Advertisement

Bounds in the Local Limit Theorem for a Random Walk Conditioned to Stay Positive

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

Abstract

Let \(\left( X_{i}\right) _{i\ge 1}\) be a sequence i.i.d.  random variables and \(S_{n}=\sum _{i=1}^{n}X_{i}\), \(n\ge 1\). For any starting point \(y>0\) denote by \(\tau _{y}\) the first moment when the random walk \( \left( y+S_{k}\right) _{k\ge 1}\) becomes negative. We give some bounds of order \(n^{-3/2}\) for the expectations \(\mathbb {E}\left( g\left( y+S_{n}\right) ;\tau _{n}>n\right) \), \(y\in \mathbb {R}_{+}^{*}\) which are valid for a large class of bounded measurable function g with constants depending on some norms of the function g.

Keywords

Exit time Random walk conditioned to stay positive Local limit theorem 

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.
    Borovkov, A.A.: On the asymptotic behavior of the distributions of first-passage times. I. Math. Notes 75(1), 23–37 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times. II. Math. Notes 75(3), 322–330 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Borovkov, A.A.: On the rate of convergence in the invariance principle. Probab. Theory Appl. 18(2), 217–234 (1973)MathSciNetGoogle Scholar
  5. 5.
    Caravenna, F.: A local limit theorem for random walks conditioned to stay positive. Probab. Theory Relat. Fields. 133(4), 508–530 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Denisov, D., Wachtel, V.: Conditional limit theorems for ordered random walks. Electron. J. Probab. 15, 292–322 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Doney, R.A.: Conditional limit theorems for asymptotically stable random walks. Z. Wahrscheinlichkeitsth. 70, 351–360 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eichelsbacher, P., König, W.: Ordered random walks. Electron. J. Probab. 13(46), 1307–1336 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gnedenko, B.V.: On a local limit theorem of the theory of probability. Uspekhi Mat. Nauk 3(3), 187–194 (1948)MathSciNetGoogle Scholar
  11. 11.
    Grama, I., Lauvergnat, R., Le Page, E.: Limit theorems for affine Markov walks conditioned to stay positive. To appear in Ann. I.H.P. (2016). arXiv:1601.02991
  12. 12.
    Iglehart, D.L.: Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2(4), 608–619 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Spitzer, F.: Principles of Random Walk, Second edn. Springer, Berlin (1976)Google Scholar
  14. 14.
    Shepp, L.A.: A local limit theorem. Ann. Math. Statist. 35(1), 419–423 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stone, L.A.: A local limit theorem for nonlattice multi-dimensional distribution functions Ann. Math. Statist. 36(2), 546–551 (1964)CrossRefzbMATHGoogle Scholar
  16. 16.
    Vatutin, V.A., Wachtel, V.: Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields 143(1–2), 177–217 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université de Bretagne SudVannesFrance

Personalised recommendations