Is there any standard notation for the product of multiple variables? For the sum of two random variables $X$ and $Y$, we can write: $$ f_{X+Y}(x) =(f_X∗f_Y)(x), $$
but I can't seem to find any standard notation for representing the sum of three or more variables.
Would the following notation be acceptable?
$$ f_{X+Y+Z+K+L}(x) =(f_X∗f_Y∗f_Z∗f_K∗f_L)(x) $$
Notice that the equation you wrote is not a definition, but a theorem about densities of the sum of independent random variables. So, $f_{X+Y+Z+K+L}$ already has a meaning: it is the density of the random variable $X+Y+Z+K+L$. Also $f_X*f_Y*f_Z*f_K*f_L$ already has a meaning: it is the convolution among the functions $f_X, f_Y, f_Z, f_K, f_L$. Then, you may ask yourself if the two sides are equal. They could very well differ from each other, but, if $X,Y,Z,K,L$ are indipendent random variables, the two sides are equal.