Mathematical backing for observations seen in adding independent random variables together

Question

I have a function $Y = F(N)$ that takes as an argument an integer number $N$ and returns a summation of $N$ sine-waves of different random parameters. I have plotted the results of two function calls where $N = 10$ and $N = 20$ below.

(Sorry don't have enough "reputation" to embed images!)

$N = 10$:

$N = 20$:

What I have observed using this function and from my general experience is that as more signals or independent random variables are added together the resulting summation has higher peaks and lower troughs.

In this case when $N=10$ was called the resulting summation had a max of $41$ and a min of $-39$, however, when $N=20$ was called the resulting summation had a max of $52$ and a min of $-50$.

Is there/What is the mathematical backing or proof that this phenomena is real and consistent? For example, does the central limit theorem say anything about the variance of the distribution as $N\to\infty$?

Answer 1

This will entirely depend on the way you create your random sine-waves. For example, if you choose the correct parameters the limiting distribution is a brownian bridge.

Answer 2

As Dominik said, the result will depend on what you mean by "random parameters". If $\lambda_1, \ldots, \lambda_n$ are a random sample from a continuous distribution, with probability $1$ they and $1$ are linearly independent over the rationals, and then by Kronecker's theorem the $n$-tuples $ [\sin(k \lambda_1), \ldots \sin(k \lambda_n)]$, $k \in \mathbb N$, are dense in $[-1,1]^n$. In particular, almost surely $\sum_{j=1}^n \sin(x \lambda_j)$ takes values arbitrarily close to $-n$ and to $n$. Of course you may need to look at very large $x$ to see such values.