I am currently starting to read through "Fundamentals of statistical and thermal physics" by Federick Reif and I got to chapter 1.10 - Calculation of the probability density function.
In the context of the random walk problem Suppose the total displacement $x$ is given by:
$$ \sum^N_{i=1}s_i $$
where $si$ is the displacement of the $i$th step that follows a probability density function $\omega (s_i)$. Then the probability of a certain sequence of steps being taken is given by:
$$ \prod_{i=1}^N \omega(s_i)s_i $$
and the probability of the total displacement $x$ lies in the range of $x$ and $x+dx$ is given by:
$$ P(x)dx=\prod_{i=1}^N \int\omega(s_i)s_i $$
where the limits of integration are subjected to the restraint $x<\sum^N_{i=1}<x+dx$. Now this mess with the limits of integration is solved with the dirac delta function by:
$$ P(x)dx=\prod_{i=1}^N \int\omega(s_i)s_i \delta(x-\sum^N_{j=1}s_j)dx $$
and this seems reasonable. But suppose I wanted to do a particular case of just two steps and not think about a random walk and just two different, independent random variables $s_1$ and $s_2$ and $x=s_1+s_2$. Now the problem is, what is the pdf of $x$?
If I just used the previous solution I would write:
$$ p(x)dx=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\omega(s_1)\omega(s_2)\delta(x-(s_1+s_2))ds_1ds_2 $$
which upon rearranging:
$$ p(x)dx=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\omega(s_1)\omega(s_2)\delta(s_1-(x-s_2))ds_1ds_2dx $$
and integrating in $s_1$:
$$ p(x)dx=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\omega(x-s_2)\omega(s_2)ds_2=\omega(s_1)\circledast \omega(s_2)dx $$
hence $p(x)=\omega(s_1)\circledast \omega(s_2)$ and this is coherent with the fourier transform part that comes next on the book, where the sum of variables just translates to a multiplication i the fourier space, but this approach doesn't seem doable for more than two variables unless I do it recursevely, that is, write always $x=s_1+s_2+s_3+\dots +s_N \to x=s_1+y$ where $y=\sum^N_{i=2}$, and use the previous result. But doing it explicitly should return the same result, but the delta dirac disapears after just one integration and I can't seem to get around that. Am I having the correct approach?