ≤ j .
4
Thus, we can identify a subsequence n 1 , n 2 ,... such that
1
2
) ] ≤ , j = 1, 2,....
E[(Y n j+1 − Y n j j
4
Define
m

T m = |Y n j +1 − Y n j |, m = 1, 2,....
j=1
Let denote the sample space of the underlying experiment so that, for each m =
1, 2,..., T m ≡ T m (ω), ω ∈ . Note that the terms in the sum forming T m are all nonnegative
so that, for each ω ∈ , either lim m→∞ T m (ω)exists or the sequence diverges. Define a
random variable T by
T (ω) = lim T m (ω),ω ∈
m→∞
if the limit exists; otherwise set T (ω) =∞.
2
Note that for any real-valued random variables Z 1 , Z 2 such that E(Z ) < ∞, j = 1, 2,
j

1 1 2 1 1
2 2 2 2 2
E Z 2 + E Z 2 − E[(Z 1 + Z 2 ) ] = 2E Z 2 E Z 2 − 2E(Z 1 Z 2 ) ≥ 0,
j 2 1 2
by the Cauchy-Schwarz inequality. Hence,
1
2

E[(Z 1 + Z 2 ) ] 2 ≤ E Z 2 1 2 + E Z 2 1 2 .
1 2