How Can the Sum of Independent Normal Random Variables Be Represented?

[Apteronotus]

For random variables Xk~N(0,1) is there any way of representing the following sum by another random variable?
$$lim_{n\rightarrow \infty}\sum_{k=0}^n X_k$$

thanks

 

[D H]

What does the central limit theorem have to say about this question?

 

[Apteronotus]

What does the central limit theorem have to say about this question?

Thanks DH for your reply, but I've already taken the CLT into consideration. However, since there is no normalizing factor (1/n) in front of the sum I didn't think it applied.

Does it?

 

[mathman]

For random variables Xk~N(0,1) is there any way of representing the following sum by another random variable?
$$lim_{n\rightarrow \infty}\sum_{k=0}^n X_k$$

thanks

The random variable for sum to n is normal with a variance = n. The distribution function does not converge to anything as n -> ∞.