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Journal of Mathematical Sciences

, Volume 133, Issue 3, pp 1308–1313 | Cite as

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

  • A. I. Martikainen
Article
  • 33 Downloads

Abstract

Let {Xi, Yi}i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y1 = y) = 0 for all y. Put Mn(j) = max0≤k≤n-j (Xk+1 + ... Xk+j)Ik,j, where Ik,k+j = I{Yk+1 < ⋯ < Yk+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let Ln be the largest index l ≤ n for which Ik,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for Mn(Ln) has been recently derived for the case where X1 has a finite moment of order 3 + ε, ε > 0. Assuming that X1 has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(Ln - a) is equivalent to EX 1 3+a I{X1 > 0} < ∞. We derive also some new results for the a.s. asymptotics of Ln. Bibliography: 5 titles.

Keywords

Asymptotic Behavior Random Vector Indicator Function Maximal Gain Large Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. I. Martikainen
    • 1
  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

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