Asymptotic Behavior of Eigenvalues of Variance-Covariance Matrix of a High-Dimensional Heavy-Tailed Lévy Process

2020 ◽  
Author(s):  
Asma Teimouri ◽  
Mahbanoo Tata ◽  
Mohsen Rezapour ◽  
Rafal Kulik ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Asymptotic Behavior ◽  
Covariance Matrix ◽  
Lévy Process ◽  
Levy Process ◽  
High Dimensional ◽  
Heavy Tailed ◽  
Variance Covariance Matrix

scholarly journals Limit Theory for High Frequency Sampled MCARMA Models

2014 ◽  
Vol 46 (3) ◽  
pp. 846-877 ◽  
Author(s):  
Vicky Fasen
Keyword(s):  
Asymptotic Behavior ◽  
High Frequency ◽  
Lévy Process ◽  
Asymptotic Properties ◽  
Domain Of Attraction ◽  
Levy Process ◽  
Sample Variance ◽  
Limit Theory ◽  
Second Moments ◽  
Time Grid

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {hn, 2hn,…, nhn}, where hn ↓ 0 and nhn → ∞ as n → ∞, or at a constant time grid where hn = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

scholarly journals Uniform Asymptotics for Discounted Aggregate Claims in Dependent Risk Models

2014 ◽  
Vol 51 (3) ◽  
pp. 669-684 ◽  
Author(s):  
Yang Yang ◽  
Kaiyong Wang ◽  
Dimitrios G. Konstantinides
Keyword(s):  
Lévy Process ◽  
Tail Probability ◽  
Levy Process ◽  
Insurance Company ◽  
Risk Models ◽  
Heavy Tailed Distributions ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Heavy Tailed ◽  
Renewal Risk

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.

scholarly journals Construction of a free Lévy process as high-dimensional limit of a Brownian motion on the unitary group

2015 ◽  
Vol 18 (03) ◽  
pp. 1550018 ◽  
Author(s):  
Michaël Ulrich
Keyword(s):  
Brownian Motion ◽  
Unitary Group ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Alternative Proof ◽  
High Dimensional ◽  
General Context ◽  
Work Done

It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally, it is quite natural to try to build free Lévy processes as high-dimensional limits of classical matricial Lévy processes. We will focus here on one specific such construction, discussing and generalizing the work done previously by Biane in Ref.2, who has shown that the (classical) Brownian motion on the Unitary group U(d) converges to the free multiplicative Brownian motion when d goes to infinity. We shall first recall that result and give an alternative proof for it. We shall then see how this proof can be adapted in a more general context in order to get a free Lévy process on the dual group (in the sense of Voiculescu) U〈n〉. This result will actually amount to a truly noncommutative limit theorem for classical random variables, of which Biane's result constitutes the case n = 1.

scholarly journals Normality Testing of High-Dimensional Data Based on Principle Component and Jarque-Bera Statistics

2021 ◽  
Author(s):  
Ya-nan Song ◽  
Xuejing Zhao
Keyword(s):  
Covariance Matrix ◽  
High Dimensional Data ◽  
Simulated Data ◽  
High Dimensional ◽  
Empirical Power ◽  
High Dimensions ◽  
Principle Component ◽  
Orthogonal Space ◽  
Reduced Data ◽  
Variance Covariance Matrix

The testing of high-dimensional normality has been an important issue and has been intensively studied in literatures, it depends on the Variance-Covariance matrix of the sample, numerous methods have been proposed to reduce the complex of the Variance-Covariance matrix. The principle component analysis(PCA) was widely used since it can project the high-dimensional data into lower dimensional orthogonal space, and the normality of the reduced data can be evaluated by Jarque-Bera(JB) statistic on each principle direction. We propose two combined statistics, the summation and the maximum of one-way JB statistics, upon the independency of each principle direction, to test the multivariate normality of data in high dimensions. The performance of the proposed methods is illustrated by the empirical power of the simulated data of normal data and non-normal data. Two real examples show the validity of our proposed methods.

