Hölderian invariance principle for Hilbertian linear processes

Račkauskas, Alfredas; Suquet, Charles

doi:10.1051/ps:2008011

Cited by 3 publications

(1 citation statement)

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“…almost surely for every t ∈ [0, ∞). The rescaled associated random walk converges (see [35], Theorem 3) to a Brownian motion (S t ) t≥0 , that is,…”

Section: Diffusion Approximationmentioning

confidence: 99%

Branching diffusions in random environment

Böinghoff,

Hutzenthaler

2011

Preprint

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We consider the diffusion approximation of branching processes in random environment (BPREs). This diffusion approximation is similar to and mathematically more tractable than BPREs. We obtain the exact asymptotic behavior of the survival probability. As in the case of BPREs, there is a phase transition in the subcritical regime due to different survival opportunities. In addition, we characterize the process conditioned to never go extinct and establish a backbone construction. In the strongly subcritical regime, mean offspring numbers are increased but still subcritical in the process conditioned to never go extinct. Here survival is solely due to an immortal individual, whose offspring are the ancestors of additional families. In the weakly subcritical regime, the mean offspring number is supercritical in the process conditioned to never go extinct. Thus this process survives with positive probability even if there was no immortal individual.

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“…almost surely for every t ∈ [0, ∞). The rescaled associated random walk converges (see [35], Theorem 3) to a Brownian motion (S t ) t≥0 , that is,…”

Section: Diffusion Approximationmentioning

confidence: 99%

Branching diffusions in random environment

Böinghoff,

Hutzenthaler

2011

Preprint

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On limit theorems for Banach-space-valued linear processes

Račkauskas

Suquet

2010

Lith Math J

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Let (X k) k∈Z be a linear process with values in a separable Hilbert space H given by X k = ∞ j=0 (j + 1) −N ε k−j for each k ∈ Z, where N : H → H is a bounded, linear normal operator and (ε k) k∈Z is a sequence of independent, identically distributed H-valued random variables with Eε 0 = 0 and Eε 0 2 < ∞. We investigate the central and the functional central limit theorem for (X k) k∈Z when the series of operator norms ∞ j=0 (j + 1) −N op diverges. Furthermore we show that the limit process in case of the functional central limit theorem generates an operator self-similar process.

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