Let $S$ be the set of all real sequences $x_i$ that are equal to zero for sufficiently high indices. Consider the set of random variables $X$ with image in $S$ such that there exists an $n$ for which $$P(\forall i\geq n,\:X_i=0)=1.$$ Denote by $n_X$ the smallest such $n$ for $X$.
Given two such random variables $X$ and $Y$, define their independent concatenation $X\otimes Y$ to be equal to $X$ for $i\leq n_X$ and equal to $Y$ thereafter, with the distribution for indices up to $n_X$ being independent to the distribution from $n_X+1$ onward. This operation is associative and the degenerate random variable which is always equal to zero is an identity, so this is a Monoid.
Does it have a name? Are there any known interesting extra properties about it?