An extension of the Kahane-Khinchine inequality in a Banach space
Let \(\omega_ 1,\omega_ 2,..\). be the Steinhaus variables (independent identically distributed random variables, uniformly distributed on [0,1]) and X a Banach space. Then there is a constant \(c>0\) such that \[ \exp {\mathcal E} \log \| \sum^{N}_{j=1}e^{2\pi i\omega_ j}x_ j\| \geq c({\mathcal E}\| \sum^{N}_{j=1}e^{2\pi i\omega_ j}x_ j\|)^{1/2} \] for any N and \(x_ 1,...,x_ N\) in X, where \({\mathcal E}\) denotes the expected value.
independent identically distributed random variables
scientific article; zbMATH DE number 4067598