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Exponential inequalities for selfnormalized martingales with applications, in "Annals of Applied Probability
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"... Abstract. We propose several exponential inequalities for selfnormalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching ..."
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Abstract. We propose several exponential inequalities for selfnormalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided. 1.
LIKELY PATH TO EXTINCTION IN SIMPLE BRANCHING MODELS WITH LARGE INITIAL POPULATION
, 2005
"... We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time we study the GaltonWatson process and in continuous time the Branching diffusion. The most likely paths are found with the help of the Large Deviation Princip ..."
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We give explicit formulae for most likely paths to extinction in simple branching models when initial population is large. In discrete time we study the GaltonWatson process and in continuous time the Branching diffusion. The most likely paths are found with the help of the Large Deviation Principle (LDP). We also find asymptotics for the extinction probability, which gives a new expression in continuous time and recovers the known formula in discrete time. Due to the nonnegativity of the processes, the proof of LDP at the point of extinction uses a nonstandard argument of independent interest.
Small positive values and lower large deviations for supercritical branching processes in random environment
, 2011
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Lower Deviations Porbabilities For Supercritical . . .
, 2005
"... There is a wellknown sequence of constants cn describing the growth of supercritical GaltonWatson processes Zn. With “lower deviation probabilities ” we refer to P(Zn = kn) with kn = o(cn) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. ..."
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There is a wellknown sequence of constants cn describing the growth of supercritical GaltonWatson processes Zn. With “lower deviation probabilities ” we refer to P(Zn = kn) with kn = o(cn) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Zn+1/Zn. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the wellknown Cramér method for proving large deviations of sums of independent variables to our needs.
Large deviations for sums . . .
, 2006
"... In this paper we study the large deviation behavior of sums of i.i.d. random variables Xi defined on a supercritical GaltonWatson process Z. We assume the finiteness of the moments EX 2 1 and EZ1 log Z1. The underlying interplay of the partial sums of the Xi and the lower deviation probabilities ..."
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In this paper we study the large deviation behavior of sums of i.i.d. random variables Xi defined on a supercritical GaltonWatson process Z. We assume the finiteness of the moments EX 2 1 and EZ1 log Z1. The underlying interplay of the partial sums of the Xi and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on
Applied Probability Trust (17 May 2012) UPPER DEVIATIONS FOR SPLIT TIMES OF BRANCHING PROCESSES HAMED AMINI, ∗ AND
"... Upper deviation results are obtained for the split time of a supercritical continuoustime Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expressio ..."
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Upper deviation results are obtained for the split time of a supercritical continuoustime Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times which shows that the scaling is completely different from the upper deviations.
5 LIKELY PATH TO EXTINCTION IN SIMPLE BRANCHING MODELS WITH LARGE INITIAL POPULATION
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LARGE DEVIATIONS FOR THE EMPIRICAL DISTRIBUTION IN THE BRANCHING RANDOM WALK
"... Abstract. We consider the branching random walk (Zn)n≥0 on R where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. It is well known that ¯ Zn(A) → ν(A) almost surely as n → ∞ for typical A’s, where ¯ Zn is the empirical pa ..."
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Abstract. We consider the branching random walk (Zn)n≥0 on R where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. It is well known that ¯ Zn(A) → ν(A) almost surely as n → ∞ for typical A’s, where ¯ Zn is the empirical particles distribution at generation n and ν is the standard Gaussian measure on R. We therefore analyze the rate at which P ( ¯ Zn(A)> ν(A) + ɛ) and P ( ¯ Zn(A) < ν(A) − ɛ) go to zero for any ɛ> 0. We show that the decay is doubly exponential in either n or √ n, depending on A and ɛ and find the leading coefficient in the top exponent. To the best of our knowledge, this is the first time such large deviation probabilities are treated in this model.
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"... Large deviations for the empirical distribution in the branching random walk* ..."
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Large deviations for the empirical distribution in the branching random walk*