According to the WLLN (Application 1 to Theorem 9), if the r.v.s X 1 ,...,X n are
Question:
According to the WLLN (Application 1 to Theorem 9), if the r.v.s X1,...,Xn are i.i.d. with finite ԐX1, then
The following example shows that it is possible for {Sn/n} to converge in probability to a finite constant, as n → ∞, even if the ԐX1 does not exist. To this effect, for j = 1, 2,..., let Xj be i.i.d. r.v.s such that P(Xj = -n) = P(Xj = n) = c/n2 log n, n ≥ 3, where
Then show that ԐX1 does not exist, but
Where
Show that ԐX1 does not exist by showing that ԐX1+ = ԐX1- = ∞. Next, set Xnj = Xj if |Xj| < n, and Xnj = 0 otherwise, j = 1,2…> 3, and let
Then show that (i)
by showing that
(ii) Ԑ(S*n/n) = 0
(iii)
(iii), conclude that
Then (i) and (iv) complete the proof.
In all Exercises 11-16, i is to be treated as a real number, subject, of course, to the requirement that i2 = -1
Step by Step Answer:
An Introduction to Measure Theoretic Probability
ISBN: 978-0128000427
2nd edition
Authors: George G. Roussas