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

Denisov, Denis; Perfilev, Elena; Wachtel, Vitali

doi:10.1017/jpr.2020.85

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…In fact we can consider the random walk {S n : n ≥ 1} defined by S n := n j=1 T j , and the bivariate sequence {(S n , S 1 + • • • + S n ) : n ≥ 1} coincides with the sequence {(τ (n), A(n)) : n ≥ 1} presented above. Among the references with asymptotic results for integrated random walks here we recall [5] and [8] for the heavy-tailed case, and [15] for the light-tailed case.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic results for certain first-passage times and areas of renewal processes

Macci¹,

Pacchiarotti²

2021

Preprint

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We consider the process {x − N (t) : t ≥ 0}, where x > 0 and {N (t) : t ≥ 0} is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of (τ (x), A(x)) where τ (x) is the first-passage time of {x − N (t) : t ≥ 0} to reach zero or a negative value, and A(x) is the corresponding first-passage area. We remark that we can define the sequence {(τ (n), A(n)) : n ≥ 1} by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as x → ∞ in the fashion of large (and moderate) deviations.

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Section: Introductionmentioning

confidence: 99%

Asymptotic results for certain first-passage times and areas of renewal processes

Macci¹,

Pacchiarotti²

2021

Preprint

View full text Add to dashboard Cite

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Asymptotic results for certain first-passage times and areas of renewal processes

Macci

Pacchiarotti

2023

Theor. Probability and Math. Statist.

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We consider the process { x − N ( t ) : t ≥ 0 } \{x-N(t):t\geq 0\} , where x ∈ R + x\in \mathbb {R}_+ and { N ( t ) : t ≥ 0 } \{N(t):t\geq 0\} is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of ( τ ( x ) , A ( x ) ) (\tau (x),A(x)) where τ ( x ) \tau (x) is the first-passage time of { x − N ( t ) : t ≥ 0 } \{x-N(t):t\geq 0\} to reach zero or a negative value, and A ( x ) ≔ ∫ 0 τ ( x ) ( x − N ( t ) ) d t A(x)≔\int _0^{\tau (x)}(x-N(t))dt is the corresponding first-passage (positive) area swept out by the process { x − N ( t ) : t ≥ 0 } \{x-N(t):t\geq 0\} . We remark that we can define the sequence { ( τ ( n ) , A ( n ) ) : n ≥ 1 } \{(\tau (n),A(n)):n\geq 1\} by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as x → ∞ x\to \infty in the fashion of large (and moderate) deviations.

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

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 heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

Cited by 2 publications

References 17 publications

Asymptotic results for certain first-passage times and areas of renewal processes

Asymptotic results for certain first-passage times and areas of renewal processes

Asymptotic results for certain first-passage times and areas of renewal processes

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