Question

Consider the probability space ([0, 1], B, IP), where P is uniform measure. Let X nlo,i/n). Determine which of the following statements hold. In each case, use the appropriate definition to verify your answer (a) E(X,] → 0 as n → oo (b) Xn →d 0 as n → oo (c) Xn, 0 as noo

Answer 1

E(X(n))=

\

1 / ·n 7t

This converges to 0 as n tends to infinity. Hence a is True.

b. X(n) → 0 in distribution if

lim Fn(x) converges to F(x) as n tends to infinity. F(x) = 0 if x<0 and =1 if x>=0 in this case.

Now,

Fn(x) =

n rdr

which equals n* x^2/2 if 0<x<1/n and 1 if x>1/n

as n tends to infinity x^2 converges to 0 at a much faster rate and hence Fn(x) converges to F(x). So b holds.

3. Since X(n) goes in distribution to a constant. So X(n) will also go in probability to 0. Hence c is true. In general

if X(n) goes in distribution to c where c is cont. then X(n) goes in probability to c.

4.X(n) will converge almost surely to 0 as we can define a R.V. X such that X is degenerated at 0. Xn follows U(0,1/n).

Thus as n tends to infinity Xn tends to degenerate at 0. hence d is also true.