TY - JOUR

T1 - On asymptotic behavior of increments of sums along head runs

AU - Frolov, A. N.

N1 - Funding Information: This research was supported by the Government of St. Petersburg, by the Ministry of Education of Russia, and by the Russian Academy of Sciences, grant M98-2.1 -173. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2002

Y1 - 2002

N2 - Let {(Xi, Yi)} be a sequence of independent equidistributed random, vectors with P(Y1 = 1) = p = 1 - P(Y 1 = 0) ∈ (0,1). Let Mn(j) = max 0≤k≤n-j(Xk+1 + ⋯ + Xk+j)I k,j , where Ik,j = I{Yk+1 = ⋯ = Y k+j = 1} and I{·} denotes the indicator function of the event in brackets. If, for example, {Xi} are the gains and {Yi} are the indicators of success in repetitions of a game of chance, then M n(j) is the maximal gain along head runs (sequences of successes without interruptions) of length j. We investigate the asymptotic behavior of the values Mn(j), j = jn ≤ Ln, where L n is the length of the longest head run in Y1, . . . ,Yn. We show that the asymptotics of the values Mn(j) depend significantly on the growth rate of j and that these asymptotics vary from the strong noninvariance (as in the Erdos-Rényi law of large numbers) to the strong invariance (as in the Csörgo-Révész strong approximation laws). We also consider the Shepp-type statistics.

AB - Let {(Xi, Yi)} be a sequence of independent equidistributed random, vectors with P(Y1 = 1) = p = 1 - P(Y 1 = 0) ∈ (0,1). Let Mn(j) = max 0≤k≤n-j(Xk+1 + ⋯ + Xk+j)I k,j , where Ik,j = I{Yk+1 = ⋯ = Y k+j = 1} and I{·} denotes the indicator function of the event in brackets. If, for example, {Xi} are the gains and {Yi} are the indicators of success in repetitions of a game of chance, then M n(j) is the maximal gain along head runs (sequences of successes without interruptions) of length j. We investigate the asymptotic behavior of the values Mn(j), j = jn ≤ Ln, where L n is the length of the longest head run in Y1, . . . ,Yn. We show that the asymptotics of the values Mn(j) depend significantly on the growth rate of j and that these asymptotics vary from the strong noninvariance (as in the Erdos-Rényi law of large numbers) to the strong invariance (as in the Csörgo-Révész strong approximation laws). We also consider the Shepp-type statistics.

UR - http://www.scopus.com/inward/record.url?scp=52649163504&partnerID=8YFLogxK

U2 - 10.1023/A:1014597820537

DO - 10.1023/A:1014597820537

M3 - Article

AN - SCOPUS:52649163504

VL - 109

SP - 2229

EP - 2240

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -