ABSTRACT
Let https://www.w3.org/1998/Math/MathML"> ∠ 1 , … , ∠ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2685.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be i.i.d. standard normal variables, https://www.w3.org/1998/Math/MathML"> M n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2686.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the extreme among their absolute values, and https://www.w3.org/1998/Math/MathML"> Q n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2687.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> their rooted sum of squares. Here we study the ratio https://www.w3.org/1998/Math/MathML"> v n = M n / Q n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2688.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We show that https://www.w3.org/1998/Math/MathML"> μ v n = μ M n / μ Q n ∼ b n / n ∼ 2 l o g ( n ) / n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2689.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> c n ∼ d n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2690.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denotes that https://www.w3.org/1998/Math/MathML"> c n / d n → 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2691.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for https://www.w3.org/1998/Math/MathML"> n → ∞ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2692.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Moreover, convergence happens in a stable manner, i.e. https://www.w3.org/1998/Math/MathML"> v n ∼ μ v n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2693.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in probability. For low dimensions we also derive explicit expressions for https://www.w3.org/1998/Math/MathML"> μ v n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2694.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and variance https://www.w3.org/1998/Math/MathML"> σ 2 v n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2695.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . These expressions are calculated by using an appealing geometrical interpretation of https://www.w3.org/1998/Math/MathML"> v n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429227714/c64932ec-bf9c-4c8e-a5d1-8a90dc5fbe07/content/eq2696.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
