On the fractional probabilistic Taylor’s and mean value theorems. (English) Zbl 1385.60043
Summary: In order to develop certain fractional probabilistic analogues of Taylor’s theorem and mean value theorem, we introduce the \(n\)th-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the \(n\)th-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The \(n\)th-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor’s theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion, we discuss the probabilistic Taylor’s theorem based on fractional Caputo derivatives.
MSC:
| 60E99 | Distribution theory |
| 26A33 | Fractional derivatives and integrals |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
Keywords:
characterization of exponential distribution; fractional calculus; fractional equilibrium distribution; generalized Taylor formula; mean value theorem; survival bounded orderReferences:
| [1] | Y. Cheng, J.S. Pai, The maintenance properties of nth stop-loss order. In: Proc. of the 30th International ASTIN Colloquium/9th International AFIR Colloquium (1999), 95-118.; Cheng, Y.; Pai, J. S., The maintenance properties of nth stop-loss order, Proc. of the 30th International ASTIN Colloquium/9th International AFIR Colloquium, 95-118 (1999) |
| [2] | A. Di Crescenzo, A probabilistic analogue of the mean value theorem and its applications to reliability theory. J. Appl. Probab. 36, No 3 (1999), 706-719; doi:10.1239/jap/1032374628.; Di Crescenzo, A., A probabilistic analogue of the mean value theorem and its applications to reliability theory, J. Appl. Probab, 36, 3, 706-719 (1999) · Zbl 0952.60083 |
| [3] | R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Heidelberg (2014); doi:10.1007/978-3-662-43930-2.; Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S. V., Mittag-Leffler Functions, Related Topics and Applications (2014) · Zbl 1309.33001 |
| [4] | W.L. Harkness, R. Shantaram, Convergence of a sequence of transformations of distribution functions. Pacific J. Math. 31, No 2 (1969), 403-415.; Harkness, W. L.; Shantaram, R., Convergence of a sequence of transformations of distribution functions, Pacific J. Math, 31, 2, 403-415 (1969) · Zbl 0186.25002 |
| [5] | E. Hewitt, K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. 3rd printing. Graduate Texts in Mathematics, No 25, Springer-Verlag, New York-Heidelberg (1975).; Hewitt, E.; Stromberg, K., Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, 25 (1975) · Zbl 0307.28001 |
| [6] | P. Guo, C. Li, G. Chen, On the fractional mean-value theorem. Int. J. Bifurcation Chaos 22 (2012), Article 1250104; doi: 10.1142/S0218127412501040.; Guo, P.; Li, C.; Chen, G., On the fractional mean-value theorem, Int. J. Bifurcation Chaos, 22 (2012) · Zbl 1258.26003 |
| [7] | S.G. Kellison, R.L. London, Risk Models and Their Estimation. ACTEX Publications, Winsted, CT (2011).; Kellison, S. G.; London, R. L., Risk Models and Their Estimation. (2011) |
| [8] | S.A. Klugman, H.H. Panjer, G.E. Willmot, Loss Models: From Data to Decisions. 4th edition, Wiley, Hoboken, NJ (2012).; Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: From Data to Decisions (2012) · Zbl 1272.62002 |
| [9] | G.D. Lin, On a probabilistic generalization of Taylor’s theorem. Statist. Probab. Lett. 19, No 3 (1994), 239-243. doi:10.1016/0167-7152(94)90110-4; Lin, G. D., On a probabilistic generalization of Taylor’s theorem, Statist. Probab. Lett, 19, 3, 239-243 (1994) · Zbl 0824.60014 |
| [10] | A.W. Marshall, I. Olkin, Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer-Verlag, New York (2007); doi:10.1007/978-0-387-68477-2.; Marshall, A. W.; Olkin, I., Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families (2007) · Zbl 1304.62019 |
| [11] | W.A. Massey, W. Whitt, A probabilistic generalization of Taylor’s theorem. Statist. Probab. Lett. 16, No 1 (1993), 51-54; doi:10.1016/0167-7152(93)90122-Y.; Massey, W. A.; Whitt, W., A probabilistic generalization of Taylor’s theorem, Statist. Probab. Lett, 16, 1, 51-54 (1993) · Zbl 0765.60032 |
| [12] | T. Mikosch, G. Samorodnitsky, L. Tafakori, Fractional moments of solutions to stochastic recurrence equations. J. Appl. Probab. 50, No 4 (2013), 969-982; doi:10.1239/jap/1389370094.; Mikosch, T.; Samorodnitsky, G.; Tafakori, L., Fractional moments of solutions to stochastic recurrence equations, J. Appl. Probab, 50, 4, 969-982 (2013) · Zbl 1303.60042 |
| [13] | Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula. Appl. Math. Comput. 186, No 1 (2007), 286-293; doi:10.1016/j.amc.2006.07.102.; Odibat, Z. M.; Shawagfeh, N. T., Generalized Taylor’s formula, Appl. Math. Comput, 186, 1, 286-293 (2007) · Zbl 1122.26006 |
| [14] | S. Ortobelli, S. Rachev, H. Shalit, F.J. Fabozzi, Orderings and risk probability functionals in portfolio theory. Probab. Math. Statist. 28, No 2 (2008), 203-234.; Ortobelli, S.; Rachev, S.; Shalit, H.; Fabozzi, F. J., Orderings and risk probability functionals in portfolio theory, Probab. Math. Statist, 28, 2, 203-234 (2008) · Zbl 1158.60322 |
| [15] | J.E. Pečari\'c, I. Peri\'c, H.M. Srivastava, A family of the Cauchy type mean-value theorems. J. Math. Anal. Appl. 306, No 2 (2005), 730-739. doi:10.1016/j.jmaa.2004.10.018; Pečari\'c, J. E.; Peri\'c, I.; Srivastava, H. M., A family of the Cauchy type mean-value theorems, J. Math. Anal. Appl, 306, 2, 730-739 (2005) · Zbl 1068.26008 |
| [16] | M. Shaked, J.G. Shanthikumar, Stochastic Orders. Springer, New York (2007); doi:10.1007/978-0-387-34675-5.; Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007) · Zbl 0806.62009 |
| [17] | D. Stoyan, Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983).; Stoyan, D., Comparison Methods for Queues and Other Stochastic Models (1983) · Zbl 0536.60085 |
| [18] | J.J. Trujillo, M. Rivero, B. Bonilla, On a Riemann-Liouville generalized Taylor’s formula. J. Math. Anal. Appl. 231, No 1 (1999), 255-265; doi:10.1006/jmaa.1998.6224.; Trujillo, J. J.; Rivero, M.; Bonilla, B., On a Riemann-Liouville generalized Taylor’s formula, J. Math. Anal. Appl, 231, 1, 255-265 (1999) · Zbl 0931.26004 |
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