ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 69,   2   (2000)
pp.   137-144

ON LARGE RANDOM ALMOST EUCLIDEAN BASES
R. VERSHYNIN

Abstract.  A new class of random proportional embeddings of $l_2^n$ into certain Banach spaces is found. Let $(\xi_i)_i=1^n$ be i.i.d. mean zero \Cramer random variables. Suppose $(x_i)_i=1^n$ is a sequence in the unit ball of a Banach space with $\E \| \sum_i \e_i x_i \| \ge \d n$. Then the system of $$cn$$ independent random vectors distributed as $\sum_i \xi_i x_i$ is well equivalent to the euclidean basis with high probability ($c$ depends on $\xi_1$ and $\d$). A connection with combinatorial discrepancy theory is presented.

AMS subject classification.  46B09, 05B20, 41A28
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