Hölder-type inequalities and their applications to concentration and correlation bounds

Let Yv,v∈V, be real-valued random variables having a dependency graph G=(V,E). We show that E∏v∈VYv≤∏v∈VEYvχbbbχb,where χb is the b-fold chromatic number of G. This inequality may be seen as a dependency-graph analogue of a generalised Hölder inequality, due to Helmut Finner. Additionally, we provide applications of the aforementioned Hölder-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables whose dependencies can be described in terms of graphs or hypergraphs.