Operator Decomposable Measures and Stochastic Difference Equations

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• 作者：C. R. E. Raja
• 关键词：Stochastic difference equation ; Probability measures ; Linear map ; Convolution product ; Operator decomposable measures ; Contraction subspace
• 刊名：Journal of Theoretical Probability
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：28
• 期：3
• 页码：785-803
• 全文大小：548 KB
• 参考文献：1.Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MATH MathSciNet CrossRef
  2.Cirel’son, B.S.: An example of a stochastic differential equation that has no strong solution (Russian). Teor. Verojatnost. i Primenen. 20, 427–430 (1975)MathSciNet
  3.Yor, M.: Tsirel’son’s equation in discrete time. Probab. Theory Rel. Fields 91, 135–152 (1992)MATH MathSciNet CrossRef
  4.Hirayama, T., Yano, K.: Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups. Stochast. Process. Appl. 120, 1404–1423 (2010)MATH MathSciNet CrossRef
  5.Akahori, J., Uenishi, C., Yano, K.: Stochastic equations on compact groups in discrete negative time. Probab. Theory Rel. Fields 140, 569–593 (2008)MATH MathSciNet CrossRef
  6.Takahashi, Y.: Time evolution with and without remote past. Adv. Discrete Dynam. Syst. 53, 347–361 (2009)
  7.Raja, C.R.E.: A stochastic difference equation with stationary noise on groups. Can. J. Math. 64, 1075–1089 (2012)MATH MathSciNet CrossRef
  8.Siebert, E.: Strongly operator-decomposable probability measures on separable Banach spaces. Math. Nachr. 154, 315–326 (1991)MATH MathSciNet CrossRef
  9.Siebert, E.: Operator-decomposability of Gaussian measures on separable Banach spaces. J. Theor. Probab. 5, 333–347 (1992)MATH MathSciNet CrossRef
  10.Zakusilo, O.: Some properties of random vectors of the form \(\sum _0 ^\infty A\xi _i\) . Theory Probab. Math. Stat. 13, 62–64 (1977)MATH
  11.Parthasarathy, K.R.: Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press Inc., New York (1967)
  12.Linde, W.: Probability in Banach Spaces: Stable and Infinitely Divisible Distributions, 2nd edn. Wiley, Chichester (1986)MATH
  13.Csiszár, I.: On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5, 279–295 (1966)MATH MathSciNet CrossRef
  14.Tortrat, A.: Convolutions dénombrables équitendues dans un groupe topologique X, Les probabilités sur les structures algébriques (Actes Colloq. Internat. CNRS, No. 186, Clermont-Ferrand, 1969), 327–343. Éditions Centre Nat. Recherche Sci., Paris (1970)
  15.Azencott, R.: Espaces de Poisson des groupes localement compacts, Lecture Notes in Mathematics, Vol. 148. Springer, Berlin (1970)CrossRef
  16.Chung, K.L.: A Course in Probability Theory, 3rd edn. Academic Press Inc., San Diego, CA (2001)
  17.Linnik, J.V., Ostrovs’kii, I.V.: Decomposition of Random Variables and Vectors. Translated from the Russian. Translations of Mathematical Monographs, vol. 48. American Mathematical Society, Providence, RI (1977)
  
• 作者单位：C. R. E. Raja (1)
  
  1. Stat-Math Unit, Indian Statistical Institute (ISI), 8th Mile Mysore Road, Bangalore, 560 059, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

