scholarly journals On the Maximum of a Random Walk with Small Negative Drift

1983 ◽  
Vol 11 (3) ◽  
pp. 491-505 ◽  
Author(s):  
Michael J. Klass
Keyword(s):  
Random Walk ◽  
Negative Drift

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.

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (2) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

scholarly journals Monte Carlo Algorithms for Finding the Maximum of a Random Walk with Negative Drift

2006 ◽  
Vol 43 (01) ◽  
pp. 74-86
Author(s):  
Ludwig Baringhaus ◽  
Rudolf Grübel
Keyword(s):  
Monte Carlo ◽  
Random Walk ◽  
Simple Random Walk ◽  
Mathematical Statistics ◽  
Global Maximum ◽  
Random Input ◽  
Monte Carlo Algorithms ◽  
Negative Drift

We discuss two Monte Carlo algorithms for finding the global maximum of a simple random walk with negative drift. This problem can be used to connect the analysis of random input Monte Carlo algorithms with ideas and principles from mathematical statistics.

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.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

1986 ◽  
Vol 23 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Michael L. Hogan
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Exponential Family ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Negative Drift ◽  
Family Case ◽  
Correction Terms

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue

1982 ◽  
Vol 14 (01) ◽  
pp. 143-170 ◽  
Author(s):  
Søren Asmussen
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Limit Theorems ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Brownian Bridge ◽  
Ruin Probabilities ◽  
Asymptotic Formulae ◽  
Negative Drift ◽  
Busy Cycle

LetSn=X1+ · · · +Xnbe a random walk with negative drift μ < 0, letF(x) =P(Xk≦x),v(u) =inf{n:Sn>u} and assume that for some γ > 0is a proper distribution with finite meanVarious limit theorems for functionals ofX1,· · ·,Xv(u)are derived subject to conditioning upon {v(u)< ∞} withularge, showing similar behaviour as if theXiwere i.i.d. with distributionFor example, the deviation of the empirical distribution function fromproperly normalised, is shown to have a limit inD, and an approximation forby means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and theGI/G/1 queue considered either within a busy cycle or in the steady state. The methods produce an alternate approach to known asymptotic formulae for ruin probabilities as well as related waiting-time approximations for theGI/G/1 queue. For exampleuniformly inN, withWNthe waiting time of the Nth customer.

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