Page 273 - Probability and Statistical Inference

250 5. Concepts of Stochastic Convergence

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continuous function g(x) = x for x > 0. Similarly, for example, as n
→ ∞, by considering the continuous function . By the

same token, we can also claim, for example, that sin as
n → ∞. !

Example 5.2.9 Let X , ..., X be iid Poisson (λ) with λ > 0 and look at ,
1 n
the sample mean. By the Weak WLLN, we immediately claim that as
n → ∞. By combining the results from the two Theorems 5.2.4 and 5.2.5, it
also immediately follows, for example, that

But then can one also claim that

Before jumping into a conclusion, one should take into consideration the fact
that 0, which is positive for every fixed
n ≥ 1 whatever be λ(> 0). One can certainly conclude that

We ask the reader to sort out the subtle difference between the question raised
in (5.2.15) and the statement made in (5.2.16). !
Under fair bit of generality, we may claim that as n → ∞.
But, inspite of the result in the Theorem 5.2.4, part (iii), we may not
be able to claim that as n → ∞ when α > 0.
The reason is that µ may be zero or P( = 0) may be positive for all n.
See the Example 5.2.9.
Example 5.2.10 Let X , ..., X be iid random variables with E(X ) = µ and
1 n 1
γ
V(X ) = σ , –∞ < µ < ∞, 0 < σ < ∞. Define U = n with 0 < γ <
2
1 n
1. Note that U > 0 w.p. 1. Now, for any arbitrary ε(> 0), by the Markov
n
inequality, we observe that
Hence, by the Definition 5.2.1, as n → ∞. !