A Characteristic Martingale Related to the Counting Process of Records |
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Authors: | Raúl Gouet F Javier López Gerardo Sanz |
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Institution: | (1) Dpto. Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile;(2) Dpto. Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain |
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Abstract: | Let (X
n
) be a sequence of nonnegative, integrable, independent and identically distributed random variables, with common distribution
function F. We consider the problem of finding all distribution functions F such that N
n
−cM
n
is a discrete time martingale, where N
n
is the counting process of upper records, M
n
=max {X
1,…,X
n
} is the process of partial maxima and c is a positive constant. We solve the problem by explicitly giving the solution with finite support and using this for constructing
the solution for the general case by a limiting process. We show that the set of solutions can be parameterized by their support
and the mass at the leftmost point of the support. |
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Keywords: | Characterization of distributions Martingales Records |
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