How many entries of a typical orthogonal matrix can be approximated by independent normals?

الأصل · EN

We solve an open problem of Diaconis that asks what are the largest orders of pₙ and qₙ such that Zₙ, the pₙ× qₙ upper left block of a random matrix Γₙ 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 Zₙ and that of pₙqₙ independent standard normals goes to zero provided pₙ=o(√n) and qₙ=o(√n). We also show that the above variation distance does not go to zero if pₙ=[x√n] and qₙ=[y√n] for any positive numbers x and y. This says that the largest orders of pₙ and qₙ are o(n¹/²) in the sense of the above approximation. Second, suppose Γₙ=(γij)ₙ× ₙ is generated by performing the Gram--Schmidt algorithm on the columns of Yₙ=(yij)ₙ× ₙ, where {yij;1≤ i,j≤ n} are i.i.d. standard normals. We show that εₙ(m):=₁≤ ᵢ≤ ₙ,₁≤ ⱼ≤ ₘ|√n·γij-yij| goes to zero in probability as long as m=mₙ=o(n/ n). We also prove that εₙ(mₙ)→ 2√α in probability when mₙ=[nα/ n] for any α>0. This says that mₙ=o(n/ n) is the largest order such that the entries of the first mₙ columns of Γₙ can be approximated simultaneously by independent standard normals.