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Moments of Some Stopping Rules
Author(s) -
Teicher Henry,
Zhang CunHui
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005912
Subject(s) - combinatorics , independent and identically distributed random variables , random variable , mathematics , moment (physics) , physics , statistics , quantum mechanics
Let X n for n ⩾1 be independent random variables with E X N 2 = 1 . SetS k , n = ∑ i 1 < … < i k ⩽ nX i 1… X i k. Define T k,c,m =inf{ n ⩾ m :∣ k ! S k,n ∣> cn k /2 }. We study critical values c k,p for k ⩾2 and p >0, such that E T k , c , m p = ∞ for c < c k,p and all m , and E T k , c , m p = ∞ for c > c k,p and all sufficiently large m . In particular, c 1,1 = c 2,1 =1, c 3,1 =2 and c 4,1 =3 under certain moment conditions on X 1 , when X n are identically distributed. We also investigate perturbed stopping rules of the form T h,m =inf{ n ⩾ m : h ( S 1, n / n 1/2 )<ζ n or >ζ n } for continuous functions h and random variables ζ n ∼ a and ζ n ∼ b with a < b . Related stopping rules of the Wiener process are also considered via the Uhlenbeck process.