Page 271 - Probability and Statistical Inference

248 5. Concepts of Stochastic Convergence

The proof in this case is similar to the one in the previous case and hence
it is left out as the Exercise 5.2.6.
Case 4: u ≠ 0, v ≠ 0
From (5.2.9), using similar arguments as before, we observe that

In the last step in (5.2.10), the first and third terms certainly converge to zero
since , ≠ 0 as n → ∞. For the middle term in the last step
of (5.2.10), let us again use the triangular inequality and proceed as follows:

which implies that P{| U | > 2 | u |} → 0 as n → ∞, since as n
n
→ ∞. Hence the lhs in (5.2.10), converges to zero as n → ∞. This completes
the proof of part (ii). "
(iii) In view of part (ii), we simply proceed to prove that as n
→ ∞. Note that | V – v | ≥ | v | – | V | which implies that
n n

Observe that the lhs of (5.2.11) converges to unity as n → ∞ because
as n → ∞. This implies

Let us denote W = | V – v | {| V || v |} . Then, for any arbitrary but
–1
n
n
n
otherwise fixed ε(> 0), we can write