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Gaming the Law of Large Numbers

Thomas R. Hoffman, Bart Snapp
2013 Mathematics Teacher  
We developed the rules for Fibber's Dice by directly adapting the rules of Liar's Dice found at [3] . These modifications make the game more suitable for the classroom.  ...  Conclusion It is our hope that you try out Fibber's Dice, have fun, be daring, and experience the law of large numbers.  ...  Fibber's Dice illustrates the connection between these ideas. The notions of probability and relative frequency are glued together via the law of large numbers.  ... 
doi:10.5951/mathteacher.106.5.0378 fatcat:ezezffrz4feklgr4aemk55opvq

Large games and the law of large numbers

Nabil I. Al-Najjar
2008 Games and Economic Behavior  
Randomness can be modeled explicitly and an exact law of large numbers holds.  ...  This paper introduces discrete large games where the set of players is a countable dense 'grid' with a finitely additive distribution.  ...  I am grateful to Ehud Kalai for many insightful conversations, and to Nenad Kos for his comments and careful reading of the paper.  ... 
doi:10.1016/j.geb.2007.11.002 fatcat:nxc5kp63srallj7wbra5agktfq

On the Law of Large Numbers

E. H. Linfoot
1928 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
LINFOOT ON THE LAW OF LARGE NUMBERS.  ...  LINPOOT ON THE LAW OP LARGE NUMBERS.  ...  v alu es of i.  ... 
doi:10.1098/rsta.1928.0011 fatcat:tpi3xrvzdvhtdpm2rzigsfibim

The Law of Large Numbers for Large Stable Matchings [article]

Jacob Schwartz, Kyungchul Song
2021 arXiv   pre-print
In an empirical study of a two-sided large matching market (such as a college admissions problem), the primary statistics of interest are often the empirical matching probabilities which represent the  ...  We introduce a notion of partial homogeneity of preferences to express the degree of alignment in the preferences of colleges over students, and show that the more the preferences are aligned, the more  ...  When the statistic of interest takes the form of a sum of random variables, concentration-of-measure naturally yields the rate of convergence for a law of large numbers.  ... 
arXiv:2101.00399v3 fatcat:ed5xmrpfsraehjvno3f2xhx5s4

On the strong law of large numbers

John Slivka, Norman C. Severo
1970 Proceedings of the American Mathematical Society  
Indeed, this law may be formulated in terms of this counting variable as in the following. Strong law of large numbers.  ...  ^(X)-Then ATM(X) = Jji" Yk(\) is precisely the "finitely many" random variable of the Strong Law of Large Numbers.  ... 
doi:10.1090/s0002-9939-1970-0259993-9 fatcat:lkbdv4hevra6laonkaasx3br2e

On the strong law of large numbers

P. Erd{ös
1949 Transactions of the American Mathematical Society  
In a recent paper Kac, Salem, and Zygmund(') prove ( (2) lim -(X/(W) = 0, JV-A7 \ 4=1 / or roughly speaking the strong law of large numbers holds lor f(nkx) (in fact the authors prove that 2~2f(nkx  ...  Hence it will suffice to show that for every e and sufficiently large k the measure of the set in x satisfying at least one of the inequalities (10) is greater than 1-e.  ... 
doi:10.1090/s0002-9947-1949-0032971-4 fatcat:ksfyb3mk6vh2xe26ie336zio54

The Essence of the Law of Large Numbers [chapter]

Michael Keane
1995 Algorithms, Fractals, and Dynamics  
Now we can state the BASIC LAW OF LARGE NUMBERS:  ...  The law of large numbers, not really a law but a mathematical theorem, is at the same time a justification for application of statistics and an essential tool for the mathematical theory of probability  ... 
doi:10.1007/978-1-4613-0321-3_11 fatcat:5vgwogwp5bbuzdc6r5r5vzjy4q

The Reverse of The Law of Large Numbers [article]

Kieran Kelly, Przemyslaw Repetowicz, Seosamh macReamoinn
2008 arXiv   pre-print
an "reverse" to the law of large numbers.  ...  The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists.  ...  We termed this result as the "Law of Many Outcomes", or alternatively, the "Reverse of the Law of Large Numbers".  ... 
arXiv:0803.3913v1 fatcat:biforam46zco7lek3xmgeuijlu

