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Results of the type of de Finetti for \(l_ p\). (Russian) Zbl 0746.60022

Let \(\vec \zeta_ n = (\zeta_ 1,\dots ,\zeta_ n)\) be uniformly distributed on the surface of the sphere \(\bigl\{ x_ n = (x_ 1,\dots ,x_ n):\;\sum_ 1^ n| x_ i| ^ p = n\bigr\} \). Let \(\vec \xi_ n = (\xi_ 1,\dots ,\xi_ n)\) be a random vector and \(\xi_ 1,\dots ,\xi_ n\) be i.i.d. random variables with density \(p(t) = \alpha \exp \{ -\beta| t| ^ p\}\). Denote by \(P_{n,k}\) the distribution of the first \(k\) coordinates of \(\zeta_ n\), by \(P_ k\) the law of \((\xi_ 1,\dots ,\xi_ k)\). Let \(\| P-Q\|\) be variation distance between two probability measures \(P\) and \(Q\). It is shown, that for \(p\geq 1\), \(1\leq k\leq n,\;n=1,2,\dots\) \[ \| P_{n,k} - Q_ k\| \leq C_{n/k}. \] This theorem is proved in [P. Diaconis and D. Freedman, Ann. Inst. Henri PoincarĂ©, Probab. Stat. 23, Suppl., 397-423 (1987; Zbl 0619.60039)] for cases \(p=1\) and \(p=2\).

MSC:

60F05 Central limit and other weak theorems
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization

Citations:

Zbl 0619.60039
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