Limit theorems of general functions of independent and identically distributed random variables
Let \(\{X_n, n\geq 1\}\) be a sequence of independent and identically random variables and \(g(X_1,\dots,X_N)\) a measurable function of \(X_1,\dots, X_N)\), where \(N= n\) or \(N\) is a positive integer-valued random variable independent of \((X_n)\). It is proved that if \(g\) fulfils a certain divisibility condition and under proper normalization \(g(X_1,\dots,X_N)\) converges in distribution to a limiting random variable \(Z\), then \(Z\) is degenerate or satisfies some stability property defined by \(g\). Limit distributions for sums, maxima and minima, both for random and non-random \(N\), are presented as applications of the main theorem.
functions of independent and identically distributed random variables
scientific article; zbMATH DE number 5634852