Page 279 - Probability and Statistical Inference

256 5. Concepts of Stochastic Convergence

is whether , some appropriate random variable, as n → ∞. For
, let us start with the mgf M (t) and write
n

Here, we use the form of the mgf of from (2.3.28), namely, E {exp (sX )}
n
= (1 – 2s) –n/2 for s < ½. Hence, with we obtain

Thus, for large n, we can write

Here, we used the series expansion of log(1 – x) from (1.6.15). We have thus
proved that M (t) → exp(½t ) as n → ∞. Hence, by the Theorem 5.3.1, we
2
n
2
conclude that as n → ∞, since M(t) = exp(½t ) hap-
pens to be the mgf of the standard normal distribution. !
5.3.1 Combination of the Modes of Convergence
In this section, we summarize some important results without giving their
proofs. One would find the proofs in Sen and Singer (1993, Chapter 2). One
may also refer to Serfling (1980) and other sources.
Theorem 5.3.2 Suppose that a sequence of real valued random variables
U converges in probability to another real valued random variable U as n →
n
∞. Then, as n → ∞.
The converse of Theorem 5.3.2 is not necessarily true.
That is, it is possible to have , but as n → ∞.
Refer to Exercise 5.3.14.