Schur-Constant and Related Dependence Models, with Application to Ruin Probabilities

Lefèvre, Claude; Simon, Matthieu

doi:10.1007/s11009-019-09744-2

Cited by 6 publications

(2 citation statements)

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“…, α N −1 are the simple roots of s N = G S N (s) in s ∈ S 1 (0) \ {1}, see [19]. If N = 3, the main matrix in (25) is…”

Section: Theorem 2 Let Us Assume That the Probabilitiesmentioning

confidence: 99%

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Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Gervė,

Grigutis

2024

Modern Stochastics: Theory and Applications

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In this paper, the distribution function \[ \varphi (u)=\mathbb{P}\Bigg(\underset{n\geqslant 1}{\sup }{\sum \limits_{i=1}^{n}}({X_{i}}-\kappa )\lt u\Bigg),\] and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.

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“…, α N −1 are the simple roots of s N = G S N (s) in s ∈ S 1 (0) \ {1}, see [19]. If N = 3, the main matrix in (25) is…”

Section: Theorem 2 Let Us Assume That the Probabilitiesmentioning

confidence: 99%

“…[42]) are the reasons which make the recent literature voluminous. Next to the mentioned references, see [11,36,4,12,10,34,26,25,32,8] as the recent ones on the subject, too.…”

Section: Introductionmentioning

confidence: 99%

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Gervė,

Grigutis

2024

Modern Stochastics: Theory and Applications

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Aging and Positive Dependence

Spizzichino

2021

Wiley StatsRef: Statistics Reference Online

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Multivariate survival models for vectors of dependent lifetimes also typically manifest a phenomena of aging. This article presents a discussion about the connections existing between these types of properties and points out some aspects that they have in common, from a conceptual point of view. On this purpose, some basic notions of positive dependence and some notions of multivariate aging are being recalled. Furthermore, some special types of multivariate dependence models are reviewed and briefly discussed.

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