Persistent convergence on randomly deleted sets
Let \(0 \leq t_ 1 < t_ 2 < \dots\) be a fixed sequence of times with \(t_ n \to \infty\) as \(n \to \infty\). At time \(t_ k\) a unit with magnitude \(X_ k\) and lifetime \(L_ k\) enters a population or is placed into a system. An observer, inspecting the system at time \(t\), sees only the units with indices in \(S(t) = \{k : t_ k \leq t < t_ k + L_ k\}\). Let \(B_ n\) denote the sequence \(\{X_ k\}_{1 \leq k \leq n}\) and \(B^*_ t\) the sequence \(\{X_ k\}_{k \in S(t)}\). Suppose \(\lambda\) is a real-valued measurable function defined on the collection of finite sequences of real numbers (including the empty sequence \(\emptyset)\), with \(\lambda (\emptyset) = 0\), and symmetric in the sense that \(\lambda (x_ 1, \dots, x_ n) = \lambda (x_{r_ 1}, \dots, x_{r_ n})\) for all permutations \((x_{r_ 1}, \dots, x_{r_ n})\) of \((x_ 1, \dots, x_ n)\). In many cases \(\lambda (B_ n)\) is convergent in probability, or with probability one, to a constant \(\varphi\), or weakly convergent to a random variable \(\varphi\). The authors present conditions under which \(\lambda (B_ t^*)\) converges or fails to converge to \(\varphi\) in the same sense. The authors have been working under the assumption that \(X_ n\), \(n \geq 1\), are i.i.d. random variables, and \(L_ n\), \(n \geq 1\), are i.i.d. nonnegative random variables, jointly independent of the \(X_ k\)'s.
scientific article; zbMATH DE number 681907