Random Variable Convergence

X_{1},...X_{n }are a random sample from U(0,theta). A sequence of random variables are defined Z_{n}=n(theta-X_{(n)}) for positive n.

I need to show that Z_{n} converges in distribution to Z, where Z~Exp(theta).

Any help in getting started?

Can I say that X_{(n)} converges in law to theta and is that useful?

Re: Random Variable Convergence

Hey Mick.

I'm wondering if you can use the distribution of the order statistic to show it converges to the right distribution.

One approach I would try is to use either the characteristic function or the moment generating function to show that the MGF has the form of an exponential distribution MGF.

Re: Random Variable Convergence

I can definetly go down the MGF route. The MGF of an Exponential(theta) random variable is 1/(1-theta*t).

The MGF of a U(0, theta) random variable is (exp(theta*t)-1)/(theta*t). I'm guessing the uniform MGF will be needed somewhere but I'm not sure how the MGF of X applies to the MGF of Z.

Re: Random Variable Convergence

I have two suggestions.

The first is to get the PDF of X(n) and then use a transformation to get the PDF of the new random variable.

The second is to get the MGF of the final random variable and prove it has the form of an exponential with that parameter.

I would personally be inclined to use the first option after thinking about it, but you could really try any (including using the characteristic function which is a third option).