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Register a new userLocal limit theory and large deviations for supercritical Branching processes
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In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:nge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n earrow infty, and use this to study conditional large deviations of {Y_{Z_n}:nge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:nge1} conditioned on Z_nge v_n.
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Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes [math.PR/0407059]
We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $log Z_n$.
Let $X^{(delta)}$ be a Wishart process of dimension $delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes ${delta^{-1} X_t^{(delta)}, t leq 1 }$ as $delta$ tends to infinity. The process $X^{(delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.
We study two one-parameter families of point processes connected to random matrices: the Sine_beta and Sch_tau processes. The first one is the bulk point process limit for the Gaussian beta-ensemble. For beta=1, 2 and 4 it gives the limit of the GOE, GUE and GSE models of random matrix theory. In particular, for beta=2 it is a determinantal point process conjectured to have similar behavior to the critical zeros of the Riemann zeta-function. The second process can be obtained as the bulk scaling limit of the spectrum of certain discrete one-dimensional random Schrodinger operators. Both processes have asymptotically constant average density, in our normalization one expects close to lambda/(2pi) points in a large interval of length lambda. Our main results are large deviation principles for the average densities of the processes, essentially we compute the asymptotic probability of seeing an unusual average density in a large interval. Our approach is based on the representation of the counting functions of these processes using stochastic differential equations. We also prove path level large deviation principles for the arising diffusions. Our techniques work for the full range of parameter values. The results are novel even in the classical beta=1, 2 and 4 cases for the Sine_beta process. They are consistent with the existing rigorous results on large gap probabilities and confirm the physical predictions made using log-gas arguments.
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann-Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann-Liouville process.
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