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Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes
| Content Provider | Scilit |
|---|---|
| Author | Löpker, Andreas Stadje, Wolfgang |
| Copyright Year | 2011 |
| Description | We consider the level hitting times $τ_{y}$= inf{t≥ 0 $|X_{t}$=y} and the running maximum $processM_{t}$= $sup{X_{s}$| 0 ≤s≤t} of a growth-collapse process $(X_{t})_{t≥0}$, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of $τ_{y}$can be determined in terms of the extended generator $ofX_{t}$and give a power series expansion of the reciprocal of Ee^{−sτ_{y}$}$. We prove asymptotic results for $τ_{y}andM_{t}$: for example, ifm(y) = $Eτ_{y}$is of rapid variation thenM_{t}$/m^{-1}$(t) $→^{w}$1 ast→ ∞, $wherem^{-1}$is the inverse function ofm, while ifm(y) is of regular variation with indexa∈ (0, ∞) $andX_{t}$is ergodic, thenM_{t}$/m^{-1}$(t) converges weakly to a Fréchet distribution with exponenta. In several special cases we provide explicit formulae. |
| Related Links | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/41449E28C81047D9939F4FA632074D71/S0021900200007889a.pdf/div-class-title-hitting-times-and-the-running-maximum-of-markovian-growth-collapse-processes-div.pdf |
| Ending Page | 312 |
| Page Count | 18 |
| Starting Page | 295 |
| ISSN | 00219002 |
| e-ISSN | 14756072 |
| DOI | 10.1017/s0021900200007889 |
| Journal | Journal of applied probability |
| Issue Number | 02 |
| Volume Number | 48 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 2011-06-01 |
| Access Restriction | Open |
| Subject Keyword | Journal of applied probability Mathematical Physics collapse Process Piecewise Deterministic Markov Process Hitting Time Running Maximum Asymptotic Behavior Regular Variation Separable Jump Measure |
| Content Type | Text |
| Resource Type | Article |
| Subject | Statistics and Probability Statistics, Probability and Uncertainty |
