Evaluating the small deviation probabilities for subordinated Lévy processes

Linde, Werner; Shi, Zhan

doi:10.1016/j.spa.2004.04.001

Cited by 21 publications

(29 citation statements)

References 16 publications

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“…However, recently, a number of works appeared where small deviations are studied for cases with rather arbitrary behaviour of entropy, see e.g. [9,10]. In particular, a slow increase of N (ε) when ε tends to zero is not excluded at all.…”

Section: Motivationmentioning

confidence: 99%

Small deviation probability via chaining

Аурзада

Lifshits

2008

Stochastic Processes and their Applications

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We obtain several extensions of Talagrand's lower bound for the small deviation probability using metric entropy. For Gaussian processes, our investigations are focused on processes with sub-polynomial and, respectively, exponential behaviour of covering numbers. The corresponding results are also proved for non-Gaussian symmetric stable processes, both for the cases of critically small and critically large entropy. The results extensively use the classical chaining technique; at the same time they are meant to explore the limits of this method.

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Section: Motivationmentioning

confidence: 99%

Small deviation probability via chaining

Аурзада

Lifshits

2008

Stochastic Processes and their Applications

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“…In this language, the process X(t) is called subordinated to the parent process X(u) via the leading process U(t) [22]. Subordinated processes find applications in finance [23] and are also studied in mathematics [24][25][26][27][28]. We need to stress here, that U(t) is not the inverse of a stable motion.…”

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confidence: 99%

Weak ergodicity breaking in an anomalous diffusion process of mixed origins

Thiel

Sokolov

2014

Phys. Rev. E

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The ergodicity breaking parameter is a measure for the heterogeneity among different trajectories of one ensemble. In this report this parameter is calculated for fractional Brownian motion with a random change of time scale, often called "subordination". We proceed to show that this quantity is the same as the known CTRW case.PACS numbers: 05.40.Fb,05.40.Fb "Weak ergodicity breaking" is a term which occurred in the spotlight a few years ago, see for instance [1][2][3][4][5][6] with respect to observables showing aging behavior. The term "aging" applies to the specific behavior pertinent to the values of an observable measured at two distinct instants of time, like the mean squared displacement (MSD) in the time interval between t 1 and t 2 , ∆X 2 (t 1 , t 2 ) as depending on the age t 1 of the system (assumed to be prepared at t = 0) at the beginning of observation. Aging here essentially refers to a non-stationarity of the observed process, in which ∆X 2 (t 1 , t 2 ) cannot be expressed via the time lag τ = t 2 − t 1 only and keeps a considerable dependence on t 1 as a stand-alone variable.Let us concentrate now on the ergodicity properties the squared displacement during the time interval τ in a measurement starting at t. In the case when the data acquisition in each single run of the process takes time from t = 0 to t = T , the time average of ∆X 2 (t, t + τ ) can be performed:The process is considered ergodic provided ∆X 2 (τ ) = ∆X 2 (t, t + τ ) . The discussion of the ergodicity implies that the corresponding limit does exist in some probabilistic sense (say, in probability) and is equal to the ensemble mean. Note that the time-average over the data acquisition interval removes the explicit t-dependence. For non-stationary processes the ensemble mean at any t depends on t, while the time-integration over the data acquisition interval removes this dependence. Therefore, even provided the integral above converges in a whatever sense, it cannot converge to all ∆X 2 (t, t + τ ) simultaneously, and our process is trivially non-ergodic.Whether the random process is stationary or not, one can ask how diverse are its different realizations, i.e. how different are two trajectories of the process with respect to some time-averaged observable O(τ ; t) (in our case the MSD, O(τ ; t) = ∆X 2 (t, t + τ )). To this purpose one considers a fixed-T approximation to eq.(1)At any finite T the value of O(τ ) T is a random variable. For both, stationary and non-stationary processes O(τ, t), one can consider a parameter describing the strength of fluctuations of O(τ ) T . As a measure of homogeneity or heterogeneity of different trajectories one can take the relative dispersion of the O(τ ) T in different realizations of the process:The parameter J (often called "ergodicity breaking parameter") shows how different are different trajectories of the process with respect to the observable O. For stationary processes, vanishing of J in the long time limit indeed implies ergodicity [7], and its non-vanishing witnesses against erg...

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“…It might be fruitful, as suggested by a referee, to try to analyse a subordinated Brownian motion along these lines (cf. also [22]). …”

Section: Lemma 45 Supposementioning

confidence: 87%

The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component

Buchmann

Maller

2009

Probab. Theory Relat. Fields

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We give a "small time" functional version of Chung's "other" law of the iterated logarithm for Lévy processes with non-vanishing Brownian component. This is an analogue of the "other" law of the iterated logarithm at "large times" for Lévy processes and random walks with finite variance, as extended to a functional version by Wichura. As one of many possible applications, we mention a functional law for a two-sided passage time process.Keywords Lévy process · Local behaviour · Almost sure convergence · Iterated logarithm laws · Other law of the iterated logarithm · Cluster sets · Functional limit theorem Mathematics Subject Classification (2000) 60G51 · 60F15 · 60F17 · 60F05 · 60J65 · 60J75

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Evaluating the small deviation probabilities for subordinated Lévy processes

Cited by 21 publications

References 16 publications

Small deviation probability via chaining

Small deviation probability via chaining

Weak ergodicity breaking in an anomalous diffusion process of mixed origins

The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component

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