When the search algorithm QuickSelect compares keys during its execution in order to find a key of target rank, it must operate on the keys' representations or internal structures, which were ignored by the previous studies that quantified the execution cost for the algorithm in terms of the number of required key comparisons. In this paper we analyze running costs for the algorithm that take into account not only the number of key comparisons, but also the cost of each key comparison. We suppose that keys are represented as sequences of symbols generated by various probabilistic sources and that QuickSelect operates on individual symbols in order to find the target key. We identify limiting distributions for the costs, and derive integral and series expressions for the expectations of the limiting distributions. These expressions are used to recapture previously obtained results on the number of key comparisons required by the algorithm.

We study estimation of the tail-decay parameter of the marginal distribution corresponding to a discrete-time, real-valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail-decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest ‘mixing’ conditions. Consistency properties of these estimators are established, as well as minimax convergence rates. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost-sure limit theory for normalized extreme values and related first-passage times in stationary sequences.

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.