any ideas for this problem !!

arindam · science forum beginner Joined: 04 Oct 2005 Posts: 2

Hi all,

Any help with this problem will be highly appreciated.

Let X_n be a sequence of random variables such that X_n --> N(0,1)
weakly (in distribution). The claim is:

sup |P(X_n <= t) - \Phi(t)| = sup |P(X_n < t) - \Phi(t)|
t t

where, the supremum is over all t in R (real line) and \Phi(t) is the
standard normal cdf at t.

I am suprised to think that this holds true for all n>=1. Any help or
counterexample will be useful.

Thanks in advance,

Arindam

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