I am reading this paper. On the paragraph below Theorem 2.2, it says
However, without the normality assumption, Pearson coefficient, ρ, may be problematic. Indeed, as shown in Frechet (1957), ρ may not be bounded by 1 in absolute value and the bounds differ for different distributions
I find difficulty to understand this statement. I am unable to read the Frechet's paper because it is not in English.
Frechet, M. (1957). Les tableaux de correlations dont les marges sont donnees, Annales de l’Universite de Lyon, Sciences Mathematiques at Astronomie, Serie A 4, 13-31.
I thought the Crachy Schawarz inequality would guarantee that Pearson Coefficient lies in [-1,1]. Why does the Pearson Coefficient may not bounded by 1?
There is a bit of ambiguous terminology in that paper. The Pearson coefficient is still guaranteed to be in $[-1, 1]$, but what the authors mean is that the bounds could be tighter for certain distributions. The example here is that the correlation between $A = e^X$ and $B = e^Y$ is bounded within $[-1/3, 1]$, where $X, Y \sim N(0,1)$.