scholarly journals Asymptotics for the maximum of a modulated random walk with heavy-tailed increments

2002 ◽  
pp. 37-52 ◽  
Author(s):  
S. Foss ◽  
S. Zachary
Keyword(s):  
Random Walk ◽  
Heavy Tailed

scholarly journals A Continuous-Time Random Walk Extension of the Gillis Model

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1431
Author(s):  
Gaia Pozzoli ◽  
Mattia Radice ◽  
Manuele Onofri ◽  
Roberto Artuso
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Hitting Times ◽  
Occupation Times ◽  
Theoretical Predictions ◽  
Normal Diffusion ◽  
Heavy Tailed ◽  
The One ◽  
Waiting Periods

We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (01) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Y n , N n ) n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n ) n≥1 defined by Z n = Y n - Σ n-1, where It is shown that the probability that the maximum M = max n≥1 Z n exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

scholarly journals Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

2021 ◽  
Vol 58 (1) ◽  
pp. 217-237
Author(s):  
Denis Denisov ◽  
Elena Perfilev ◽  
Vitali Wachtel
Keyword(s):  
Random Walk ◽  
Tail Asymptotics ◽  
Tail Probabilities ◽  
Tail Behaviour ◽  
Negative Drift ◽  
Heavy Tailed

AbstractWe study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

The Uniform Local Asymptotics of the Overshoot of a Random Walk with Heavy-Tailed Increments

2009 ◽  
Vol 25 (3) ◽  
pp. 508-521 ◽  
Author(s):  
Guochun Chen ◽  
Yuebao Wang ◽  
Fengyang Cheng
Keyword(s):  
Random Walk ◽  
Local Asymptotics ◽  
Heavy Tailed

Some asymptotic results for transient random walks with applications to insurance risk

2001 ◽  
Vol 38 (01) ◽  
pp. 108-121 ◽  
Author(s):  
Aleksandras Baltrūnas
Keyword(s):  
Random Walk ◽  
Ruin Probability ◽  
Risk Theory ◽  
Insurance Risk ◽  
Asymptotic Results ◽  
First Passage ◽  
Tail Behaviour ◽  
Heavy Tailed ◽  
Exact Rate Of Convergence ◽  
Passage Times

We consider a real-valued random walk which drifts to -∞ and is such that the step distribution is heavy tailed, say, subexponential. We investigate the asymptotic tail behaviour of the distribution of the upwards first passage times. As an application, we obtain the exact rate of convergence for the ruin probability in finite time. Our result supplements similar theorems in risk theory.

scholarly journals Asymptotics of the density of the supremum of a random walk with heavy-tailed increments

2006 ◽  
Vol 43 (03) ◽  
pp. 874-879 ◽  
Author(s):  
Yuebao Wang ◽  
Kaiyong Wang
Keyword(s):  
Random Walk ◽  
Ladder Height ◽  
Heavy Tailed ◽  
Equivalent Conditions

Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a random walk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.

scholarly journals On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold

2009 ◽  
Vol 46 (2) ◽  
pp. 559-570 ◽  
Author(s):  
Xuemiao Ha ◽  
Qihe Tang ◽  
Li Wei
Keyword(s):  
Random Walk ◽  
Asymptotic Formula ◽  
Recent Literature ◽  
Random Variables ◽  
Applied Probability ◽  
Tail Behavior ◽  
Heavy Tailed ◽  
Precise Asymptotic ◽  
Independent And Identically Distributed

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

scholarly journals Estimating tails of independently stopped random walks using concave approximations of hazard functions

2021 ◽  
Vol 58 (3) ◽  
pp. 773-793
Author(s):  
Jaakko Lehtomaa
Keyword(s):  
Random Walk ◽  
Asymptotic Analysis ◽  
Sufficient Conditions ◽  
Hazard Functions ◽  
Logarithmic Asymptotics ◽  
Randomly Stopped Sums ◽  
Heavy Tailed ◽  
The Moment ◽  
Stopped Random Walks ◽  
Stopped Sums

AbstractThis paper considers logarithmic asymptotics of tails of randomly stopped sums. The stopping is assumed to be independent of the underlying random walk. First, finiteness of ordinary moments is revisited. Then the study is expanded to more general asymptotic analysis. Results are applicable to a large class of heavy-tailed random variables. The main result enables one to identify if the asymptotic behaviour of a stopped sum is dominated by its increments or the stopping variable. As a consequence, new sufficient conditions for the moment determinacy of compounded sums are obtained.

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