T. E. Harris and branching processes

T. E. Harris was a pioneer par excellence in many fields of probability theory. In this paper, we give a brief survey of the many fundamental contributions of Harris to the theory of branching processes, starting with his doctoral work at Princeton in the late forties and culminating in his fundamental book "The Theory of Branching Processes," published in 1963.Comment: Published in at http://dx.doi.org/10.1214/10-AOP599 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

From the mathematics olympiad to the maximum principle

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The Shrinking unit ball

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The vacillating mathematician: 2. A stochastic version

In the first part of this article, the author described the deterministic version of the Vacillating Mathematician. Stochastic generalizations of this idea lead to interesting Markov chain problems

Entropy maximization

It is shown that (i) every probability density is the unique maximizer of relative entropy in an appropriate class and (ii) in the class of all pdf f that satisfy ∫ fhi dμ = λi for i = 1, 2,..., ... k the maximizer of entropy is an f0 that is proportional to exp(∑ci hi ) for some choice of ci . An extension of this to a continuum of constraints and many examples are presented

The vacillating mathematician: 1. Where does she end up?

The problem of a mathematician who walks from her home to her office and changes her mind repeatedly during this walk is discussed. Stochastic generalizations of this problem can be used to model many real-life situations

Growth of preferential attachment random graphs via continuous-time branching processes

A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time branching scheme and, using the branching process apparatus, several results on the graph model asymptotics are obtained, some extending previous results, such as growth rates for a typical degree and the maximal degree, behavior of the vertex where the maximal degree is attained, and a law of large numbers for the empirical distribution of degrees which shows certain ``scale-free'' or ``power-law'' behaviors.Comment: 20 page

Critical age-dependent branching Markov processes and their scaling limits

This paper studies: (i) the long-time behaviour of the empirical distribution of age and normalized position of an age-dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions

Gibbs measures asymptotics

Let (Ω, B, ν) be a measure space and H:Ω→ R+ be B measurable. Let ∫ Ωe-Hdν < ∞. For 0 < T < 1 let μH,T (·) be the probability measure defined by μH,T (A) = (∫A e-H/T dν)/(∫Ω e-H/T dν ), A ∈ B. In this paper, we study the behavior of μH,T (·) as T ↓ 0 and extend the results of Hwang (1980, 1981). When Ω is R and H achieves its minimum at a single value x0 (single well case) and H(·) is Holder continuous at x0 of order α, it is shown that if XT is a random variable with probability distribution μH,T (·) then as T ↓ 0, i) XT→ x0 in probability; ii) (Xt - x0)T-1/α converges in distribution to an absolutely continuous symmetric distribution with density proportional to e-cα|x|α for some 0 < cα < ∞. This is extended to the case when H achieves its minimum at a finite number of points (multiple well case). An extension of these results to the case H : Rn→ R+ is also outlined

Mathematics of risk taking

A simple mathematical model for an investor's gains and losses over time shows that, in the long run, those with large sums to invest have an excellent chance of reaching their goal while the marginal investors have a high probability of going bankrupt. A greedy investor, rich or poor, will hit the bottom in the long run, with probability one. Consequences for the population at large are discussed