186 research outputs found
Hoeffding's inequality for supermartingales
We give an extension of Hoeffding's inequality to the case of
supermartingales with differences bounded from above. Our inequality
strengthens or extends the inequalities of Freedman, Bernstein, Prohorov,
Bennett and Nagaev.Comment: 20 pages, accepted; Stochastic Processes and their Applications
(2012), Vol. 122, pages 3545-355
Asymptotic of the distribution and harmonic moments for a supercritical branching process in a random environment
Let be a supercritical branching process in an independent and
identically distributed random environment . We show the exact decay rate
of the probability as , for each assuming that . We also determine the
critical value for the existence of harmonic moments of the random variable
under a simple moment
condition
Berry-Esseen's bound and Cram\'er's large deviation expansion for a supercritical branching process in a random environment
Let be a supercritical branching process in a random environment . We establish a Berry-Esseen bound and a Cram\'er's type large
deviation expansion for under the annealed law . We also
improve some earlier results about the harmonic moments of the limit variable
, where is the
normalized population size
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