A dependent \(F^ \alpha\)-scheme (Q1336937)

An \(F^ \alpha\)-scheme is a sequence of random variables \(\{X_ n\), \(n \geq 1\}\), where \(X_ n\) has cdf \(F^{\alpha_ n}\) for some sequence \(\{\alpha_ n\), \(n \geq 1\}\). For the independent \(F^ \alpha\)-scheme where the \(X_ n\)'s are independent, the sequence \{Maximum\((X_ 1, \dots, X_ n)\), \(n \geq 1\}\) and the sequence of record indicators are mutually independent. By using Archimedean copula function a dependent \(F^ \alpha\)-scheme is constructed that retains this property. Thus some of the results for the independent \(F^ \alpha\) scheme hold for such a dependent scheme. This has some implications on the secretary problem.