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Subordinator
About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.
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TL;DR: In this paper, it was shown that there exists a function β(t) such that ==================περγεραπεγεγαπαγε απε β(τ) = 1 a.s under certain conditions on at, and iterated logarithm results for mint(X(t+a1)-X( t)>d) as d→∞ are discussed.
6 citations
TL;DR: In this article, a new derivation of Kingman's Kingman formula is given, which contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t] = inf{s: x(t s)= b} where X is a time homogeneous continuous parameter, Markov process and b is a fixed point in its state space.
Abstract: We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {bl, ..., bn}. MARKOV PROCESS; LOCAL TIME; ZERO SET; SUBORDINATOR; EMBEDDED PROCESS
6 citations
TL;DR: In this paper, the authors identify the class of distributions to which the maximum of a Levy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs, and obtain an explicit new distributional identity for the case where the Levy process is an independent sum of a Brownian motion and a general subordinator in terms of a geometrically distributed sum of independent random variables.
Abstract: The goal is to identify the class of distributions to which the distribution of the maximum of a Levy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Levy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Levy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.
6 citations
TL;DR: In this article, an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential function up to time t which is shown to be equivalent to an infinite series under very minor restrictions is given.
6 citations
Dissertation•
01 Jul 2017
TL;DR: In this paper, the biased random walk on the subcritical Galton-Watson trees with leaves is studied and conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring.
Abstract: In this thesis we study biased randomly trapped random walks. As our main motivation, we apply these results to biased walks on subcritical Galton-Watson trees conditioned to survive. This application was initially considered model in its own right.
We prove conditions under which the biased randomly trapped random walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. We also study the regime in which the walk is sub-ballistic; in this case we prove convergence to a stable subordinator.
Furthermore, we study the fluctuations of the walk in the ballistic but sub-diffusive regime. In this setting we show that the walk can be properly centred and rescaled so that it converges to a stable process.
The biased random walk on the subcritical GW-tree conditioned to survive fits suitably into the randomly trapped random walk model; however, due to a lattice effect, we cannot obtain such strong limiting results. We prove conditions under which the walk is ballistic, satisfies an annealed invariance principle and a quenched central limit theorem with environment dependent centring. In these cases the trapping is weak enough that the lattice effect does not have an influence; however, in the sub-ballistic regime it is only possible to obtain converge along specific subsequences.
We also study biased random walks on infinite supercritical GW-trees with leaves. In this setting we determine critical upper and lower bounds on the bias such that the walk satisfies a quenched invariance principle.
6 citations