Sums Of Exchangeable Random Variables Research Articles

Hoeffding and Bernstein inequalities for weighted sums of exchangeable random variables

Hoeffding and Bernstein inequalities for weighted sums of exchangeable random variables

The limit theorem for maximum of partial sums of exchangeable random variables

We obtain the analogue of the classical result by Erdös and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central limit theorem of Blum et al. hold, the limit coincides with the classical one. Under more general assumptions, the probability of the random variables having conditional negative drift appears in the limit.

Confidence intervals for nonhomogeneous branching processes and polymerase chain reactions

We extend in two directions our previous results about the sampling and the empirical measures of immortal branching Markov processes. Direct applications to molecular biology are rigorous estimates of the mutation rates of polymerase chain reactions from uniform samples of the population after the reaction. First, we consider nonhomogeneous processes, which are more adapted to real reactions. Second, recalling that the first moment estimator is analytically known only in the infinite population limit, we provide rigorous confidence intervals for this estimator that are valid for any finite population. Our bounds are explicit, nonasymptotic and valid for a wide class of nonhomogeneous branching Markov processes that we describe in detail. In the setting of polymerase chain reactions, our results imply that enlarging the size of the sample becomes useless for surprisingly small sizes. Establishing confidence intervals requires precise estimates of the second moment of random samples. The proof of these estimates is more involved than the proofs that allowed us, in a previous paper, to deal with the first moment. On the other hand, our method uses various, seemingly new, monotonicity properties of the harmonic moments of sums of exchangeable random variables.

Book Review: Limit theorems for sums of exchangeable random variables

Book Review: Limit theorems for sums of exchangeable random variables

Limit Theorems for Sums of Exchangeable Random Variables.

Limit Theorems for Sums of Exchangeable Random Variables.

Book Reviews

Book reviewed in this article: Differential‐Geometrical Methods in Statistics. By Shun‐Ichi Amari. Limit Theorems for Sums of Exchangeable Random Variables. By R. L. Taylor, P. Z. Daffer and R. F. Patterson. Applied Regression Analysis and Experimental Design. By Richard J. Brook and Gregory C. Arnold. Essays in Time Series and Allied Process. Papers in honour of E. J. Hannan. Edited by J. Gani and M. B. Priestley.

Weak convergence of the adjusted range of cumulative sums of exchangeable random variables

Let be the adjusted range of the cumulative sums of a sequence , where . Weak convergence results for random functions constructed from cumulative sums of {Xs } are used to obtain the asymptotic distribution and moments of when {Xs } are exchangeable, or symmetrically dependent, random variables.

Weak convergence of the adjusted range of cumulative sums of exchangeable random variables

Let be the adjusted range of the cumulative sums of a sequence , where . Weak convergence results for random functions constructed from cumulative sums of {Xs} are used to obtain the asymptotic distribution and moments of when {Xs} are exchangeable, or symmetrically dependent, random variables.

On the expected range and expected adjusted range of partial sums of exchangeable random variables

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables. In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed.

On the expected range and expected adjusted range of partial sums of exchangeable random variables

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables.In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed.