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

  • Ion GramaEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 208)

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]).

Keywords

Conditioned Markov walks General linear group Affine Markov walk Limit theorems 

References

  1. 1.
    Bertoin, J., Doney, R.A.: On conditioning a random walk to stay nonnegative. Ann. Probab. 22(4), 2152–2167 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bolthausen, E.: On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4(3), 480–485 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, I. Math. Notes 75(1–2), 23–37 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, II. Math. Notes 75(3–4), 322–330 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schödinger Operators. Birghäuser, Boston-Basel-Stuttgart (1985)CrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Denisov, D., Wachtel, V.: Conditional limit theorems for ordered random walks. Electron. J. Probab. 15, 292–322 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Doney, R.A.: Conditional limit theorems for asymptotically stable random walks. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 70(3), 351–360 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eichelsbacher, P., König, W.: Ordered random walks. Electron. J. Probab. 13, 1307–1336 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gordin, M.I.: The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176 (1969)zbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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. 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. 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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Iglehart, D.L.: Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2(4), 608–619 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Le Page, E.: Théorèmes limites pour les produits de matrices aléatoires. Springer Lecture Notes, vol. 928, pp. 258–303 (1982)Google Scholar
  20. 20.
    Lévy, P.: Théorie de l’addition des variables aléatoires. Gauthier-Villars, Paris (1937)zbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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. 23.
    Varopoulos, N.Th.: Potential theory in conical domains. Math. Proc. Camb. Philos. Soc. 125(2), 335–384 (1999)Google Scholar
  24. 24.
    Varopoulos, N.Th.: Potential theory in conical domains. II. Math. Proc. Camb. Philos. Soc. 129(2), 301–320 (2000)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université de Bretagne SudVannesFrance

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