The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after $nin mathbb {N}$ periods. The accuracy of order $O(n^{-1})$ and $O(n^{-1/2})$ is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order $O(n^{-1})$. The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium.
{"title":"Asymptotics for the sum of three state Markov dependent random variables","authors":"Gabija Liaudanskait.e, V. vCekanavivcius","doi":"10.15559/18-VMSTA123","DOIUrl":"https://doi.org/10.15559/18-VMSTA123","url":null,"abstract":"The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after $nin mathbb {N}$ periods. The accuracy of order $O(n^{-1})$ and $O(n^{-1/2})$ is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order $O(n^{-1})$. The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"20 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80625697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a low-frequency sample of the infinitely divisible moving average random field ${int_{mathbb{R}^d}f(t-x)Lambda (dx), tin mathbb{R}^d}$, in [13] we proposed an estimator $hat{uv_0}$ for the function $mathbb{R}ni xmapsto u(x)v_0(x)=(uv_0)(x)$, with $u(x)=x$ and $v_0$ being the L'{e}vy density of the integrator random measure $Lambda$. In this paper, we study asymptotic properties of the linear functional $L^2(mathbb{R})ni vmapsto left langle v,hat{uv_0}right rangle_{L^2(mathbb{R})}$, if the (known) kernel function $f$ has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.
给定无限可分移动平均随机场${int_{mathbb{R}^d}f(t-x)Lambda (dx), tin mathbb{R}^d}$的一个低频样本，在[13]中，我们对函数$mathbb{R}ni xmapsto u(x)v_0(x)=(uv_0)(x)$提出了一个估计量$hat{uv_0}$，其中$u(x)=x$和$v_0$是积分器随机测度$Lambda$的lsamvy密度。本文研究了(已知)核函数$f$具有紧支持的线性泛函$L^2(mathbb{R})ni vmapsto left langle v,hat{uv_0}right rangle_{L^2(mathbb{R})}$的渐近性质。我们给出了保证一致性(均值)的条件，并证明了它的中心极限定理。
{"title":"On a linear functional for infinitely divisible moving average random fields","authors":"Stefan Roth","doi":"10.15559/19-VMSTA143","DOIUrl":"https://doi.org/10.15559/19-VMSTA143","url":null,"abstract":"Given a low-frequency sample of the infinitely divisible moving average random field ${int_{mathbb{R}^d}f(t-x)Lambda (dx), tin mathbb{R}^d}$, in [13] we proposed an estimator $hat{uv_0}$ for the function $mathbb{R}ni xmapsto u(x)v_0(x)=(uv_0)(x)$, with $u(x)=x$ and $v_0$ being the L'{e}vy density of the integrator random measure $Lambda$. In this paper, we study asymptotic properties of the linear functional $L^2(mathbb{R})ni vmapsto left langle v,hat{uv_0}right rangle_{L^2(mathbb{R})}$, if the (known) kernel function $f$ has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"90 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79738716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.
{"title":"Ruin probability for the bi-seasonal discrete time risk model with dependent claims","authors":"Olga Navickien.e, Jonas Sprindys, Jonas vSiaulys","doi":"10.15559/18-VMSTA118","DOIUrl":"https://doi.org/10.15559/18-VMSTA118","url":null,"abstract":"The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"281 3 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86575889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note gathers what is known about, and provides some new results concerning the operations of intersection, of ``generated $sigma$-field'', and of ``complementation'' for (independent) complete $sigma$-fields on probability spaces.
{"title":"Arithmetic of (independent) sigma-fields on probability spaces","authors":"M. Vidmar","doi":"10.15559/19-VMSTA135","DOIUrl":"https://doi.org/10.15559/19-VMSTA135","url":null,"abstract":"This note gathers what is known about, and provides some new results concerning the operations of intersection, of ``generated $sigma$-field'', and of ``complementation'' for (independent) complete $sigma$-fields on probability spaces.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"6 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79091251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.
{"title":"Moderate deviations for a stochastic Burgers equation","authors":"R. Belfadli, Lahcen Boulanba, M. Mellouk","doi":"10.15559/19-VMSTA134","DOIUrl":"https://doi.org/10.15559/19-VMSTA134","url":null,"abstract":"A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"33 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88530586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this estimation procedure may fail in some special cases. In this article we provide a detailed analysis of these special cases. In particular, we prove that these cases correspond to degenerate processes.
{"title":"Note on AR(1)-characterisation of stationary processes and model fitting","authors":"M. Voutilainen, L. Viitasaari, Pauliina Ilmonen","doi":"10.15559/19-VMSTA132","DOIUrl":"https://doi.org/10.15559/19-VMSTA132","url":null,"abstract":"It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this estimation procedure may fail in some special cases. In this article we provide a detailed analysis of these special cases. In particular, we prove that these cases correspond to degenerate processes.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"123 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74684517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to study the laws of the exponential functionals of the processes $X$ with independent increments, namely $$I_t= int _0^texp(-X_s)ds, ,, tgeq 0,$$ and also $$I_{infty}= int _0^{infty}exp(-X_s)ds.$$ Under suitable conditions we derive the integro-differential equations for the density of $I_t$ and $I_{infty}$. We give sufficient conditions for the existence of smooth density of the laws of these functionals. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.
本文的目的是研究具有独立增量的过程$X$，即$$I_t= int _0^texp(-X_s)ds, ,, tgeq 0,$$和$$I_{infty}= int _0^{infty}exp(-X_s)ds.$$的指数泛函的规律，在适当的条件下，我们推导了$I_t$和$I_{infty}$的密度的积分微分方程。给出了这些泛函律光滑密度存在的充分条件。在列维过程的特殊情况下，这些方程可以简化，并且在许多情况下可以显式求解。
{"title":"On distributions of exponential functionals of the processes with independent increments","authors":"L. Vostrikova","doi":"10.15559/20-vmsta159","DOIUrl":"https://doi.org/10.15559/20-vmsta159","url":null,"abstract":"The aim of this paper is to study the laws of the exponential functionals of the processes $X$ with independent increments, namely $$I_t= int _0^texp(-X_s)ds, ,, tgeq 0,$$ and also $$I_{infty}= int _0^{infty}exp(-X_s)ds.$$ Under suitable conditions we derive the integro-differential equations for the density of $I_t$ and $I_{infty}$. We give sufficient conditions for the existence of smooth density of the laws of these functionals. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"28 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83635189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the $OS$ property, while for the growth of species we use nonlinear time-fractional pure birth processes. Further, in two specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time $t$. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.
{"title":"Studies on generalized Yule models","authors":"F. Polito","doi":"10.15559/18-VMSTA125","DOIUrl":"https://doi.org/10.15559/18-VMSTA125","url":null,"abstract":"We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the $OS$ property, while for the growth of species we use nonlinear time-fractional pure birth processes. Further, in two specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time $t$. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"190 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83058674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).
{"title":"Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates","authors":"L. Beghin, C. Macci, B. Martinucci","doi":"10.15559/20-vmsta169","DOIUrl":"https://doi.org/10.15559/20-vmsta169","url":null,"abstract":"We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"203 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88029874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigration) at any fixed time is regularly varying provided that the initial sizes of the population are independent and regularly varying.
{"title":"On tail behaviour of stationary second-order Galton–Watson processes with immigration","authors":"M. Barczy, Z. Bősze, G. Pap","doi":"10.15559/20-VMSTA161","DOIUrl":"https://doi.org/10.15559/20-VMSTA161","url":null,"abstract":"A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigration) at any fixed time is regularly varying provided that the initial sizes of the population are independent and regularly varying.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"414 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84893557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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