Abstract
We derive a sufficient condition on the symmetric norm ||·|| such that the probability distribution associated with the density function f (x) ∝exp(−λ ||x||) is conditionally independent and identically distributed in the sense of de Finetti’s seminal theorem. The criterion is mild enough to comprise the ℓp-norms as special cases, in which f is shown to correspond to a polynomially tilted stable mixture of products of transformed Gamma densities. In another special case of interest f equals the density of a time-homogeneous load sharing model, popular in reliability theory, whose motivation is a priori unrelated to the concept of conditional independence. The de Finetti structure reveals a surprising link between time-homogeneous load sharing models and the concept of Lévy subordinators.
References
[1] Aldous, D.J. (1985). Exchangeability and related topics. In P.L. Hennequin (Ed.), École d’Été de Probabilités de Saint-Flour XIII-1983, pp. 1–198. Springer, New York.10.1007/BFb0099421Search in Google Scholar
[2] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12(4), 1194–1204.Search in Google Scholar
[3] de Haan, L. and S.I. Resnick (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40(4), 317–337.10.1007/BF00533086Search in Google Scholar
[4] Devroye, L. (2009). Random variate generation for exponentially and polynomially tilted stable distributions. ACM Trans. Model. Comput. Simul. 19(4), no. 18, 20 pp.10.1145/1596519.1596523Search in Google Scholar
[5] Esary, J.D. and A.W. Marshall (1974). Multivariate distributions with exponential minimums. Ann. Statist. 2(1), 84–98.10.1214/aos/1176342615Search in Google Scholar
[6] Freund, J.E. (1961). A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56(296), 971–977.10.1080/01621459.1961.10482138Search in Google Scholar
[7] Gnedin, A.V. (1995). On a class of exchangeable sequences. Statist. Probab. Lett. 25(4), 351–355.10.1016/0167-7152(94)00240-3Search in Google Scholar
[8] Herbertsson, A. and H. Rootzén (2008). Pricing k-th-to-default swaps under default contagion: the matrix analytic approach. J. Comput. Finance 12(1), 49–78.10.21314/JCF.2008.183Search in Google Scholar
[9] Jarrow, R.A. and F. Yu (2001). Counterparty risk and the pricing of defaultable securities. J. Finance 56(5), 1765–1799.10.1111/0022-1082.00389Search in Google Scholar
[10] Johnson, C.R. and P. Nylen (1991). Monotonicity properties of norms. Linear Algebra Appl. 148, 43–58.10.1016/0024-3795(91)90085-BSearch in Google Scholar
[11] Karlin, S. and L.S. Shapley (1953). Geometry of Moment Sequences. American Mathematical Society, Providence RI.Search in Google Scholar
[12] Konstantopoulos, T. and L. Yuan (2019). On the extendibility of finitely exchangeable probability measures. Trans. Amer. Math. Soc. 371(10), 7067–7092.10.1090/tran/7701Search in Google Scholar
[13] Kopp, C. and I. Molchanov (2018). Series representations of time-stable stochastic processes. Probab. Math. Statist. 38(2), 299–315.10.19195/0208-4147.38.2.4Search in Google Scholar
[14] Mai, J.-F. and M. Scherer (2009). Lévy-frailty copulas. J. Multivariate Anal. 100(7), 1567–1585.10.1016/j.jmva.2009.01.010Search in Google Scholar
[15] Mai, J.-F. (2010). Extendibility of Marshall-Olkin Distributions via Lévy Subordinators and an Application to Portfolio Credit Risk. PhD thesis, Technische Universität München, Germany.Search in Google Scholar
[16] Mai, J.-F. (2020). Canonical spectral representation for exchangeable max-stable sequences. Extremes 23(1), 151–169.10.1007/s10687-019-00361-3Search in Google Scholar
[17] Mai, J.-F. and M. Scherer (2019). Subordinators which are infinitely divisible w.r.t. time: construction, properties, and simulation of max-stable sequences and infinitely divisible laws. ALEA Lat. Am. J. Probab. Math. Statist. 16(2), 977–1005.Search in Google Scholar
[18] Molchanov, I. (2008). Convex geometry of max-stable distributions. Extremes 11(3), 235–259.10.1007/s10687-008-0055-5Search in Google Scholar
[19] Norros, I. (1985). Systems weakened by failures. Stochastic Process. Appl. 20(2), 181–196.10.1016/0304-4149(85)90209-1Search in Google Scholar
[20] Pickands, J. (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49(2), pp. 859–878.Search in Google Scholar
[21] Ressel, P. (2013). Homogeneous distributions – And a spectral representation of classical mean values and stable tail dependence functions. J. Multivariate Anal. 117, 246–256.10.1016/j.jmva.2013.02.013Search in Google Scholar
[22] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Search in Google Scholar
[23] Segers, J. (2012). Max-stable models for multivariate extremes. REVSTAT 10(1), 61–82.Search in Google Scholar
[24] Shaked, M., F.L. Spizzichino, and F. Suter (2002). Nonhomogeneous birth processes and lℓ-spherical densities, with applications in reliability theory. Probab. Engrg. Inform. Sci. 16(3), 271–288.10.1017/S0269964802163017Search in Google Scholar
[25] Spizzichino, F.L. (2019). Reliability, signature, and relative quality functions of systems under time-homogeneous load-sharing models. Appl. Stoch. Models Bus. Ind. 35(2), 158–176.10.1002/asmb.2397Search in Google Scholar
[26] Stacy, E.W. (1962). A generalization of the Gamma distribution. Ann. Math. Statist. 33(3), 1187–1192.10.1214/aoms/1177704481Search in Google Scholar
[27] Wolfe, S.J. (1975). On moments of probability distribution functions. In B. Ross (Ed.), Fractional Calculus and Its Applications, pp. 306–316. Springer, Berlin.10.1007/BFb0067116Search in Google Scholar
[28] Yu, F. (2007). Correlated defaults in intensity-based models. Math. Finance 17(2), 155–173.10.1111/j.1467-9965.2007.00298.xSearch in Google Scholar
© 2020 Jan-Frederik Mai, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
