Limit theorems for a supercritical Poisson random indexed branching process
TL;DR: A law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt are shown.
Abstract: Let {Zn, n = 0, 1, 2, . . .} be a supercritical branching process, {Nt, t ≥ 0} be a Poisson process independent of {Zn, n = 0, 1, 2, . . .}, then {ZNt, t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt.
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TL;DR: This work improves the traditional ICF approach by integrating the locality-sensitive hashing (LSH) technique, to realize secure and reliable data publishing and shows that ICFLSH performs better than the competitive approaches in terms of service recommendation accuracy, efficiency, and the capability of privacy-preservation.
Abstract: Item-based collaborative filtering (i.e., ICF) technique has been widely recruited to make service recommendations in the big data environment. However, the ICF technique only performs well when the data for service recommendation decision-making are stored in a physically centralized manner, while they often fail to recommend appropriate services to a target user in the distributed environment where the involved multiple parties are reluctant to release their data to each other due to privacy concerns. Considering this drawback, we improve the traditional ICF approach by integrating the locality-sensitive hashing (LSH) technique, to realize secure and reliable data publishing. Furthermore, through integrating the published data with little privacy across different platforms, appropriate services are recommended based on our suggested recommendation approach named ICF LSH . At last, simulated experiments are conducted on WS-DREAM data set. Experiment results show that ICF LSH performs better than the competitive approaches in terms of service recommendation accuracy, efficiency, and the capability of privacy-preservation.
33 citations
Cites background from "Limit theorems for a supercritical ..."
...With the mapping relationships between services and their complete indices (in the form of vector [38], [39]), a hash table is formed offline....
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TL;DR: In this article, the authors investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes and show that the solutions to MSDEs driven by point processes can be approximated by solutions to averaged SDEs in the sense of both convergence in mean square and convergence in probability.
Abstract: The purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.
12 citations
TL;DR: It is proved that Poisson noises can induce synchronization and sufficient conditions are established to achieve complete synchronization with probability 1.0 based on the stability theory of stochastic differential equations driven by Poisson process.
Abstract: The different Poisson noise-induced complete synchronization of the global coupled dynamical network is investigated. Based on the stability theory of stochastic differential equations driven by Poisson process, we can prove that Poisson noises can induce synchronization and sufficient conditions are established to achieve complete synchronization with probability 1. Furthermore, numerical examples are provided to show the agreement between theoretical and numerical analysis.
8 citations
TL;DR: In this paper, the large and moderate deviations for a renewal randomly indexed branching process (ZNt) were derived, where Zn is a Galton-Watson process and Nt is a renewal process which is independent of Zn.
7 citations
TL;DR: In this paper, the authors consider a Galton-Watson process and an independent Poisson process and show the large deviation results for P (Z N t ≤ e c t ) and P ( Z N t ≥ e c T ).
6 citations
Cites background from "Limit theorems for a supercritical ..."
...On the other hand, the authors of [8] studied the asymptotic properties of the probabilities...
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...1 when p1 > 0, the case p1 = 0 is given in [8]....
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...2 of [8]....
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References
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Book•
27 Mar 1998
TL;DR: The LDP for Abstract Empirical Measures and applications-The Finite Dimensional Case and Applications of Empirically Measures LDP are presented.
Abstract: LDP for Finite Dimensional Spaces.- Applications-The Finite Dimensional Case.- General Principles.- Sample Path Large Deviations.- The LDP for Abstract Empirical Measures.- Applications of Empirical Measures LDP.
5,578 citations
TL;DR: From a statistical inference point of view the first quantity above is of some interest and the probability that the running time of this algorithm deviates from its expected value corresponds to the tail probabilities and large deviations associated with branching processes.
Abstract: Let $\{Z_n\}^\infty_0$ be a Galton-Watson branching process with offspring distribution $\{p_j\}^\infty_0$. We assume throughout that $p_0 = 0, p_j
eq 1$ for any $j \geq 1$ and $1 \varepsilon\big),\quad P(|W_n - W| > \varepsilon)$, $P\big(\big|\frac{Z_{n+1}}{Z_n} - m \mid > \varepsilon\big|W \geq a\big)$ for $\varepsilon > 0$ and $a > 0$ under various moment conditions on $\{p_j\}$. It is shown that the rate for the first one is geometric if $p_1 > 0$ and supergeometric if $p_1 = 0$, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.
105 citations
TL;DR: In this article, the authors show large and moderate deviation principles for the sequence log Z n (with appropriate normalization) of a supercritical branching process in a random environment ξ, and W is the limit of the normalized population size Z n /E [ Z n | ξ ].
61 citations
"Limit theorems for a supercritical ..." refers background in this paper
...According to the Gärtner–Ellis theorem (see [2]) and the results in [6], we have our first main result....
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...The following result is a general result on large deviations; see [6]....
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TL;DR: In this paper, the authors studied the large deviation aspects of supercritical branching process convergence and showed that these large deviation probabilities for these two sequences decay geometrically and under appropriate conditioning supergeometrinically.
Abstract: Let $\{Z_n: n \geq 0\}$ be a $p$-type $(p \geq 2)$ supercritical branching process with mean matrix $M$. It is known that for any $l$ in $R^p$, $\big(\frac{l\cdot Z_n}{1\cdot Z_n} - \frac{l\cdot (Z_n M)}{1\cdot Z_n}\big) \text{and} \big(\frac{l\cdot Z_n}{1\cdot Z_n} - \frac{l\cdot v^{(1)}}{1 \cdot v^{(1)}}\big)$ converge to 0 with probability 1 on the set of nonextinction, where $v^{(1)}$ is the left eigenvector of $M$ corresponding to its maximal eigenvalue $\rho$ and 1 is the vector with all components equal to one. In this paper we study the large deviation aspects of this convergence. It is shown that the large deviation probabilities for these two sequences decay geometrically and under appropriate conditioning supergeometrically.
43 citations
"Limit theorems for a supercritical ..." refers methods in this paper
...References [1] Athreya, K. B. (1994)....
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...In [13], based on the idea of Athreya [1], the author derived the large deviation for ZNt+1/ZNt ....
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TL;DR: In this article, an extension of the Bienayme-Galton-Watson branching process is proposed to model the short-term behavior of stock prices, and the model is applied to daily closing prices of a sample of common stocks.
Abstract: An extension of the Bienayme-Galton-Watson branching process is proposed to model the short-term behavior of stock prices. Measured in units of $1/8, prices are integer-valued, yet they have many of the characteristics of the multiplicative random walk: e.g., uncorrelated increments. Unlike the random walk higher moments of returns (price relatives) depend on initial price. Conditional distributions of returns over short periods, such as one day, are thick-tailed, but tail thickness decreases as either initial price or the length of the period increases. As initial price approaches infinity, the normalized return approaches a compound-Poisson process-the compound-events model. The model is applied to daily closing prices of a sample of common stocks.
24 citations
"Limit theorems for a supercritical ..." refers methods in this paper
...[3] Dion, J. P. and Epps, T. W. (1999)....
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...The statistical investigation on various estimates and some parameters of the process were performed by Dion and Epps [3]....
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...Introduction The model of random indexed branching process is one of the important extensions of the Galton–Watson branching process, which was introduced by Epps [4] in order to study the evolution of stock prices....
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...[4] Epps, T. W. (1996)....
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...Introduction The model of random indexed branching process is one of the important extensions of the Galton–Watson branching process, which was introduced by Epps [4] in order to study the evolution of stock prices....
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