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We consider a branching random walk on the real line with a stationary and ergodic environment (ξ n) indexed by time, in which a particle of generation n gives birth to a random number of particles of the next generation, which move on the real line; the joint distribution of the number of children and their displacements on the real line depends on the environment ξ n at time n. Let Z n be the counting measure at time n , which counts the number of particles of generation n situated in a Borel set of the real line. For the case where the corresponding branching process is supercritical, we establish limit theorems such as large and moderate deviation principles, central and local limit theorems on the counting measures Z n , convergence of the free energy, law of large numbers on the leftmost and rightmost positions at time n , and the convergence to infinite divisible laws. The varying environment case is also considered. [ABSTRACT FROM AUTHOR]

Let (N , A 1 , A 2 , ...) be a sequence of random variables with N ∈ N ∪ { ∞ } and A i ∈ R + . We are interested in asymptotic properties of non-negative solutions of the distributional equation Z = (d) ∑ i = 1 N A i Z i , where Z i are non-negative random variables independent of each other and independent of (N , A 1 , A 2 , ...) , each having the same distribution as Z which is unknown. For a solution Z with finite mean, we prove that for a given α > 1 , P (Z > x) is a function regularly varying at ∞ of index − α if and only if the same is true for P (Y 1 > x) , where Y 1 = ∑ i = 1 N A i . The result completes the sufficient condition obtained by Iksanov & Polotskiy (2006) on the branching random walk. A similar result on sufficient condition is also established for the case where α = 1. [ABSTRACT FROM AUTHOR]

Let ( Z n ) be a supercritical branching process in a random environment ξ = ( ξ n ) . We establish a Berry–Esseen bound and a Cramér’s type large deviation expansion for log Z n under the annealed law P . We also improve some earlier results about the harmonic moments of the limit variable W = lim n → ∞ W n , where W n = Z n / E ξ Z n is the normalized population size. [ABSTRACT FROM AUTHOR]

Chen (2001) derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen’s results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen’s by relaxing the second moment condition used by Chen to a moment condition of the form E X ( ln + X ) 1 + λ < ∞ . In the rate functions that we find for a branching random walk, we figure out some new terms which did not appear in Chen’s work. The results are established in the more general framework, i.e. for a branching random walk with a random environment in time. The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment. [ABSTRACT FROM AUTHOR]

Consider the random walk G n : = g n ... g 1 , n ⩾ 1 , where (g n) n ⩾ 1 is a sequence of independent and identically distributed random elements with law μ on the general linear group GL (V) with V = R d . Under suitable conditions on μ , we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients 〈 f , G n v 〉 , where v ∈ V and f ∈ V ∗ . Our approach is based on the Hölder regularity of the invariant measure of the Markov chain G n ⋅ x = R G n v on the projective space of V with the starting point x = R v , under the changed measure. [ABSTRACT FROM AUTHOR]

Abstract: Let be a supercritical branching process in a random environment , and be the limit of the normalized population size . We show large and moderate deviation principles for the sequence (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of , and show an equivalence for all the moments of . Central limit theorems on and are also established. [Copyright &y& Elsevier]

Abstract: We first obtain exponential inequalities for martingales. Let be a sequence of martingale differences relative to a filtration and set . We prove that if for some , and all , a.s., then for some constant (depending only on and ) and all , , where if and if ; the converse also holds if are independent and identically distributed. This extends Bernstein’s inequality for and Hoeffding’s inequality for . We then apply the preceding result to establish exponential concentration inequalities for the free energy of directed polymers in a random environment and obtain upper bounds for its rates of convergence (in probability, almost surely and in ); we also give an expression for the free energy in terms of those of some multiplicative cascades, which improves an inequality of Comets and Vargas [Francis Comets, Vincent Vargas, Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267–277 (electronic)] to an equality. [Copyright &y& Elsevier]

Let (g n) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d (R) , with law μ. Consider the random walk G n : = g n ... g 1 . Denote respectively by ‖ G n ‖ and ρ (G n) the operator norm and the spectral radius of G n. For log ‖ G n ‖ and log ρ (G n) , we prove moderate deviation principles under exponential moment and strong irreducibility conditions on μ ; we also establish moderate deviation expansions in the normal range [ 0 , o (n 1 / 6) ] and Berry–Esseen bounds under the additional proximality condition on μ. Similar results are found for the couples (X n x , log ‖ G n ‖) and (X n x , log ρ (G n)) with target functions, where X n x : = G n ⋅ x is a Markov chain and x is a starting point on the projective space P (R d). [ABSTRACT FROM AUTHOR]

Let (g n) n ⩾ 1 be a sequence of independent identically distributed d × d real random matrices with Lyapunov exponent λ. For any starting point x on the unit sphere in R d , we deal with the norm | G n x | , where G n ≔ g n ... g 1 . The goal of this paper is to establish precise asymptotics for large deviation probabilities P (log | G n x | ⩾ n (q + l)) , where q > λ is fixed and l is vanishing as n → ∞. We study both invertible matrices and positive matrices and give analogous results for the couple (X n x , log | G n x |) with target functions, where X n x = G n x ∕ | G n x |. As applications we improve previous results on the large deviation principle for the matrix norm ‖ G n ‖ and obtain a precise local limit theorem with large deviations. [ABSTRACT FROM AUTHOR]

Let (X i) i ≥ 1 be a stationary sequence. Denote m = ⌊ n α ⌋ , 0 < α < 1 , and k = ⌊ n ∕ m ⌋ , where ⌊ a ⌋ stands for the integer part of a. Set S j ∘ = ∑ i = 1 m X m (j − 1) + i , 1 ≤ j ≤ k , and (V k ∘) 2 = ∑ j = 1 k (S j ∘) 2. We prove a Cramér type moderate deviation expansion for P (∑ j = 1 k S j ∘ ∕ V k ∘ ≥ x) as n → ∞. Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed. [ABSTRACT FROM AUTHOR]

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales. [ABSTRACT FROM AUTHOR]

Abstract: We give an extension of Hoeffding’s inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev. [Copyright &y& Elsevier]