Exponential Convergence Rates for the Law of Large Numbers

Baum, Leonard E.; Katz, Melvin; Read, Robert R.

doi:10.2307/1993673

Cited by 10 publications

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“…Theorem 1 can be extended to Au's that are independent (but not necessarily identically distributed) in} for fixed i and in i for fixed} by estimating the right sides of inequalities (Al) using known results about sums of independent random variables. For example, the 548 THE AMERICAN NATURALIST estimate (A6) can be replaced by one given by Petrov (1975, p. 288, a result credited to Baum et al 1962).…”

Section: Appendixmentioning

confidence: 99%

Host-Parasite Relations and Random Zero-Sum Games: The Stabilizing Effect of Strategy Diversification

Cohen¹,

Newman²

1989

The American Naturalist

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In many host-parasite relations, the parasitic species has numerous variants, antigenic strains, or types. The host species also has many types of reactions and defenses, such as specific antibodies, cellular defense mechanisms, and genetically determined resistances or susceptibilities. We propose here a simple phenomenological model, based on a game with random payoffs, to explain how the evolution by natural selection of antagonistic hosts and parasites may lead to large and roughly comparable numbers of strategies of parasitic attack and host defense. Diversification in strategies of parasitic attack and host defense may lead to, be accompanied by, or be a consequence of speciation. Examples of parasitemediated speciation were reviewed by Price et al. (1986, pp. 498-499). Because the model we propose relates to interspecific diversity as well as to intraspecific diversity, the model provides a theoretical basis for Eichler's rule. Eichler's rule asserts that "when a large taxonomic group, for example, family, of hosts consisting of many species is compared with an equivalent taxonomic group consisting of few representatives, the large group has the greater diversity of parasitic fauna" (Noble and Noble 1982, p. 461). In support of Eichler's rule, Price reviewed "many examples ... of a significant relationship between number of species in each host category and number of [species of] parasites that exploit members of that category" (1980, pp. 27-29). After we describe the game-theoretical model and analyze its properties mathematically, we interpret it in greater detail. We describe what the model achieves and what the model omits or ignores. We consider other explanations of the diversity of host and parasitic adaptations and review some other uses of game theory to describe host-parasite relations. Kinds of Models of Host-Parasite Coevolution Models of host-parasite coevolution fall into three broad classes (Holmes 1983, pp. 176-177). In mutual-aggression models, selection acting on the parasite diametrically opposes selection acting on the host. The parasite is selected to

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Section: Appendixmentioning

confidence: 99%

Host-Parasite Relations and Random Zero-Sum Games: The Stabilizing Effect of Strategy Diversification

Cohen¹,

Newman²

1989

The American Naturalist

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“…Furthermore, the faster z n goes to °°, the higher is the order of those moments that must be bounded. To insure this degree of increasingly normal-like behavior, we require that the underlying mgf exist on the whole real line, and we extend a theorem of Baum, Katz, and Read (1962) to show that this requirement is necessary.…”

Section: P(n^s N I= Z) = (2n)-'z-'ex P {-L 2 Z 2 N}{\ + O(z~2)}mentioning

confidence: 99%

“…/ / {X n : 1 g n < <»} is a sequence of iid random variables satisfying the conditions of Theorem (2.1), and </>(x) is a function of a real variable having the properties (3.2a) <K X ) increases monotonically and continuously to °° as x-»°°; 4. The necessity of the existence of the mgf for all < > 0 For a sequence of iid random variables having partial sums {S n : 1 S n < oo}, Baum, Katz, and Read (1962) proved that, if there are numbers A > 0, C > 0 , and 0< p < 1 such that P(n~'S n^ \V~n)^ Cp" for all sufficiently large n, then there must exist a number B, 0< B <»>, such that 4>(t) = E(expfX,)<3° for 0 s / < B. Their result was improved and sharpened by Petrov and Shirokova (1973).…”

Section: = Max {K^(k)}-\s N+k -S N )mentioning

confidence: 99%

Probabilities of very large deviations

Book

1978

J. Aust. Math. Soc.

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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.

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“…Mas, segundo o item (iv) da definição (1.2.1), se P(A c n ) decair para zero em n, então podemos construir o campo aleatório X. O teorema a seguir, principal resultado desta tese, diz como P(A c n ) deve ir para zero quando n → ∞, de tal forma que possamos construir X e também obter, para esse campo, taxa exponencial de convergência na lei multidimensional dos grandes números. Teorema 1.2.1 (Resultado Principal) 4 Seja X um campo aleatório assumindo valores em R Z d e f X a função de construção de X. Se f X for limitada e P(A c n ) = O(n −d−δ ) , para algum δ > 0, então a taxa de convergência na Lei Multidimensional dos Grandes Números será exponencial para o campo aleatório X.…”

Section: Definições Preliminaresunclassified

“…Taxas exponenciais de convergência na LMGN podem ser encontradas, por exemplo, em [4] para campos aleatórios indexados por Z; em [5] para campos aleatórios distribuídos segundo medidas estacionárias de sistemas de partículas atrativos; e em [6] para sistemas percolativos de longo alcance. Nesta presente tese taxas exponenciais de convergência na LMGN são provadas a partir da técnica usada em [2] para provar o decaimento super-exponencial das correlações espaciais do limite de saturação do processo de estacionamento (ver Capítulo 2 desta tese).…”

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Taxas exponenciais de convergência na lei multidimensional dos grandes números: uma abordagem construtiva

Bosco¹

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Exponential Convergence Rates for the Law of Large Numbers

Cited by 10 publications

References 1 publication

Host-Parasite Relations and Random Zero-Sum Games: The Stabilizing Effect of Strategy Diversification

Host-Parasite Relations and Random Zero-Sum Games: The Stabilizing Effect of Strategy Diversification

Probabilities of very large deviations

Taxas exponenciais de convergência na lei multidimensional dos grandes números: uma abordagem construtiva

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