$\delta$-records and martingales

M. Lafuente Blasco, R. Gouet, F. J. López Lorente, G. Sanz Sáiz

The concept of $\delta$-record is a natural extension of the notion of record, where $\delta$-records are observations greater than all previous observations plus a fixed real quantity (positive or negative) $\delta$. These observations have shown interesting probabilistic and statistical properties in previous works. In this talk, we study which probability distributions satisfy a martingale condition defined through a linear function of the partial maxima and the number of $\delta$-records. The corresponding problem for records was fully solved in Gouet, R., López, F. J., & Sanz, G. (2007). A characteristic martingale related to the counting process of records. Journal of Theoretical Probability, 20, 443-455. However, the case of $\delta$-records is considerably more intricate as it is connected to open problems in the fields of delayed differential equations and recurrence relations.

Palabras clave: Records, $\delta$-records, Martingales, Characterization of distributions

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