An Upper Bound of the Large Deviation Probability in Multi-server Constant Retrial Rate System

2019 ◽  
pp. 325-337 ◽  
Author(s):  
Evsey Morozov ◽  
Ksenia Zhukova
Keyword(s):  
Upper Bound ◽  
Large Deviation ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Rate System ◽  
Constant Retrial Rate ◽  
Multi Server

scholarly journals Secretary problem with vanishing objects

2020 ◽  
Vol 12 (2) ◽  
pp. 63-81
Author(s):  
Сергей Иванович Доценко ◽  
Sergey Dotsenko ◽  
Георгий Шевченко ◽  
Georgiy Shevchenko
Keyword(s):  
Large Deviation ◽  
Geometric Distribution ◽  
Secretary Problem ◽  
Independent Interest ◽  
Selection Strategy ◽  
Large Deviation Probability ◽  
Auxiliary Result ◽  
Deviation Probability ◽  
Independent Variables ◽  
Probability Estimates

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we establish large deviation probability estimates for sums of independent variables with distinct geometric distribution.

scholarly journals The Lower Tail of the Random Minimum Spanning Tree

10.37236/1004 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Abraham D. Flaxman
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Large Deviation ◽  
Random Variable ◽  
Independent Random Variables ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Lower Tail ◽  
The Subject ◽  
The Cost

Consider a complete graph $K_n$ where the edges have costs given by independent random variables, each distributed uniformly between 0 and 1. The cost of the minimum spanning tree in this graph is a random variable which has been the subject of much study. This note considers the large deviation probability of this random variable. Previous work has shown that the log-probability of deviation by $\varepsilon$ is $-\Omega(n)$, and that for the log-probability of $Z$ exceeding $\zeta(3)$ this bound is correct; $\log {\rm Pr}[Z \geq \zeta(3) + \varepsilon] = -\Theta(n)$. The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear, $\log {\rm Pr}[Z \leq \zeta(3) - \varepsilon] = -\Theta(n)$.

scholarly journals Probabilities of very large deviations

1978 ◽  
Vol 25 (3) ◽  
pp. 332-347 ◽  
Author(s):  
Stephen A. Book
Keyword(s):  
Generating Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Absolutely Continuous ◽  
The Central Limit Theorem ◽  
The Subject ◽  
Elementary Form ◽  
Independent Identically Distributed

If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

scholarly journals Matrix Liberation Process II: Relation to Orbital Free Entropy

2020 ◽  
pp. 1-49
Author(s):  
Yoshimichi Ueda
Keyword(s):  
Mutual Information ◽  
Upper Bound ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
New Approach ◽  
Free Entropy ◽  
The Matrix ◽  
Orbital Free ◽  
Definition Of

Abstract We investigate the concept of orbital free entropy from the viewpoint of the matrix liberation process. We will show that many basic questions around the definition of orbital free entropy are reduced to the question of full large deviation principle for the matrix liberation process. We will also obtain a large deviation upper bound for a certain family of random matrices that is essential to define the orbital free entropy. The resulting rate function is made up into a new approach to free mutual information.

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