Use Ferguson’s Theorem 3 (A course in Large Sample Theory)to show that if Xn converges in distribution to X, then for any ε > 0, there exists a number B and some integer N such that if n ≥ N, then
Pr[|Xn| ≥ B] < ε.
The property you are showing here is ”tightness”. Hint: we discussed the fact that if E[g(Xn)] converges to a number for any g in a separating class and if the sequence Xn is tight, then there exist a random variable X such that Xn converges in law to X. If we know there is a random variable X for which E[g(Xn)] converges to E[g(X)] for all g in a separating class, then Xn converges in law to X. Theorem 3 is basically identifying three separating classes. Also, we did not prove the results for separating classes. Recall a separating class is a collection of functions g such that Eg(X) = Eg(Y ) for all g implies that X and Y have the same distribution.
Let the sequence {xn} denote a listing of the rational numbers in (0,1). Let X define a discrete random variable with Pr[X = xn] = pn. For concreteness you can assume that pn = 1/2n but the arguments do not depend on that assumption in any crucial way. F(u) denotes the cdf of X. Also let the 0 < x0 < 1 be an irrational number. (a) Regardless of the probability distribution, show that for any N there exists δ > 0 such that |xi − x0 | > δ for i = 1, . . . , N .
(b) Showthatforanyε>0,thereexistsδ>0suchthatPr[|X−x0|<δ]<ε. Hint: Pick N so that there is very little probability close to x0.
(c) With δ defined as above, show that if x < y with both in (x0 −δ,x0 +δ), then F (y) − F (x) < ε
(d) Show that F(u) is continuous at x0.
This is weird stuff! Every open interval around x0 contains an infinite number of points with positive probability and thus the cdf has an infinite number of discontinu- ities in the interval. But still the cdf satisfies the definition of continuity at x0.
