Suppose that $\theta\_1$ and $\theta\_2$ are independent and identically distributed (i.i.d.) random variables and that $\theta\_j$ has probability density function (PDF) $f\_j = \frac{1}{2\pi}$ ($i.e.$, the uniform distribution) for $j = 1$ and $2$. Next, we define the following random variables $C = \cos \theta\_1 + \cos \theta\_2$ and $S = \sin \theta\_1 + \sin \theta\_2$.


My question is, can I say, based on the rotational invariance, that the random point $(C,S)$ is the same as that of $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, where $U$ is uniformly distributed on $[0,2\pi]$? If so, how can I show that?



Is the random point $(C,S)$ the same as $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, with $U$ a uniform r.v.?
These two scatter plots illustrate the difference, the first is for the points $(C,S)=(\cos\theta\_1+\cos\theta\_2,\sin\theta\_1+\sin\theta\_2)$, the second for the points $(1+\cos U,\sin U)=(1+\cos\theta\_3,\sin\theta\_3)$, where all angles $\theta\_i$ are uniformly distributed in $(0,2\pi)$.


![](https://ilorentz.org/beenakker/MO/disc_scatter.png)
![](https://ilorentz.org/beenakker/MO/circle_scatter.png)


The second distribution uniformly fills a unit circle with center at $(1,0)$. The first distribution fills a disc with radius 2, but not uniformly, the density diverges as $1/r$ at a distance $r$ from the origin.


To calculate the radial distribution $P(r)$, it is helpful to think of a random walk on the plane with unit step size and random orientation. The desired $P(r)$ is the distribution of the distance from the origin after two steps. 


After the first step the random walker is at some arbitrary point on the unit circle. For the distribution $P(r)$ it does not matter where on the unit circle, we may place the point at $(1,0)$. Then the second step brings the random walker to the point $(1+\cos\phi,\sin\phi)$, with $\phi$ uniformly in $(0,2\pi)$. The distance from the origin is $r=\sqrt{2+2\cos\phi}$, and the probability distribution is
$$P(r)=\frac{2/\pi}{\sqrt{4-r^2}},\;\;0<r<2.$$
The corresponding density $\rho(r)=P(r)/2\pi r$ indeed diverges $\propto 1/r$ when $r\rightarrow 0$.


For the generalization to $(C\_n,S\_n)=(\sum\_{i=1}^n\cos\theta\_i,\sum\_{i=1}^n\sin\theta\_i)$ see this [MO posting.](https://mathoverflow.net/a/347336/11260)


