# Size bias for one and all.

@article{Arratia2013SizeBF,
title={Size bias for one and all.},
author={Richard Arratia and Larry Goldstein and Fred Kochman},
journal={arXiv: Probability},
year={2013}
}
• Published 2013
• Mathematics
• arXiv: Probability
Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein's method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, and number theory. In this paper we review the basics and survey some of these unexpected connections.
48 Citations

#### Figures from this paper

Card guessing and the birthday problem for sampling without replacement
• Mathematics
• 2021
Consider a uniformly random deck consisting of cards labelled by numbers from 1 through n, possibly with repeats. A guesser guesses the top card, after which it is revealed and removed and the gameExpand
Geometric sums, size biasing and zero biasing
• Mathematics
• 2021
The geometric sum plays a significant role in risk theory and reliability theory [Kalashnikov (1997)] and a prototypical example of the geometric sum is Rényi’s theorem [Rényi (1956)] saying aExpand
On the number of leaves in a random recursive tree
This paper studies the asymptotic behavior of the number of leaves Ln in a random recursive tree Tn with n nodes. By utilizing the size-bias method, we derive an upper bound on the WassersteinExpand
New Berry-Esseen and Wasserstein bounds in the CLT for non-randomly centered random sums by probabilistic methods
We prove abstract bounds on the Wasserstein and Kolmogorov dis- tances between non-randomly centered random sums of real i.i.d. random variables with a nite third moment and the standard normalExpand
New Berry-Esseen and Wasserstein bounds in the CLT for non-randomly centered random sums by probabilistic methods
Abstract. We prove abstract bounds on the Wasserstein and Kolmogorov distances between non-randomly centered random sums of real i.i.d. random variables with a finite third moment and the standardExpand
Multivariate Concentration Inequalities with Size Biased Couplings
• Mathematics
• 2013
Let $\mathbf{W}=(W_1,W_2,...,W_k)$ be a random vector with nonnegative coordinates having nonzero and finite variances. We prove concentration inequalities for $\mathbf{W}$ using size biasedExpand
New Berry-Esseen and Wasserstein bounds in the CLT for non-randomly centered random sums by probabilistic methods
We prove abstract bounds on the Wasserstein and Kolmogorov distances between non-randomly centered random sums of real i.i.d. random variables with a finite third moment and the standard normalExpand
Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models
• Mathematics
• 2014
Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results forExpand
Non uniform exponential bounds on normal approximation by Stein’s method and monotone size bias couplings
• Mathematics
• 2018
ABSTRACT It is known that the normal approximation is applicable for sums of non negative random variables, W, with the commonly employed couplings. In this work, we use the Stein’s method to obtainExpand
Distances between probability distributions via characteristic functions and biasing
• Mathematics
• 2016
In a spirit close to classical Stein's method, we introduce a new technique to derive first order ODEs on differences of characteristic functions. Then, using concentration inequalities and FourierExpand

#### References

SHOWING 1-10 OF 86 REFERENCES
EXPLOITING THE WAITING TIME PARADOX: APPLICATIONS OF THE SIZE-BIASING TRANSFORMATION
• Mark Brown
• Mathematics
• Probability in the Engineering and Informational Sciences
• 2006
We consider the transformation T that takes a distribution F into the distribution of the length of the interval covering a fixed point in the stationary renewal process corresponding to F. ThisExpand
Characterization of distributions with the length-bias scaling property
This paper characterizes the density functions of absolutely continuous positive random variables with finite expectation whose respective distribution functions satisfy the so-called length-biasExpand
Normal Approximation by Stein ’ s Method
The aim of this paper is to give an overview of Stein’s method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums ofExpand
Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models
• Mathematics
• 2014
Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results forExpand
Exponential approximation for the nearly critical Galton-Watson process and occupation times of Markov chains
• Mathematics
• 2010
In this article we provide new applications for exponential approximation using the framework of Peköz and Röllin (in press), which is based on Stein’s method. We give error bounds for the nearlyExpand
On the variance of the ratio estimator for midzuno-sen sampling scheme
SummaryFor the sampling scheme ofMidzuno [3] andSen [4], which provides unbiased ratio estimators an expression for the variance of the estimator does not seem to be available in literature. AnExpand
Normal approximation for coverage models over binomial point processes
• Mathematics
• 2010
We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over aExpand
On the infinite divisibility of the lognormal distribution
Summary In the present paper the author proves that the lognormal distribution is infinitely divisible. This is achieved by showing that the lognormal is the weak limit of a sequence of probabilityExpand
Local limits of conditioned Galton-Watson trees: the infinite spine case
• Mathematics
• 2014
We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well asExpand
Lévy processes and infinitely divisible distributions
Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5.Expand