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By Bingham M.S.

A crucial restrict theorem is given for uniformly infinitesimal triangular arrays of random variables during which the random variables in each one row are exchangeable and take values in a in the community compact moment countable abclian team. The proscribing distribution within the theorem is Gaussian.

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Step 1 y1 ≥ √ 3 2 for i = 1, 2, . . , √ 3 . 2 ⎛ ⎞ In−2 0 −1 ⎠ on γ ◦ z. Here In−2 1 0 denotes the identity (n − 2) × (n − 2)–matrix. First of all This follows from the action of α := ⎝ φ(α ◦ γ ◦ z) = ||en · α ◦ γ ◦ z|| = ||en−1 · x · y|| = ||(en−1 + xn−1,n en ) · y|| 2 . = d y12 + xn−1,n Since |xn−1,n | ≤ 12 we see that φ(αγ z)2 ≤ d 2 (y12 + 41 ). On the other hand, the assumption of minimality forces φ(γ z)2 = d 2 ≤ d 2 y12 + 14 . This implies that y1 ≥ √ 3 . 2 g′ 1 . Then φ(gγ z) = φ(γ z). 0 1 This follows immediately from the fact that en · g = en .

N − 2. 9) ∈ S L(n, Z). 4 Haar measure where y ′′ = y′d ′ 0 y ′′ = imply that yi ≥ 0 d y′d ′ 0 √ 3 2 x′ 0 , x ′′ = 0 d ⎛ ⎜ ⎜ =⎜ ⎝ 19 ∗ . 9) applied to 1 y1 y2 · · · yn−1 d .. ⎞ . y1 d d ⎟ ⎟ ⎟, ⎠ for i = 2, 3, . . , n − 1. Step 2 insures that multiplying by g on the left does not change the value of φ(γ z). 4 √ 3 . 2 Haar measure Let n ≥ 2. The discrete subgroup S L(n, Z) acts on S L(n, R) by left multiplication. The quotient space S L(n, Z)\S L(n, R) turns out to be of fundamental importance in number theory.

We follow the exposition of Garret (2002). 1 Let n ≥ 2. 3, fix n−1 d∗z = yk−k(n−k)−1 dyk d xi, j 1≤i< j≤n k=1 to be the left S L(n, R)–invariant measure on hn = S L(n, R)/S O(n, R). Then n S L(n,Z)\hn d ∗ z = n 2n−1 · ℓ=2 ζ (ℓ) , Vol(S ℓ−1 ) where Vol(S ℓ−1 ) = √ 2( π)ℓ Ŵ (ℓ/2) denotes the volume of the (ℓ − 1)–dimensional sphere S ℓ−1 and ζ (ℓ) = denotes the Riemann zeta function. ∞ n=1 n −ℓ Discrete group actions 28 Proof for the case of S L(2, R) We first prove the theorem for S L(2, R). The more general result will follow by induction.

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A central limit theorem for exchangeable random variables on a locally compact abelian group by Bingham M.S.


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