## 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 Poincaré, 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

Zbl 0619.60039
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