Convolution of probabilities constant in a subgroup and outside a subgroup

Abstract

Let be $P$ the probability on a finite group $G$, i.e. the function $P\left(g\right)$ takes non-negative values and $\sum _{g}P\left(g\right) =1$ $\left(g\in G\right)$. For any two functions $F_{1} \left(g\right)$ and $F_{2} \left(g\right)$ their $G$ convolution
\[\left(F_{1} *F\right)_{2} \left(t\right)=\sum _{h\in G}F_{1} \left(h\right)F_{2} \left(h^{-1} t\right), t\in G\]
is also a function on $G$.

In recent years, the topic of studying random walks (and not only on groups) has become very popular. From an analytical point of view, the study of random walks is only the study of their transition function, that is, the n-fold convolution of probability measures. It is well known that under simple conditions, imposed on the probability support $P$ on the group $G$, n-fold convolution $P^{(n)} =P*...*P$ ($n$ times) with $n\to \infty $ converges to uniform probability $U\left(g\right)=\frac{1}{\left|G\right|} $( \textbf{$g\in G$) }, which is obviously a constant on $G$.

We study convolution of functions that may be different but are constant in or outside some subgroup. In short, we study the cases where the convolution of such functions has the same properties of constancy with respect to some subgroup.

The article considers the convolution of probabilities (and, in general, real functions) constant outside (or inside) a subgroup of $H$ the finite group $G$. Let $D=G\backslash H$. For a given function $F$ on $G$ the subgroup $H$, for which $F$ is constant on $D$, is unique in the following sense: there is at most one such number $c$ and one smallest subgroup $H$ of the group $G$ such that $F\left(g\right)=c$ for all elements $g\in D$.

It is proved that if the functions $F_{1} ,\ldots $, $F_{n} $ are constants on $D$, $F_{i} (x)=c_{i} $ for arbitrary $x\in D$ $\left(i=1,\ldots ,n\right)$, then their convolution $F_{1} *\ldots *F_{n} $ is also a constant on $D$. An expression for this constant is found in terms of the numbers $c_{i} ,\; \; i=1,\ldots ,n$. Functions on the group $G$ that are constant on $D$ form a semigroup with respect to the convolution.

In previous statements we studied the case, when all functions $F_{1},\ldots, F_{n}$, except one, are constant on the subgroup $H$, and this one is constant on $D=G\backslash H$. Under these conditions it is proved that the convolution $F_{1} *\ldots *F_{n} $ is constant on $H$.

In the above statements about a convolution, at least one of its factors is constant outside the subgroup. But if all factors are constant on some subgroup, then their convolution does not necessarily have the same property. An example is given of two functions that are constant on subgroup $H$, but their convolution is not constant on $H$.

This example is not presented in the language of probabilities (or functions) on a group (as in all other parts of the article), but in the language of group algebras. The group algebra $KG$ of the group $G$ over the field $K$ appears in questions related to convolutions of functions on the group $G$ in the following way: each function $F\left(g\right)$ on $G$ with values in an arbitrary field $K$ determines an element $\sum _{g}F\left(g\right)g$ $\left(g\in G\right)$ of the algebra $KG$; the convolution of probabilities corresponds to the product of elements of the algebra $KG$. If the function $F\left(g\right)$ is a probability, then $K$ is the field of real numbers.