Probability and statistics

Question

Let Zn,n 2 1, be a sequence of random variables and ca constant such that for each ɛ > 0, P{|Z, – c| > ɛ}→ 0 as n → o. Show that for any bounded continuous function g, E[g(Zn)] → g(c) as n → 00

Answer 1

First let me apologize for the incomplete answer.

I have some thoughts

E[g(Z_n)] = ∫g(Z_n)f_n(Z_n)dZ_n,-∞, c-ε + ∫g(Z_n)f_n(Z_n)dZ_n,c-ε,c+ε + ∫g(Z_n)f_n(Z_n)dZ_n,c+ε, ∞

(The limits of integration follow each term)

I believe that when you now take the limit as n->∞ , you can show that the first and last terms approach 0. Also, f_n(Z_n)->1/(2ε) in the middle integral (I am unclear about how to bring the limit inside the integral, but I believe there is a theorem which can handle this.

Following this, I believe you can use the Law of Large Numbers

That's all I have for tonight. I will try to follow up later.

1. Jason