On Random Extrema with Applications to Renewal Processes

Considering the maximum and minimum of a random number N of independent and identically distributed random variables \(X_1, X_2, \ldots ,\) $X_1,X_2,\dots ,$ with distribution function F,  N independent of \(X_1, X_2, \ldots ,\) $X_1,X_2,\dots ,$ and imposing conditions like concave/convex, log-concave/log-convex on F and/or on the random summand N,  a study is done if the properties are preserved by the random extrema. The results are illustrated through two applications to renewal processes.