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On generalized transforms of distribution functions

  • Erio Castagnoli
  • Pietro Muliere
Article

Abstract

In this note we discuss the following problem.

LetX andY to be two real valued independent r.v.'s with d.f.'sF and ϕ. Consider the d.f.F*ϕ of the r.v.X oY, being o a binary operation among real numbers. We deal with the following equation:
$$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$
where\(\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 \) are real or complex functionals, т another binary operation ands a parameter.
We give a solution, that under stronger assumptions (Aczél 1966), is the only one, of the problem. Such a solution is obtained in two steps. First of all we give a solution in the very special case in whichX andY are degenerate r.v.'s. Secondly we extend the result to the general case under the following additional assumption:
$$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$
.

Keywords

Distribution Function Real Number Economic Theory Public Finance Additional Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Sunto

In questa nota discutiamo il seguente problema.

SianoX eY due v.a. indipendenti e a valori reali con f.r.F e ϕ. Detta o un'operazione binaria tra numeri reali, siaF*ϕ la f.r. della v.a.X oY.

Ci occupiamo della seguente equazione:
$$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$
dove\(\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 \) sono funzionali reali o complessi e т è un'altra operazione binaria es è un parametro.

Diamo una soluzione dell'equazione che, sotto opportune ipotesi (Aczél 1966), è unica. Tale soluzione è ottenuta in due passi.

Dapprima diamo una soluzione nel caso speciale nel qualeX eY sono degeneri. Successivamente estendiamo il risultato al caso generale con l'ipotesi aggiuntiva:
$$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$
.

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References

  1. J. Aczél:Lectures on Functional Equations and their Applications, Academic press, New York, (1966).Google Scholar
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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Erio Castagnoli
    • 1
  • Pietro Muliere
    • 2
  1. 1.Università BocconiMilanoItaly
  2. 2.Università di PaviaPaviaItaly

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