Explicit results on conditional distributions of generalized exponential mixtures

For a set of independent exponentially distributed random variables $X_i$, $i\in \N$ with distinct rates~$\l_i$ we consider sums $\sum_{i\in\A} X_i$ for $\A\subseteq \N$ which follow generalized exponential mixture (GEM) distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in \N}X_i$ given that a subset sum $\sum_{j\in \A}X_j$ exceeds a certain threshold value $t>0$, and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$. Finally, we illustrate how our probabilistic results could be applied in practice by providing examples from both reliability theory and risk management.