Abstract
Let (Yt)t>0 be a stochastic process on a proper cone K associated with a family of distributions (ℱt)t>0. We investigate the asymptotic behavior of the distribution of 〈Yt, u〉−t for u ∈ K as the parameter t approaches zero. Next, we consider the case that ℱt belongs to an exponential family. Under some conditions, we prove that the limiting distribution of 〈Yt, u〉−t as t → 0+ is a Pareto type law, which is independent of u. Such a process is important for applications in risk management or pricing multivariate options. Hence the obtained results should be important, since by projecting Yt in all directions u ∈ K we give an approximation of the distribution of 〈Yt, u〉 for small values of the corresponding dispersion parameter t. Moreover, we prove that this limit distribution is independent of u. Finally, illustrative examples are provided at the end of this paper.
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References
B.C. Arnold, Pareto Distributions, International Cooperative Publishing House, Fairland MD, 1983.
S.K. Bar-Lev and P. Enis, Existence of moments and an asymptotic result based on a mixture of exponential distributions, Stat. Probab. Lett., 5(4):273–277, 1987.
S.K. Bar-Lev and G. Letac, The limiting behavior of some infinitely divisible exponential dispersion models, Stat. Probab. Lett., 80(23–24):1870–1874, 2010.
O. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, Wiley, New York, 1978.
T. Benarab, A. Masmoudi, and M. Zribi, Trace of the variance-covariance matrix in natural exponential families, Commun. Stat., Theory Methods, 44(6):1241–1254, 2015.
N. Bensalah and A. Masmoudi, Poisson limit laws for exponential dispersion models, Commun. Stat., Theory Methods, 41(3):393–402, 2010.
G.M. Cordeiro and M.G. Andradeb, Transformed generalized linear models, J. Stat. Plann. Inference, 92:239–256, 2005.
J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo, Palo Alto, CA, 2005.
W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1971.
B. Jørgensen, The Theory of Dispersion Models, Chapman & Hall, London, 1997.
B. Jørgensen, JR. Martinez, and M. Tsao, The asymptotic behavior of the variance function, Scand. J. Stat., 21:223–243, 1994.
G. Letac, Lectures on Natural Exponential Families and Their Variance Functions, Monogr. Mat., Vol. 50, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1992.
G. Letac and M. Mora, Natural exponential families with cubic variance functions, Ann. Stat., 81:21–37, 1990.
A. Masmoudi, Conditional natural exponential families, Stat. Probab. Lett., 76:1882–1888, 2006.
V. Pérez-Abreu and R. Stelzer, Infinitely divisiblemultivariate and matrix gamma distributions, J. Multivariate Anal., 130:155–175, 2014.
A.V. Skorohod, Random Processes with Independent Increments, Kluwer, Dordrecht, 1991.
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Rejeb, H., Masmoudi, A. Asymptotic behavior of some cone-valued infinitely divisible stochastic process through projection. Lith Math J 62, 500–508 (2022). https://doi.org/10.1007/s10986-022-09578-5
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DOI: https://doi.org/10.1007/s10986-022-09578-5