Mathematics – Probability
Scientific paper
2005-10-03
Mathematics
Probability
Scientific paper
We study the harmonic moments of Galton-Watson processes, possibly non homogeneous, with positive values. Good estimates of these are needed to compute unbiased estimators for non canonical branching Markov processes, which occur, for instance, in the modeling of the polymerase chain reaction. By convexity, the ratio of the harmonic mean to the mean is at most 1. We prove that, for every square integrable branching mechanisms, this ratio lies between 1-A/k and 1-B/k for every initial population of size k greater than A. The positive constants A and B, such that B is at most A, are explicit and depend only on the generation-by-generation branching mechanisms. In particular, we do not use the distribution of the limit of the classical martingale associated to the Galton-Watson process. Thus, emphasis is put on non asymptotic bounds and on the dependence of the harmonic mean upon the size of the initial population. In the Bernoulli case, which is relevant for the modeling of the polymerase chain reaction, we prove essentially optimal bounds that are valid for every initial population. Finally, in the general case and for large enough initial populations, similar techniques yield sharp estimates of the harmonic moments of higher degrees.
No associations
LandOfFree
Harmonic moments of non homogeneous branching processes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Harmonic moments of non homogeneous branching processes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Harmonic moments of non homogeneous branching processes will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-134486