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On exact maximal Khinchine inequalities. (English) Zbl 0971.60012

Giné, Evarist (ed.) et al., High dimensional probability II. 2nd international conference, Univ. of Washington, DC, USA, August 1-6, 1999. Boston, MA: Birkhäuser. Prog. Probab. 47, 49-63 (2000).
S. E. Graversen and G. Peškir [Math. Proc. Camb. Philos. Soc. 123, No. 1, 169-177 (1998; Zbl 0899.60013)] conjectured that \(E\max_{1\leq k\leq n}|\sum^n_{i=1} a_i\varepsilon_i |^p\leq E\max_{0\leq t\leq 1}|B_t|^p\), where \(p>0\), \(n\) is fixed, \(\varepsilon_1, \dots, \varepsilon_n\) are independent Rademacher random variables, \(B_t\) is standard Brownian motion, and \(a^2_1+\cdots +a^2_n=1\). It is now shown that the conjecture fails for some \(p\) (including \(p=1)\), but it is true when \(p\geq 3\) and \(a_1= \cdots =a_n\). Some inequalities which are less precise but more general than the one above are established, using ideas of G. A. Hunt [Proc. Am. Math. Soc. 6, 506-510 (1955; Zbl 0064.13004)].
For the entire collection see [Zbl 0948.00040].

MSC:

60E15 Inequalities; stochastic orderings
60J65 Brownian motion
60E05 Probability distributions: general theory
60G50 Sums of independent random variables; random walks
60G15 Gaussian processes
60G51 Processes with independent increments; Lévy processes
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