kth records from discrete distributions

Dziubdziela and Kopociński (Zastosowania Matematyki 15 (1976) 187-190) introduced a definition of kth records as a strictly increasing sequence of kth largest observations yet seen. Later in the literature appeared another definition of kth records which allowed ties between succesive kth record values. These two definitions are considered to be equivalent. This note shows that it is not true if the underlying distribution is discrete. Also a third definition is possible: weak kth records. We investigate properties of the distribution of kth records from discrete distributions in the three cases.