Decomposition of Exponential Distributions on Positive Semigroups
Tóm tắt
Let (S,·) be a positive semigroup and T a sub-semigroup of S. In many natural cases, an element
$$x\in S$$
can be factored uniquely as x=yz, where
$$y \in T$$
and where z is in an associated “quotient space” S/T. If X has an exponential distribution on S, we show that Y and Z are independent and that Y has an exponential distribution on T. We prove a converse when the sub-semigroup is
$$S_t =\{t^n : n \in\mathbb{N}\}$$
for
$$t\in S$$
. Specifically, we show that if Y
t
and Z
t
are independent and Y
t
has an exponential distribution on S
t
for each
$$t\in S$$
, then X has an exponential distribution on S. When applied to ([0,∞), +) and
$$(\mathbb{N}, +)$$
, these results unify and extend known results on the quotient and remainder when X is divided by t.
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