On read-once transformations of random variables over finite fields

Abstract

Transformations of independent random variables over a finite field by read-once formulas are considered. Subsets of probability distributions that are preserved by read-once transformations are constructed. Also we construct a family of distributions that may be arbitrarily closely approximated by a read-once combination of independent identically distributed random variables, whose distributions have no zero components.

On convex polytopes of distributions preserved by finite field operations

“…By virtue of distributive property for multiplication over addition we easily obtain that any combination of permutations reduces to one multiplication of by an element of the field and one addition of a field element, which implies that the group contains ( − 1) elements. In author's earlier papers [1,2] there have been constructed other sets preserved by + and × operations. We shall now compare previously constructed families of sets with the ones that are subject of the present work.…”

“…, ) belongs to as well. In this case the set shall be referred to as preserved by operations from ℬ (see also [1,2]). Usually the set is constructed for some given set of initial distributions, so as to have ⊆ .…”