Analysis of the Luria–Delbrück distribution using discrete convolution powers

The Luria–Delbrück distribution arises in birth-and-mutation processes in population genetics that have been systematically studied for the last fifty years. The central result reported in this paper is a new recursion relation for computing this distribution which supersedes all past results in simplicity and computational efficiency: p0 = e–m; where m is the expected number of mutations. A new relation for the asymptotic behavior of pn (≈ c/n2) is also derived. This corresponds to the probability of finding a very large number of mutants. A formula for the z-transform of the distribution is also reported.

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Waiting times for multiple occurrences of several events

Journal of Applied Probability ◽

10.1017/s0021900200041334 ◽

1988 ◽

Vol 25 (03) ◽

pp. 624-629

Author(s):

Stephen Scheinberg

Keyword(s):

Asymptotic Behavior ◽

Finite Number ◽

Waiting Times ◽

Expected Number

Consider an ‘experiment' which can be repeated indefinitely often resulting in independent random outcomes. Fix attention on a finite number of possible (sets of) outcomes E 1, E 2, … and define W = W(N 1, N 2, …) to be the expected number of repetitions needed to ensure that E 1 has occurred (at least) N 1 times, E 2 has occurred (at least) N 2 times, etc. This article examines the asymptotic behavior of W as a function of the sum Σ j N j, as the latter grows without bound.

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Asymptotic behavior of a generalization of Bailey's simple epidemic

Advances in Applied Probability ◽

10.1017/s0001867800037873 ◽

1971 ◽

Vol 3 (02) ◽

pp. 220-221

Author(s):

George H. Weiss ◽

Menachem Dishon

Keyword(s):

Asymptotic Behavior ◽

Differential Equations ◽

Rate Constants ◽

Deterministic Model ◽

Stochastic Theory ◽

Expected Number ◽

Deterministic Theory ◽

Mean Values ◽

Image Position ◽

Fixed Population

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

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Subdivided populations: the characteristic values and rate of loss of alleles

Journal of Applied Probability ◽

10.1017/s0021900200104929 ◽

1977 ◽

Vol 14 (02) ◽

pp. 241-248 ◽

Cited By ~ 3

Author(s):

Keith Gladstien

Keyword(s):

Population Genetics ◽

Asymptotic Behavior ◽

Multiple Allele ◽

Subdivided Populations ◽

Characteristic Values

The characteristic values of certain matrices (underlying multiple allele models in population genetics) are investigated and related to the asymptotic behavior of Pr(at leastkdistinct alleles in the population at timet).

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Asymptotic behavior of a generalization of Bailey's simple epidemic

Advances in Applied Probability ◽

10.2307/1426164 ◽

1971 ◽

Vol 3 (2) ◽

pp. 220-221 ◽

Cited By ~ 2

Author(s):

George H. Weiss ◽

Menachem Dishon

Keyword(s):

Asymptotic Behavior ◽

Differential Equations ◽

Rate Constants ◽

Deterministic Model ◽

Stochastic Theory ◽

Expected Number ◽

Deterministic Theory ◽

Mean Values ◽

Image Position ◽

Fixed Population

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

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A note on the asymptotic properties of correlated random walks

Journal of Applied Probability ◽

10.2307/3213964 ◽

1985 ◽

Vol 22 (4) ◽

pp. 951-956 ◽

Cited By ~ 2

Author(s):

J. B. T. M. Roerdink

Keyword(s):

Asymptotic Behavior ◽

Random Walks ◽

Asymptotic Properties ◽

Simple Relation ◽

Finite Variance ◽

Correlated Random Walk ◽

Expected Number ◽

Similar Relation ◽

Correlated Random Walks ◽

Multistate Random Walks

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.

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Simulating the Emergence of Mutations and Their Subsequent Evolution in an Age-Structured Stochastic Self-Regulating Process with Two Sexes

International Journal of Stochastic Analysis ◽

10.1155/2013/826321 ◽

2013 ◽

Vol 2013 ◽

pp. 1-23 ◽

Cited By ~ 1

Author(s):

Charles J. Mode ◽

Candace K. Sleeman ◽

Towfique Raj

Keyword(s):

