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=Mn(Ln)=max0≤k≤n-Ln(X k+1++Xk+Ln)Ik,Ln, where Ik,ℓ=I{Yk+1≤≤Yk+ℓ} denotes the indicator function of the event in brackets, Ln is the largest ℓ≤n, for which Ik,ℓ=1 for some k=0,1,...,n-ℓ. If, for example, Xi=Yi, i≥1, and Xi denotes the gain in the ith repetition of a game of chance, then Mn is the maximal gain over increasing runs of maximal length Ln. We derive a strong law of large numbers and a law of iterated logarithm type result for Mn.

Original languageEnglish
Pages (from-to)305-312
Number of pages8
JournalStatistics and Probability Letters
Volume50
Issue number3
DOIs
StatePublished - 15 Nov 2000

    Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

    Research areas

  • Increasing run, Law of iterated logarithm, Primary 60F15, Secondary 60F10, Strong law of large numbers

ID: 75020743