Laws of large numbers for the number of weak records

Raúl Gouet, F. Javier López, Gerardo Sanz
2008 Statistics and Probability Letters  
Acknowledgements The first author thanks the Departamento de Métodos Estadísticos of Universidad de Zaragoza for kind hospitality.  ...  Strong laws of large numbers for the number of (ordinary) records in discrete models were given in Gouet et al. (2001) .  ...  Now, the idea of the proof is to check (14) and (15) of Proposition 4.1 to obtain a strong law of large numbers for S n defined in Proposition 2.1 (a) As lim sup k→∞ r k < 1, there exists δ > 0 such  ... 
doi:10.1016/j.spl.2008.01.067 fatcat:ta6ruue2erdctlvjzynetp5hhq

Schoenberg's Theorem via the law of large numbers [article]

Davar Khoshnevisan
2005 arXiv   pre-print
We present a non-technical derivation of of Schoenberg's theorem that relies chiefly on the de Finneti theorem and the law of large numbers of classical probability theory.  ...  A classical theorem of S. Bochner states that a function $f:R^n \to C$ is the Fourier transform of a finite Borel measure if and only if $f$ is positive definite. In 1938, I.  ...  The simplest form of the law of large numbers dictates that n i=1 X 2 i /n → VarX 1 = t 2 in probability.  ... 
arXiv:math/0504603v2 fatcat:53udmhsf4ndifildy6y3er2c7a

Aggregation and the law of large numbers in large economies

Nabil I. Al-Najjar
2004 Games and Economic Behavior  
In this model, the law of large numbers is meaningful and holds on all subintervals.  ...  This framework provides, among other things, a new interpretation of the measurability problem and the failure of the law of large numbers in the continuum.  ...  That is, the conclusion of the law of large numbers for the discrete model, Eq. (7) , fails.  ... 
doi:10.1016/s0899-8256(03)00175-1 fatcat:673k2wndpvfoppfgbpoiqqc3ee

A version of the law of large numbers

Katusi Fukuyama
2001 Colloquium Mathematicum  
By the method of Rio [10  ...  By using the following law of large numbers for a quasi-orthogonal sequence (cf. e.g. Stout [11, Th. 3.7 .3]), we have the law of large numbers on [0, 1] × [b, b + 1]. Theorem A.  ...  In this paper we are concerned with the law of large numbers (1) lim N →∞ 1 N N k=1 f (λ k x) = 1 0 f (y) dy for a.e. x, where f is a function with period 1.  ... 
doi:10.4064/cm90-2-9 fatcat:ma24heudpvento4jm42beenzrm

On the strong law of large numbers for φ-subgaussian random variables [article]

Krzysztof Zajkowski
2021 arXiv   pre-print
This result is a generalization of the SLLN for independent subgaussian random variables (Taylor and Hu \cite{TayHu}) to the case of dependent $\varphi_p$-subgaussian random variables.  ...  We prove that if for a sequence $(\xi_n)\subset Sub_{\varphi_p}$ ($p>1$) there exist positive constants $c$ and $\alpha$ such that for every natural number $n$ the following inequality $\tau_{\varphi_p  ...  The strong law of large numbers for various classes of many type associated random variables one can find for instance in Bulinski and Shashkin [2, Chap. 4] .  ... 
arXiv:1607.03035v4 fatcat:35sbii2sjfdddiki4aleydftti

Functional strong law of large numbers for Betti numbers in the tail [article]

Takashi Owada, Zifu Wei
2021 arXiv   pre-print
The nature of the obtained law of large numbers is determined by the decay rate of a probability density.  ...  We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius $R_n$, such that $R_n\to\infty  ...  Functional strong law of large numbers We develop two main theorems in this section.  ... 
arXiv:2103.05799v2 fatcat:yjiohcmsxjecvlhtaadoofy66a

Convergence rates in the law of large numbers

Leonard E. Baum, Melvin Katz
1963 Bulletin of the American Mathematical Society  
Since ^1, this theorem concerns sequences of independent and identically distributed random variables for which the Strong Law of Large Numbers holds and thus it is natural to study the rate of convergence  ...  Let {X i :i=l 1 2, • • • } be a sequence of independent and identically distributed random variables and let S n = XXi ^*« ^e" cently a number of papers [l], [2] , [3] and [5] have considered the  ... 
doi:10.1090/s0002-9904-1963-11027-7 fatcat:zgrk3sklqnf7dd3vp7kb34puqi
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