Population Genetics ◽

Natural Selection ◽

Branching Process ◽

Initial Population ◽

Expected Number ◽

Parametric Function ◽

Simulation Experiments ◽

Multitype Branching Process ◽

Male Sexual Partner ◽

Age Structured

The stochastic process under consideration is intended to be not only part of the working paradigm of evolutionary and population genetics but also that of applied probability and stochastic processes with an emphasis on computer intensive methods. In particular, the process is an age-structured self-regulating multitype branching process with a genetic component consisting of an autosomal locus with two alleles for females and males. It is within this simple context that mutation will be quantified in terms of probabilities that a given allele mutates to the other per meiosis. But, unlike many models that are currently being used in mathematical population genetics, in which natural selection is often characterized in terms of parameters called fitness by genotype or phenotype, in this paper the parameterization of submodules of the model provides a framework for characterizing natural selection in terms of some of its components. One of these modules consists of reproductive success that is quantified in terms of the total expected number of offspring a female contributes to the population throughout her fertile years. Another component consists of survival probabilities that characterize an individual’s ability to compete for limited environmental resources. A third module consists of a parametric function that expresses the probabilities of survival in a birth cohort of individuals by age for both females and males. A forth module of the model as an acceptance matrix of conditional probabilities such female may show a preference for the genotype or phenotype as her male sexual partner. It is assumed that any force of natural selection acts at the level of the three genotypes under consideration for each sex. By assigning values of the parameters in each of the modules under consideration, it is possible to conduct Monte Carlo simulation experiments designed to study the effects of each component of selection separately or in any combination on a population evolving from a given initial population over some specified period of time.

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Waiting times for multiple occurrences of several events

Journal of Applied Probability ◽

10.2307/3213990 ◽

1988 ◽

Vol 25 (3) ◽

pp. 624-629

Author(s):

Stephen Scheinberg

Keyword(s):

Asymptotic Behavior ◽

Finite Number ◽

Waiting Times ◽

Expected Number

Consider an ‘experiment' which can be repeated indefinitely often resulting in independent random outcomes. Fix attention on a finite number of possible (sets of) outcomes E1, E2, … and define W = W(N1, N2, …) to be the expected number of repetitions needed to ensure that E1 has occurred (at least) N1 times, E2 has occurred (at least) N2 times, etc. This article examines the asymptotic behavior of W as a function of the sum ΣjNj, as the latter grows without bound.

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Exchanges of "Quanta" Among Distinguishable and Indistinguishable Objects

International Journal of Modern Physics B ◽

10.1142/s0217979297001167 ◽

1997 ◽

Vol 11 (19) ◽

pp. 2281-2301

Author(s):

Antonio Bonelli ◽

Stefano Ruffo

Keyword(s):

Asymptotic Behavior ◽

Recursion Relation ◽

Combinatorial Problems ◽

Physical Problem ◽

Explicit Formulas ◽

Recursive Solution ◽

Theoretic Problem ◽

Recursive Formulas ◽

Bose Einstein ◽

Positive Integers

Beginning with a physical problem of exchange of n indistinguishable "quanta" of energy in an ensemble of k oscillators we define a new wide class of combinatorial problems, which also contains statistics intermediate between Fermi–Dirac and Bose–Einstein. One such problem is related to the number theoretic problem of computing the partitions of positive integers. After establishing such a connection, we give explicit formulas for the partitions M(n,k) of an integer n into k parts with k ≤ 4. Moreover, we derive a recursion relation for fixed n and varying k which is valid for any k. A formula which relates partitions to the cardinality of the partition set taking order into account is also derived. The leading asymptotic behavior for n large is obtained for any k. A suggestive interpretation of this formulas is proposed in terms of simplicial lattices. Recursive formulas at fixed k and varying n are then written for k ≤ 5 using the concept of factorial triangle, which is amenable for generalizations to larger k's. The problem of distinct partitions is mapped onto the probability problem of ball removal with replacement, for which we give again recursive solution formulas. Finally, the method of generalized Tartaglia triangle allows the derivation of recursive formulas for limited partitions which take order into account. This latter result is related to the problem of finding the number of distinct ways of dividing n indistinguishable objects into k distinguishable groups, for which explicit summations had been previously found.

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On a conjecture on sparse binomial-type polynomials by Brown, Dilcher and Manna

Analysis and Applications ◽

10.1142/s0219530514500304 ◽

2014 ◽

Vol 12 (05) ◽

pp. 511-522

Author(s):

Wolfgang Gawronski ◽

Thorsten Neuschel

Keyword(s):

Asymptotic Behavior ◽

Independent Sets ◽

Expected Number ◽

Theoretical Context ◽

Binomial Type

We prove a conjecture by Brown, Dilcher and Manna on the asymptotic behavior of sparse binomial-type polynomials arising naturally in a graph-theoretical context in connection with the expected number of independent sets of a graph.

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