Re: Normal distribution and convergence help please

In article <[EMAIL PROTECTED]>,
 "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)

S_n has the normal distribution with mean 0 and variance n.

Therefore,

 E[Z_n] = exp(a^2*n/2 - bn),

and more generally,

 E[(Z_n)^p] = exp(a^2p^2*n/2 - bpn).

 
> Show that Z_n ---> 0  with probability 1 if and only if b > 0, and show that
> for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a

It should now be clear that E[(Z_n)^p) --> 0 if and only if
a^2p < 2b.

Also, that Z_n --> 0 w.p. 1 if and only if a^2 < 2b.

-- 
A.