Speed of convergence of complementary probabilities on finite group

Keywords: probability, finite group, convergence, convolution, group algebra


Let function P be a probability on a finite group G, i.e. $P(g)\geq0\ $ $(g\in G),\ \sum\limits_{g}P(g)=1$ (we write $\sum\limits_{g}$ instead of
$\sum\limits_{g\in G})$. Convolution of two functions $P, \; Q$ on group $G$ is \linebreak
$ (P*Q)(h)=\sum\limits_{g}P(g)Q(g^{-1}h)\ \ (h\in G)$. Let
$E(g)=\frac{1}{|G|}\sum\limits_{g}g$ be the uniform (trivial) probability on the group $G$, $P^{(n)}=P*...*P$ ($n$ times) an $n$-fold convolution
of $P$. Under well known mild condition probability $P^{(n)}$ converges to $E(g)$ at $n\rightarrow\infty$. A lot of papers are devoted to
estimation the rate of this convergence for different norms.
Any probability (and, in general, any function with values in the field $R$ of real numbers) on a group can be associated with an element of
the group algebra of this group over the field $R$. It can be done as follows.
Let $RG$ be a group algebra of a finite group $G$ over the field $R$. A probability $P(g)$ on the group $G$ corresponds to the element $ p =
\sum\limits_{g} P(g)g $ of the algebra RG. We denote a function on the group $G$ with a capital letter and the corresponding element of $RG$ with
the same (but small) letter, and call the latter a probability on $RG$. For instance, the uniform probability $E(g)$ corresponds to
the element $e=\frac{1}{|G|}\sum\limits_{g}g\in RG. $
The convolution of two functions $P, Q$ on $G$ corresponds to product $pq$ of corresponding elements $p,q$ in the group algebra $RG$. For a
natural number $n$, the $n$-fold convolution of the probability $P$ on $G$ corresponds to the element $p^n \in RG$. In the article we study
the case when a linear combination of two probabilities in algebra $RG$ equals to the probability $e\in RG$. Such a linear
combination must be convex. More exactly, we correspond to a probability $p \in RG$ another probability $p_1 \in RG$ in the following way.
Two probabilities $p, p_1 \in RG$ are called complementary if their convex linear combination is $e$, i.e. $ \alpha p + (1- \alpha) p_1 = e$
for some number $\alpha$, $0 <\alpha <1$. We find conditions for existence of such $\alpha$ and compare $\parallel p ^ n-e
\parallel$ and $\parallel {p_1} ^ n-e \parallel$ for an arbitrary norm ǁ·ǁ.


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How to Cite
Vyshnevetskiy, A. (2021). Speed of convergence of complementary probabilities on finite group. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 93, 12-17. https://doi.org/10.26565/2221-5646-2021-93-02