We consider the following convolution equation (or equivalently stochastic difference equation) $$\begin{aligned} \lambda _k = \mu _k*\phi (\lambda _{k-1}),\quad k \in {\mathbb Z}\end{aligned}$$ (1)for a given bi-sequence \((\mu _k)\) of probability measures on \({\mathbb R}^d\) and a linear map \(\phi \) on \({\mathbb R}^d\). We study the solutions of Eq. (1) by realizing the process \((\mu _k)\) as a measure on \(({\mathbb R}^d)^{\mathbb Z}\) and rewriting the stochastic difference equation as \(\lambda = \mu *\tau (\lambda )\)-any such measure \(\lambda \) on \(({\mathbb R}^d)^{\mathbb Z}\) is known as \(\tau \)-decomposable measure with co-factor \(\mu \) where \(\tau \) is a suitable weighted shift operator on \(({\mathbb R}^d)^{\mathbb Z}\). This enables one to study the solutions of (1) in the settings of \(\tau \)-decomposable measures. A solution \((\lambda _k)\) of (1) will be called a fundamental solution if any solution of (1) can be written as \(\lambda _k*\phi ^k(\rho )\) for some probability measure \(\rho \) on \({\mathbb R}^d\). Motivated by the splitting/factorization theorems for operator decomposable measures, we address the question of existence of fundamental solutions when a solution exists and answer affirmatively via a one–one correspondence between fundamental solutions of (1) and strongly \(\tau \)-decomposable measures on \(({\mathbb R}^d)^{\mathbb Z}\) with co-factor \(\mu \). We also prove that fundamental solutions are extremal solutions and vice versa. We provide a necessary and sufficient condition in terms of a logarithmic moment condition for the existence of a (fundamental) solution when the noise process is stationary and when the noise process has independent \(\ell _p\)-paths. Keywords Stochastic difference equation Probability measures Linear map Convolution product Operator decomposable measures Contraction subspace Mathematics Subject Classification (2010) 60B15 60G50 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (17) References1.Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MATHMathSciNetCrossRef2.Cirel’son, B.S.: An example of a stochastic differential equation that has no strong solution (Russian). Teor. Verojatnost. i Primenen. 20, 427–430 (1975)MathSciNet3.Yor, M.: Tsirel’son’s equation in discrete time. Probab. Theory Rel. Fields 91, 135–152 (1992)MATHMathSciNetCrossRef4.Hirayama, T., Yano, K.: Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups. Stochast. Process. Appl. 120, 1404–1423 (2010)MATHMathSciNetCrossRef5.Akahori, J., Uenishi, C., Yano, K.: Stochastic equations on compact groups in discrete negative time. Probab. Theory Rel. Fields 140, 569–593 (2008)MATHMathSciNetCrossRef6.Takahashi, Y.: Time evolution with and without remote past. Adv. Discrete Dynam. Syst. 53, 347–361 (2009)7.Raja, C.R.E.: A stochastic difference equation with stationary noise on groups. Can. J. Math. 64, 1075–1089 (2012)MATHMathSciNetCrossRef8.Siebert, E.: Strongly operator-decomposable probability measures on separable Banach spaces. Math. Nachr. 154, 315–326 (1991)MATHMathSciNetCrossRef9.Siebert, E.: Operator-decomposability of Gaussian measures on separable Banach spaces. J. Theor. Probab. 5, 333–347 (1992)MATHMathSciNetCrossRef10.Zakusilo, O.: Some properties of random vectors of the form \(\sum _0 ^\infty A\xi _i\). Theory Probab. Math. Stat. 13, 62–64 (1977)MATH11.Parthasarathy, K.R.: Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press Inc., New York (1967)12.Linde, W.: Probability in Banach Spaces: Stable and Infinitely Divisible Distributions, 2nd edn. Wiley, Chichester (1986)MATH13.Csiszár, I.: On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5, 279–295 (1966)MATHMathSciNetCrossRef14.Tortrat, A.: Convolutions dénombrables équitendues dans un groupe topologique X, Les probabilités sur les structures algébriques (Actes Colloq. Internat. CNRS, No. 186, Clermont-Ferrand, 1969), 327–343. Éditions Centre Nat. Recherche Sci., Paris (1970)15.Azencott, R.: Espaces de Poisson des groupes localement compacts, Lecture Notes in Mathematics, Vol. 148. Springer, Berlin (1970)CrossRef16.Chung, K.L.: A Course in Probability Theory, 3rd edn. Academic Press Inc., San Diego, CA (2001)17.Linnik, J.V., Ostrovs’kii, I.V.: Decomposition of Random Variables and Vectors. Translated from the Russian. Translations of Mathematical Monographs, vol. 48. American Mathematical Society, Providence, RI (1977) About this Article Title Operator Decomposable Measures and Stochastic Difference Equations Journal Journal of Theoretical Probability Volume 28, Issue 3 , pp 785-803 Cover Date2015-09 DOI 10.1007/s10959-013-0534-8 Print ISSN 0894-9840 Online ISSN 1572-9230 Publisher Springer US Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Probability Theory and Stochastic Processes Statistics, general Keywords Stochastic difference equation Probability measures Linear map Convolution product Operator decomposable measures Contraction subspace 60B15 60G50 Industry Sectors IT & Software Telecommunications Authors C. R. E. Raja (1) Author Affiliations 1. Stat-Math Unit, Indian Statistical Institute (ISI), 8th Mile Mysore Road, Bangalore, 560 059, India Continue reading... To view the rest of this content please follow the download PDF link above.