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A Survey on Conditioned Limit Theorems for Products of Random Matrices and Affine Random Walks

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 208))

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Abstract

This paper is a survey of results on the asymptotics of the exit time from certain domains and conditioned limit theorems to stay in the same domains for two type of Markov walks studied in Grama et al. (Prob Theory Rel Fields, 2016, [15]) and Grama et al. (Ann I.H.P, 2016, [16]).

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References

  1. Bertoin, J., Doney, R.A.: On conditioning a random walk to stay nonnegative. Ann. Probab. 22(4), 2152–2167 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bolthausen, E.: On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4(3), 480–485 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, I. Math. Notes 75(1–2), 23–37 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, II. Math. Notes 75(3–4), 322–330 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schödinger Operators. Birghäuser, Boston-Basel-Stuttgart (1985)

    Book  MATH  Google Scholar 

  6. Dembo, A., Ding, J., Gao, F.: Persistence of iterated partial sums. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 49(3), 873–884 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Denisov, D., Vatutin, V., Wachtel, V.: Local probabilities for random walks with negative drift conditioned to stay nonnegative. Electron. J. Probab. 19(88), 1–17 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Denisov, D., Wachtel, V.: Conditional limit theorems for ordered random walks. Electron. J. Probab. 15, 292–322 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Denisov, D., Wachtel, V.: Exit times for integrated random walks. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 51(1), 167–193 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doney, R.A.: Conditional limit theorems for asymptotically stable random walks. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 70(3), 351–360 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eichelsbacher, P., König, W.: Ordered random walks. Electron. J. Probab. 13, 1307–1336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gordin, M.I.: The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176 (1969)

    MATH  Google Scholar 

  14. Grama, I., Le Page, E., Peigné, M.: On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains. Colloq. Math. 134, 1–55 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grama, I., Le Page, E., Peigné, M.: Conditioned limit theorems for products of random matrices. Prob. Theory Rel. Fields 168(3–4), 601–639 (2017)

    Google Scholar 

  16. Grama, I., Lauvergnat, R., Le Page, E.: Limit theorems for affine Markov walks conditioned to stay positive. Ann. I.H.P. (2016). arXiv:1601.02991. (to appear)

  17. Guivarc’h, Y., Le Page, E.: On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Ergod. Theory Dyn. Syst. 28(02), 423–446 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Iglehart, D.L.: Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2(4), 608–619 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Le Page, E.: Théorèmes limites pour les produits de matrices aléatoires. Springer Lecture Notes, vol. 928, pp. 258–303 (1982)

    Google Scholar 

  20. Lévy, P.: Théorie de l’addition des variables aléatoires. Gauthier-Villars, Paris (1937)

    MATH  Google Scholar 

  21. Presman, E.: Boundary problems for sums of lattice random variables, defined on a finite regular Markov chain. Theory Probab. Appl. 12(2), 323–328 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  22. Presman, E.: Methods of factorization and a boundary problems for sums of random variables defined on a markov chain. Izvestija Akademii Nauk SSSR 33, 861–990 (1969)

    Google Scholar 

  23. Varopoulos, N.Th.: Potential theory in conical domains. Math. Proc. Camb. Philos. Soc. 125(2), 335–384 (1999)

    Google Scholar 

  24. Varopoulos, N.Th.: Potential theory in conical domains. II. Math. Proc. Camb. Philos. Soc. 129(2), 301–320 (2000)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Université de Bretagne Sud, 56017, Vannes, France

    Ion Grama

Authors
  1. Ion Grama
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Correspondence to Ion Grama .

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Editors and Affiliations

  1. Department of Statistics and Data Analysis, Higher School of Economics, Moscow, Russia

    Vladimir Panov

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Grama, I. (2017). A Survey on Conditioned Limit Theorems for Products of Random Matrices and Affine Random Walks. In: Panov, V. (eds) Modern Problems of Stochastic Analysis and Statistics. MPSAS 2016. Springer Proceedings in Mathematics & Statistics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-319-65313-6_5

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