On generalized transforms of distribution functions

  • Erio Castagnoli
  • Pietro Muliere


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:
$$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$


Distribution Function Real Number Economic Theory Public Finance Additional Assumption 
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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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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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