TY - JOUR
T1 - How many entries of a typical orthogonal matrix can be approximated by independent normals?
AU - Jiang, Tiefeng
PY - 2006/7
Y1 - 2006/7
N2 - We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n, the p n × q n upper left block of a random matrix Γ n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of Z n and that of p nq n independent standard normals goes to zero provided p n = o(√/n) and q n = o(√/n). We also show that the above variation distance does not go to zero if p n = [x√/n] and q n = [y√/n] for any positive numbers x and y. This says that the largest orders of p n and q n are o(n 1/2) in the sense of the above approximation. Second, suppose Γ n = (γij)n×n is generated by performing the GramSchmidt algorithm on the columns of Y n = (yij)n×n, where {yij; 1 ≤ i, j ≤ n} are i.i.d. standard normals. We show that ε n(m):=max 1≤i≤n,1≤j≤m |√n· γij - yij | goes to zero in probability as long as m = m n = o(n/ log n). We also prove that ε n (m n) → 2√α in probability when m n = [nα/ log n] for any α > 0. This says that m n = o(n/log n) is the largest order such that the entries of the first m n columns of Γ n can be approximated simultaneously by independent standard normals.
AB - We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n, the p n × q n upper left block of a random matrix Γ n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of Z n and that of p nq n independent standard normals goes to zero provided p n = o(√/n) and q n = o(√/n). We also show that the above variation distance does not go to zero if p n = [x√/n] and q n = [y√/n] for any positive numbers x and y. This says that the largest orders of p n and q n are o(n 1/2) in the sense of the above approximation. Second, suppose Γ n = (γij)n×n is generated by performing the GramSchmidt algorithm on the columns of Y n = (yij)n×n, where {yij; 1 ≤ i, j ≤ n} are i.i.d. standard normals. We show that ε n(m):=max 1≤i≤n,1≤j≤m |√n· γij - yij | goes to zero in probability as long as m = m n = o(n/ log n). We also prove that ε n (m n) → 2√α in probability when m n = [nα/ log n] for any α > 0. This says that m n = o(n/log n) is the largest order such that the entries of the first m n columns of Γ n can be approximated simultaneously by independent standard normals.
KW - Gram-Schmidt algorithm
KW - Haar measure
KW - Large deviation
KW - Maxima
KW - Product distribution
KW - Random matrix theory
KW - Variation distance
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UR - http://www.scopus.com/inward/citedby.url?scp=33750157608&partnerID=8YFLogxK
U2 - 10.1214/009117906000000205
DO - 10.1214/009117906000000205
M3 - Article
AN - SCOPUS:33750157608
SN - 0091-1798
VL - 34
SP - 1497
EP - 1529
JO - Annals of Probability
JF - Annals of Probability
IS - 4
ER -