Generate random movement for a point in 3D space

I want to simulate a point that moves with random vibration around a mean position (let's say around the position [X, Y, Z] = [0,0,0]). The first solution that I found is to sum a couple of sinusoids for each axis based on the following equation:

<a href="https://www.codecogs.com/eqnedit.php?latex=\sum_{i&space;=&space;1}^n&space;A_i&space;\sin(\omega_i&space;t&plus;\phi)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\sum_{i&space;=&space;1}^n&space;A_i&space;\sin(\omega_i&space;t&plus;\phi)" title="\sum_{i = 1}^n A_i \sin(\omega_i t+\phi)" /></a>

where A_i is a normal random amplitude, and omega_i is a normal random frequency. I have not tested the phase yet, so I leave it to zero for now. I generated figures of the expect normal distribution and equation results with the following approach. I tried multiple values of N and I'm not sure that the equation is giving a normally distributed results. Is my approach correct? Is there a better way to generate random vibration?