scholarly journals Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims

2009 ◽  
Vol 41 (1) ◽  
pp. 206-224 ◽  
Author(s):  
C. C. Heyde ◽  
Dingcheng Wang
Keyword(s):  
Finite Time ◽  
Lévy Process ◽  
Financial Risk ◽  
Risk Model ◽  
Levy Process ◽  
Insurance Risk ◽  
Total Risk ◽  
Investment Return ◽  
Exponential Lévy Process ◽  
Heavy Tailed

By expressing the discounted net loss process as a randomly weighted sum, we investigate the finite-time ruin probabilities for the Poisson risk model with an exponential Lévy process investment return and heavy-tailed claims. It is found that in finite time, however, the extreme of insurance risk dominates the extreme of financial risk, but, for the case of dangerous investment (see Klüppelberg and Kostadinova (2008) for an accurate definition of dangerous investment), the extreme of financial risk has more and more of an effect on the total risk, and as time passes, the extreme of financial risk finally dominates the extreme of insurance risk.

Uniform Asymptotics for Discounted Aggregate Claims in Dependent Risk Models

2014 ◽  
Vol 51 (03) ◽  
pp. 669-684 ◽  
Author(s):  
Yang Yang ◽  
Kaiyong Wang ◽  
Dimitrios G. Konstantinides
Keyword(s):  
Lévy Process ◽  
Tail Probability ◽  
Levy Process ◽  
Insurance Company ◽  
Risk Models ◽  
Heavy Tailed Distributions ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Heavy Tailed ◽  
Renewal Risk

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.

scholarly journals Limit Theory for High Frequency Sampled MCARMA Models

2014 ◽  
Vol 46 (03) ◽  
pp. 846-877 ◽  
Author(s):  
Vicky Fasen
Keyword(s):  
Asymptotic Behavior ◽  
High Frequency ◽  
Lévy Process ◽  
Asymptotic Properties ◽  
Domain Of Attraction ◽  
Levy Process ◽  
Sample Variance ◽  
Limit Theory ◽  
Second Moments ◽  
Time Grid

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {h n , 2h n ,…, nh n }, where h n ↓ 0 and nh n → ∞ as n → ∞, or at a constant time grid where h n = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

scholarly journals Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims

2009 ◽  
Vol 41 (01) ◽  
pp. 206-224 ◽  
Author(s):  
C. C. Heyde ◽  
Dingcheng Wang
Keyword(s):  
Finite Time ◽  
Lévy Process ◽  
Financial Risk ◽  
Risk Model ◽  
Levy Process ◽  
Insurance Risk ◽  
Total Risk ◽  
Investment Return ◽  
Exponential Lévy Process ◽  
Heavy Tailed

By expressing the discounted net loss process as a randomly weighted sum, we investigate the finite-time ruin probabilities for the Poisson risk model with an exponential Lévy process investment return and heavy-tailed claims. It is found that in finite time, however, the extreme of insurance risk dominates the extreme of financial risk, but, for the case of dangerous investment (see Klüppelberg and Kostadinova (2008) for an accurate definition of dangerous investment), the extreme of financial risk has more and more of an effect on the total risk, and as time passes, the extreme of financial risk finally dominates the extreme of insurance risk.

scholarly journals Normality Testing of High-Dimensional Data Based on Principle Component and Jarque-Bera Statistics

2021 ◽  
Author(s):  
Ya-nan Song ◽  
Xuejing Zhao
Keyword(s):  
Covariance Matrix ◽  
High Dimensional Data ◽  
Simulated Data ◽  
High Dimensional ◽  
Empirical Power ◽  
High Dimensions ◽  
Principle Component ◽  
Orthogonal Space ◽  
Reduced Data ◽  
Variance Covariance Matrix

The testing of high-dimensional normality has been an important issue and has been intensively studied in literatures, it depends on the Variance-Covariance matrix of the sample, numerous methods have been proposed to reduce the complex of the Variance-Covariance matrix. The principle component analysis(PCA) was widely used since it can project the high-dimensional data into lower dimensional orthogonal space, and the normality of the reduced data can be evaluated by Jarque-Bera(JB) statistic on each principle direction. We propose two combined statistics, the summation and the maximum of one-way JB statistics, upon the independency of each principle direction, to test the multivariate normality of data in high dimensions. The performance of the proposed methods is illustrated by the empirical power of the simulated data of normal data and non-normal data. Two real examples show the validity of our proposed methods.

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