We establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.
{"title":"Exponential control of the trajectories of iterated function systems with application to semi-strong GARCH models","authors":"Baye Matar Kandji","doi":"10.1017/jpr.2023.13","DOIUrl":"https://doi.org/10.1017/jpr.2023.13","url":null,"abstract":"\u0000 We establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45204156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.
{"title":"Distribution of consensus in a broadcast-based consensus algorithm with random initial opinions","authors":"S. Shioda, Dai Kato","doi":"10.1017/jpr.2023.9","DOIUrl":"https://doi.org/10.1017/jpr.2023.9","url":null,"abstract":"\u0000 We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43885289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a stochastic SIR (susceptible $rightarrow$ infective $rightarrow$ removed) model in which the infectious periods are modulated by a collection of independent and identically distributed Feller processes. Each infected individual is associated with one of these processes, the trajectories of which determine the duration of his infectious period, his contamination rate, and his type of removal (e.g. death or immunization). We use a martingale approach to derive the distribution of the final epidemic size and severity for this model and provide some general examples. Next, we focus on a single infected individual facing a given number of susceptibles, and we determine the distribution of his outcome (number of contaminations, severity, type of removal). Using a discrete-time formulation of the model, we show that this distribution also provides us with an alternative, more stable method to compute the final epidemic outcome distribution.
{"title":"SIR epidemics driven by Feller processes","authors":"Matthieu Simon","doi":"10.1017/jpr.2023.2","DOIUrl":"https://doi.org/10.1017/jpr.2023.2","url":null,"abstract":"\u0000 We consider a stochastic SIR (susceptible \u0000 \u0000 \u0000 \u0000$rightarrow$\u0000\u0000 \u0000 infective \u0000 \u0000 \u0000 \u0000$rightarrow$\u0000\u0000 \u0000 removed) model in which the infectious periods are modulated by a collection of independent and identically distributed Feller processes. Each infected individual is associated with one of these processes, the trajectories of which determine the duration of his infectious period, his contamination rate, and his type of removal (e.g. death or immunization). We use a martingale approach to derive the distribution of the final epidemic size and severity for this model and provide some general examples. Next, we focus on a single infected individual facing a given number of susceptibles, and we determine the distribution of his outcome (number of contaminations, severity, type of removal). Using a discrete-time formulation of the model, we show that this distribution also provides us with an alternative, more stable method to compute the final epidemic outcome distribution.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44878983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<jats:p>We consider a one-dimensional superprocess with a supercritical local branching mechanism <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline1.png" /> <jats:tex-math> $psi$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where particles move as a Brownian motion with drift <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline2.png" /> <jats:tex-math> $-rho$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and are killed when they reach the origin. It is known that the process survives with positive probability if and only if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline3.png" /> <jats:tex-math> $rho<sqrt{2alpha}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline4.png" /> <jats:tex-math> $alpha=-psi'(0)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. When <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline5.png" /> <jats:tex-math> $rho<sqrt{2 alpha}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, Kyprianou <jats:italic>et al.</jats:italic> [18] proved that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline6.png" /> <jats:tex-math> $lim_{tto infty}R_t/t =sqrt{2alpha}-rho$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> almost surely on the survival set, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline7.png" /> <jats:tex-math> $R_t$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the rightmost position of the support at time <jats:italic>t</jats:italic>. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0021900223000013_inline8.png" /> <jats:tex-math> $mathbb{P}_{delta_x} (R_t >gamma t+theta)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w
{"title":"A large deviation theorem for a supercritical super-Brownian motion with absorption","authors":"Ya-Jie Zhu","doi":"10.1017/jpr.2023.1","DOIUrl":"https://doi.org/10.1017/jpr.2023.1","url":null,"abstract":"\u0000\t <jats:p>We consider a one-dimensional superprocess with a supercritical local branching mechanism <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$psi$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, where particles move as a Brownian motion with drift <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$-rho$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> and are killed when they reach the origin. It is known that the process survives with positive probability if and only if <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$rho<sqrt{2alpha}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, where <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$alpha=-psi'(0)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. When <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$rho<sqrt{2 alpha}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, Kyprianou <jats:italic>et al.</jats:italic> [18] proved that <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$lim_{tto infty}R_t/t =sqrt{2alpha}-rho$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> almost surely on the survival set, where <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$R_t$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is the rightmost position of the support at time <jats:italic>t</jats:italic>. Motivated by this work, we investigate its large deviation, in other words, the convergence rate of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223000013_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$mathbb{P}_{delta_x} (R_t >gamma t+theta)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> as <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43817177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Large deviations of the largest and smallest eigenvalues of $mathbf{X}mathbf{X}^top/n$ are studied in this note, where $mathbf{X}_{ptimes n}$ is a $ptimes n$ random matrix with independent and identically distributed (i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size p and the sample size n is $p=p(n)rightarrowinfty$ with $p(n)={mathrm{o}}(n)$ . This study generalizes one result obtained in [3].
{"title":"Large deviations of extremal eigenvalues of sample covariance matrices","authors":"Denise Uwamariya, Xiangfeng Yang","doi":"10.1017/jpr.2022.130","DOIUrl":"https://doi.org/10.1017/jpr.2022.130","url":null,"abstract":"\u0000\t <jats:p>Large deviations of the largest and smallest eigenvalues of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900222001309_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$mathbf{X}mathbf{X}^top/n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> are studied in this note, where <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900222001309_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$mathbf{X}_{ptimes n}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is a <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900222001309_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$ptimes n$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> random matrix with independent and identically distributed (i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size <jats:italic>p</jats:italic> and the sample size <jats:italic>n</jats:italic> is <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900222001309_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$p=p(n)rightarrowinfty$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> with <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900222001309_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$p(n)={mathrm{o}}(n)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. This study generalizes one result obtained in [3].</jats:p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42613765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.
{"title":"Characterization theorems for pseudo cross-variograms","authors":"Christopher Dörr, M. Schlather","doi":"10.1017/jpr.2022.133","DOIUrl":"https://doi.org/10.1017/jpr.2022.133","url":null,"abstract":"\u0000 Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48099626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract 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 Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.
{"title":"Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process","authors":"Xiequan Fan, Qi-Man Shao","doi":"10.1017/jpr.2022.134","DOIUrl":"https://doi.org/10.1017/jpr.2022.134","url":null,"abstract":"Abstract 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 Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135223522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson (Statist. Comput. 8, 1998). The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed the Haar mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the $Delta$ -guided Metropolis–Haar kernel, is constructed by using the Haar mixture kernel as a proposal kernel. The proposed non-reversible kernel is at least 10 times better than the random-walk Metropolis kernel and Hamiltonian Monte Carlo kernel for the logistic regression and a discretely observed stochastic process in terms of effective sample size per second.
{"title":"Non-reversible guided Metropolis kernel","authors":"K. Kamatani, Xiaolin Song","doi":"10.1017/jpr.2022.109","DOIUrl":"https://doi.org/10.1017/jpr.2022.109","url":null,"abstract":"Abstract We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson (Statist. Comput. 8, 1998). The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed the Haar mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the \u0000$Delta$\u0000 -guided Metropolis–Haar kernel, is constructed by using the Haar mixture kernel as a proposal kernel. The proposed non-reversible kernel is at least 10 times better than the random-walk Metropolis kernel and Hamiltonian Monte Carlo kernel for the logistic regression and a discretely observed stochastic process in terms of effective sample size per second.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"955 - 981"},"PeriodicalIF":1.0,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48071905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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