4 problems

Let $X$ be a centred, non-degenerate random vector, and let $R(t)={\mathbb E}[\cos(t^\top X)]$ and $M(t)={\mathbb E}[\exp(t^\top X)]$ denote its real characteristic-function part a…

Let $U_{\mathcal{S}}$ be a binary sequence generated by an ergodic stationary source $\mathcal{S}$ with finite memory, and let $H_{\mathcal{S}}$ denote the per-bit entropy generate…

Let $\hat{C}_n$ and $\hat{r}_n$ be the statistics discussed above, and consider tests of independence that are consistent against all alternatives. A statistic has the stated extre…

Conjecture on real-life data compressors. The same results can be proven for some particular real-life data compressors, for example those based on the measure $R$ or on the LZ alg…