On domination of tail probabilities of (super)martingales: Explicit bounds

2006 ◽  
Vol 46 (1) ◽  
pp. 1-43 ◽  
Author(s):  
V. Bentkus ◽  
N. Kalosha ◽  
M. van Zuijlen
Keyword(s):  
Tail Probabilities ◽  
Explicit Bounds

scholarly journals The critical window in random digraphs

2021 ◽  
pp. 1-19
Author(s):  
Matthew Coulson
Keyword(s):  
Tail Probabilities ◽  
Component Structure ◽  
Critical Window ◽  
Random Digraphs ◽  
Explicit Bounds

Abstract We consider the component structure of the random digraph D(n,p) inside the critical window $p = n^{-1} + \lambda n^{-4/3}$ . We show that the largest component $\mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In particular we give explicit bounds on the tail probabilities of $|\mathcal{C}_1|n^{-1/3}$ .

scholarly journals Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities

1985 ◽  
Vol 13 (1) ◽  
pp. 179-195 ◽  
Author(s):  
Richard Davis ◽  
Sidney Resnick
Keyword(s):  
Random Variables ◽  
Moving Averages ◽  
Tail Probabilities ◽  
Limit Theory ◽  
Regularly Varying

scholarly journals Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

2021 ◽  
Vol 53 (1) ◽  
pp. 162-188
Author(s):  
Krzysztof Bartoszek ◽  
Torkel Erhardsson
Keyword(s):  
Normal Distribution ◽  
Phenotypic Trait ◽  
Phenotypic Traits ◽  
Normal Distributions ◽  
Average Value ◽  
Conditional Moments ◽  
Mixture Of Normal Distributions ◽  
Explicit Bounds ◽  
Mixtures Of Normal Distributions ◽  
Parameter Values

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

scholarly journals Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 675
Author(s):  
Xuze Zhang ◽  
Saumyadipta Pyne ◽  
Benjamin Kedem
Keyword(s):  
Disease Modeling ◽  
Empirical Distribution ◽  
Density Ratio ◽  
Washington State ◽  
Ratio Model ◽  
Multiple Sources ◽  
Tail Probabilities ◽  
Data Set ◽  
Density Ratio Model ◽  
Regional Prediction

In disease modeling, a key statistical problem is the estimation of lower and upper tail probabilities of health events from given data sets of small size and limited range. Assuming such constraints, we describe a computational framework for the systematic fusion of observations from multiple sources to compute tail probabilities that could not be obtained otherwise due to a lack of lower or upper tail data. The estimation of multivariate lower and upper tail probabilities from a given small reference data set that lacks complete information about such tail data is addressed in terms of pertussis case count data. Fusion of data from multiple sources in conjunction with the density ratio model is used to give probability estimates that are non-obtainable from the empirical distribution. Based on a density ratio model with variable tilts, we first present a univariate fit and, subsequently, improve it with a multivariate extension. In the multivariate analysis, we selected the best model in terms of the Akaike Information Criterion (AIC). Regional prediction, in Washington state, of the number of pertussis cases is approached by providing joint probabilities using fused data from several relatively small samples following the selected density ratio model. The model is validated by a graphical goodness-of-fit plot comparing the estimated reference distribution obtained from the fused data with that of the empirical distribution obtained from the reference sample only.

scholarly journals Minimal periods for ordinary differential equations in strictly convex Banach spaces and explicit bounds for someLp-spaces

2014 ◽  
Vol 256 (8) ◽  
pp. 2846-2857 ◽  
Author(s):  
Michaela A.C. Nieuwenhuis ◽  
James C. Robinson ◽  
Stefan Steinerberger
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Banach Spaces ◽  
Strictly Convex ◽  
Explicit Bounds ◽  
Strictly Convex Banach Spaces

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

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