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Register a new userSelf-normalized Cramer moderate deviations for a supercritical Galton-Watson process
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Added by
Xiequan Fan
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Let $(Z_n)_{ngeq0}$ be a supercritical Galton-Watson process. Consider the Lotka-Nagaev estimator for the offspring mean. In this paper, we establish self-normalized Cram{e}r type moderate deviations and Berry-Esseens bounds for the Lotka-Nagaev estimator. The results are believed to be optimal or near optimal.
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Let $(xi_i,mathcal{F}_i)_{igeq1}$ be a sequence of martingale differences. Set $S_n=sum_{i=1}^nxi_i $ and $[ S]_n=sum_{i=1}^n xi_i^2.$ We prove a Cramer type moderate deviation expansion for $mathbf{P}(S_n/sqrt{[ S]_n} geq x)$ as $nto+infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.
Let $(X _i)_{igeq1}$ be a stationary sequence. Denote $m=lfloor n^alpha rfloor, 0< alpha < 1,$ and $ k=lfloor n/m rfloor,$ where $lfloor a rfloor$ stands for the integer part of $a.$ Set $S_{j}^circ = sum_{i=1}^m X_{m(j-1)+i}, 1leq j leq k,$ and $ (V_k^circ)^2 = sum_{j=1}^k (S_{j}^circ)^2.$ We prove a Cramer type moderate deviation expansion for $mathbb{P}( sum_{j=1}^k S_{j}^circ /V_k^circ geq x)$ as $nto infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
At each site of a supercritical Galton-Watson tree place a parking spot which can accommodate one car. Initially, an independent and identically distributed number of cars arrive at each vertex. Cars proceed towards the root in discrete time and park in the first available spot they come to. Let $X$ be the total number of cars that arrive to the root. Goldschmidt and Przykucki proved that $X$ undergoes a phase transition from being finite to infinite almost surely as the mean number of cars arriving to each vertex increases. We show that $EX$ is finite at the critical threshold, describe its growth rate above criticality, and prove that it increases as the initial car arrival distribution becomes less concentrated. For the canonical case that either 0 or 2 cars arrive at each vertex of a $d$-ary tree, we give improved bounds on the critical threshold and show that $P(X = 0)$ is discontinuous.
We are concerned with exploring the probabilities of first order statements for Galton-Watson trees with $Poisson(c)$ offspring distribution. Fixing a positive integer $k$, we exploit the $k$-move Ehrenfeucht game on rooted trees for this purpose. Let $Sigma$, indexed by $1 leq j leq m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $Sigma$. Let $x_{j}(c)$ denote the true probability of the class $j in Sigma$ under $Poisson(c)$ regime, and $vec{x}(c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $Gamma$, and a map $Psi = Psi_{c}: D rightarrow D$ such that $vec{x}(c)$ is a fixed point of $Psi_{c}$, and starting with any distribution $vec{x} in D$, we converge to this fixed point via $Psi$ because it is a contraction. We show this both for $c leq 1$ and $c > 1$, though the techniques for these two ranges are quite different.
The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $lambda_2=0$ and (ii) when it is geometric($p$) we have $lambda_2 le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $lambda_c(n)$ for prolonged persistence on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $lambda_c(n) to 0$ we give estimates on the rate of convergence. Physicists tell us that $lambda_c(n) sim 1/Lambda(n)$ where $Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
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