TY - JOUR
T1 - ON THE UPPER BOUND OF THE Lp DISCREPANCY OF HALTON’S SEQUENCE AND THE CENTRAL LIMIT THEOREM FOR HAMMERSLEY’S NET, II
AU - Levin, Mordechay B.
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.
PY - 2024/4
Y1 - 2024/4
N2 - Let Hs+1,N=(Hs(n),n/N)n=0N-1 be the s+1-dimensional Hammersley’s point set. Let D(x¯,(Hs+1,N)n=0N-1) be the local discrepancy of (Hs+1,N)n=0N-1, and let Ds,p((Hs+1,N)n=0N-1) be the Lp discrepancy of (Hs+1,N)n=0N-1. In this Part II of our paper, we prove the Central Limit Theorem for Hammersley’s net: (Formula presented.) where x¯ is a uniformly distributed random variable in [0,1]s+1 and we claim that the lower bound of Lp discrepancy for p>0 is of the same order as the upper bound: (Formula presented.)
AB - Let Hs+1,N=(Hs(n),n/N)n=0N-1 be the s+1-dimensional Hammersley’s point set. Let D(x¯,(Hs+1,N)n=0N-1) be the local discrepancy of (Hs+1,N)n=0N-1, and let Ds,p((Hs+1,N)n=0N-1) be the Lp discrepancy of (Hs+1,N)n=0N-1. In this Part II of our paper, we prove the Central Limit Theorem for Hammersley’s net: (Formula presented.) where x¯ is a uniformly distributed random variable in [0,1]s+1 and we claim that the lower bound of Lp discrepancy for p>0 is of the same order as the upper bound: (Formula presented.)
KW - Central limit theorem
KW - Halton’s sequence
UR - http://www.scopus.com/inward/record.url?scp=85205935924&partnerID=8YFLogxK
U2 - 10.1007/s10958-024-07312-9
DO - 10.1007/s10958-024-07312-9
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AN - SCOPUS:85205935924
SN - 1072-3374
VL - 280
SP - 1030
EP - 1064
JO - Journal of Mathematical Sciences
JF - Journal of Mathematical Sciences
IS - 6
ER -