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


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.


Asymptotic Behavior Random Vector Indicator Function Maximal Gain Large Index 
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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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