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 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 Local asymptotics of the cycle maximum of a heavy-tailed random walk

2007 ◽  
Vol 39 (1) ◽  
pp. 221-244 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Random Walk ◽  
Positive Constant ◽  
Random Variables ◽  
Heavy Tailed Distribution ◽  
Local Asymptotics ◽  
Cycle Maximum ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

scholarly journals Local asymptotics of the cycle maximum of a heavy-tailed random walk

2007 ◽  
Vol 39 (01) ◽  
pp. 221-244 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Random Walk ◽  
Positive Constant ◽  
Random Variables ◽  
Heavy Tailed Distribution ◽  
Local Asymptotics ◽  
Cycle Maximum ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let S n =ξ1+⋯+ξ n and M n =max k≤n S k . Let τ=min{n≥1: S n ≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)&lt;0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

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

2009 ◽  
Vol 46 (02) ◽  
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 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 &gt; 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).

On the validity of Wald's equation

1994 ◽  
Vol 31 (04) ◽  
pp. 949-957 ◽  
Author(s):  
Markus Roters
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Stieltjes Transform ◽  
Positive Part ◽  
Negative Drift ◽  
The Given ◽  
Independent Identically Distributed

In this paper we review conditions under which Wald's equation holds, mainly if the expectation of the given stopping time is infinite. As a main result we obtain what is probably the weakest possible version of Wald's equation for the case of independent, identically distributed (i.i.d.) random variables. Moreover, we improve a result of Samuel (1967) concerning the existence of stopping times for which the expectation of the stopped sum of the underlying i.i.d. sequence of random variables does not exist. Finally, we show by counterexamples that it is impossible to generalize a theorem of Kiefer and Wolfowitz (1956) relating the moments of the supremum of a random walk with negative drift to moments of the positive part of X 1 to the case where the expectation of X 1 is —∞. Here, the Laplace–Stieltjes transform of the supremum of the considered random walk plays an important role.

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).

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 &gt; 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).

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