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.

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.

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 On Exceedance Times for Some Processes with Dependent Increments

2014 ◽  
Vol 51 (01) ◽  
pp. 136-151 ◽  
Author(s):  
Søren Asmussen ◽  
Sergey Foss
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Risk Process ◽  
Fluid Models ◽  
Insurance Risk ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Time Position ◽  
Heavy Tailed ◽  
Markov Modulated

Let {Z n } n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = sup n≥0 Z n be its supremum. Asmussen and Klüppelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → ∞, for the distribution of the quadruple that includes the time τ = τ(x) to exceed level x, position Z τ at this time, position Z τ-1 at the prior time, and the trajectory up to it (similar results were obtained for the Cramér-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of τ. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of τ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).

scholarly journals Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

2010 ◽  
Vol 47 (2) ◽  
pp. 301-322 ◽  
Author(s):  
Jose Blanchet ◽  
Jingchen Liu
Keyword(s):  
Random Walk ◽  
Importance Sampling ◽  
Mean Squared Error ◽  
First Passage Time ◽  
Passage Time ◽  
Natural Generalization ◽  
Efficient Estimation ◽  
State Dependent ◽  
Negative Drift ◽  
Heavy Tailed

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

scholarly journals Local asymptotics for the area under the random walk excursion

2018 ◽  
Vol 50 (2) ◽  
pp. 600-620
Author(s):  
Elena Perfilev ◽  
Vitali Wachtel
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Tail Behaviour ◽  
Local Asymptotics ◽  
Negative Drift ◽  
Local Central Limit Theorem

Abstract We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and light-tailed increments. We determine the asymptotics for local probabilities for the area and prove a local central limit theorem for the duration of the excursion conditioned on the large values of its area.

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

2001 ◽  
Vol 38 (1) ◽  
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 Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (1) ◽  
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 (Yn, Nn)n≥1 be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean ν and the Yn have a common heavy-tailed distribution F. We consider the process (Zn)n≥1 defined by Zn = Yn - Σn-1, where It is shown that the probability that the maximum M = maxn≥1Zn 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 Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

2010 ◽  
Vol 47 (02) ◽  
pp. 301-322 ◽  
Author(s):  
Jose Blanchet ◽  
Jingchen Liu
Keyword(s):  
Random Walk ◽  
Importance Sampling ◽  
Mean Squared Error ◽  
First Passage Time ◽  
Passage Time ◽  
Natural Generalization ◽  
Efficient Estimation ◽  
State Dependent ◽  
Negative Drift ◽  
Heavy Tailed

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.

scholarly journals On Exceedance Times for Some Processes with Dependent Increments

2014 ◽  
Vol 51 (1) ◽  
pp. 136-151
Author(s):  
Søren Asmussen ◽  
Sergey Foss
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Risk Process ◽  
Fluid Models ◽  
Insurance Risk ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Time Position ◽  
Heavy Tailed ◽  
Markov Modulated

Let {Zn}n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = supn≥0Zn be its supremum. Asmussen and Klüppelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → ∞, for the distribution of the quadruple that includes the time τ = τ(x) to exceed level x, position Zτ at this time, position Zτ-1 at the prior time, and the trajectory up to it (similar results were obtained for the Cramér-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of τ. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of τ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).

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