Re: Normal distribution and convergence help please

"David C. Ullrich" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]
> On Tue, 2 Dec 2003 23:39:17 -0500, "Steven Rossi" <[EMAIL PROTECTED]>
> wrote:
>
> >Dear Math experts,
> >
> >How can I prove this?
> >
> >Suppose that {X_i  : 1 <= i < infinity} is a sequence of independent
random
> >variables with the standard normal distribution.  Let S_n = X_1 + X_2 +
...
> >+ X_n and let
> >Z_n = exp(a*S_n - bn)
> >
> >Show that Z_n ---> 0  with probability 1 if and only if b > 0,
>
> If I recall what the "Law of the Iterated Logarithm" says correctly
> then this follows immediately from that. (No doubt it also follows
> by easy arguments from the Central Limit Theorem, but if I'm
> recalling LIL correctly then there's no argument needed.)
>

I guess my problem is that I don't know the Law of Iterated Logarithms and
would much rather
solve this using CLT arguments....how could I go about this?  It seems like
a variation on CLT but
what kind of variation?  I just don't see it...

Steven


> >and show that
> >for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
>
> Well, you have a product of independent random variables;
> the expected value of the product is the product of the
> expected values...
>
> >
> >
> >Thank you very very much for your help,
> >
> >Steven Rossi
> >
>
> ************************
>
> David C. Ullrich