Critical branching processes with immigration

Ispány, Márton; Pap, Gyula

doi:10.1007/978-3-642-11156-3_10

Cited by 7 publications

(9 citation statements)

References 23 publications

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“…We here analytically derive the novel MR estimator for branching processes (BP) [60,14,61]. We expect that analogous derivations apply to any process with a first order autoregressive representation (PAR) [62], because these processes fulfill Eq. (S5).…”

Section: Supp 1 Applicability Of Mr Estimationmentioning

confidence: 99%

Inferring collective dynamical states from widely unobserved systems

2018

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When assessing spatially extended complex systems, one can rarely sample the states of all components. We show that this spatial subsampling typically leads to severe underestimation of the risk of instability in systems with propagating events. We derive a subsampling-invariant estimator, and demonstrate that it correctly infers the infectiousness of various diseases under subsampling, making it particularly useful in countries with unreliable case reports. In neuroscience, recordings are strongly limited by subsampling. Here, the subsampling-invariant estimator allows to revisit two prominent hypotheses about the brain’s collective spiking dynamics: asynchronous-irregular or critical. We identify consistently for rat, cat, and monkey a state that combines features of both and allows input to reverberate in the network for hundreds of milliseconds. Overall, owing to its ready applicability, the novel estimator paves the way to novel insight for the study of spatially extended dynamical systems.

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Section: Supp 1 Applicability Of Mr Estimationmentioning

confidence: 99%

Inferring collective dynamical states from widely unobserved systems

2018

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“…[5,10] . Ispàny [7] has proven functional limit theorems for the sequence of nearly critical processes with a stationary immigration under weaker conditions on the offspring and immigration variances.…”

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confidence: 99%

Approximation of Fluctuations in a Sequence of Nearly Critical Branching Processes

Rahimov¹

2009

Stochastic Models

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We consider a sequence of discrete time branching processes with generation-dependent immigration, where the offspring mean tends to its critical value 1. Using a martingale approach, we prove functional limit theorems for suitable normalized fluctuations of the process around its mean when the mean number of immigrating individuals tends to infinity. The limiting processes are deterministically time-changed Wiener processes with three different non-linear time change functions, depending on the behavior of the mean and the variance of the number of immigrants. For the normalized sequence of processes we obtain a deterministic approximation. Consequences related to the maxima and the total progeny of the process will be discussed.

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“…Recently, Khusanbaev [9] obtained the rate of convergence in CLT for fluctuation of process (1). Motivated by that paper we continue studying the rate of convergence in one-dimensional functional limit theorem 2.9 of Ispány [7].…”

Section: Introductionmentioning

confidence: 92%

“…The process {X(t), t ∈ R} turns out to be an Ornstein-Uhlenbeck type process driven by a time changed Wiener process. Further systematic investigations are due to Ispány [7], Györfi et al [5], Li [14], [15] Rahimov [18], [19], Khusanbaev [10], [11], [12], [13] and see also references therein. From these papers, it follows that the necessary norming factor and the possible limit process strongly depend on the various conditions that are supposed to hold for the offspring and immigration processes.…”

Section: Introductionmentioning

confidence: 99%