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 -> ∞.

 

[blue_raver22]

Yes, there is a way of representing this sum by another random variable. This sum can be represented by a Normal random variable with mean 0 and variance n. This is because the sum of independent Normal random variables with mean 0 and variance 1 is also a Normal random variable with mean 0 and variance equal to the sum of the variances of the individual random variables. As n approaches infinity, the variance of this new Normal random variable also approaches infinity, representing the fact that the sum of an infinite number of Normal random variables becomes increasingly spread out.