summation of values of a uniform random variable

Question

Say we have a random variable Y belonging to {-1,1}. Each time an ideal random number code-simulation generates a value for Y, using "Uniform distribution", let us give that value a symbol yi. So, each yi belongs to [-1,1]. eg: 10 random numbers generated can be [-0.9, -0.892, 0.23,...etc].If I want to sum =∑yi for very large i, can I say that X goes to '0'? Intuitively I feel so, but mathematically, I do not know what it should be. Can anyone help derive an analytical form here? Thanks a lot.

Answer 1

Assume that $S_n=\sum_{i=1}^nY_i$ indeed "goes to $0$" if $n$ increases.

Then similarly it must be true that $S_n-Y_1=\sum_{i=2}^nY_i$ "goes to $0$" if $n$ increases.

This however implies that $Y_1=0$ which definitely does not have to be the case.

So the assumption must be wrong.