In my research I need an upper bound for chi-square divergence in terms KL divergence which works for general alphabets. To make this precise, note that for two probability measures $P$ and $Q$ defined over a general alphabet $\mathcal{X}$, if $P\ll Q$, then $$\chi ^2(P||Q):=\int_{\mathcal{X}}\Big(\frac{dP}{dQ}\Big)^2dQ$$ and $$D(P||Q):=\int_{\mathcal{X}}dP\log\frac{dP}{dQ}.$$I am looking for an upper bound of $\chi^2(P||Q)$ in terms of $D(P||Q)$ which works wven if $\mathcal{X}$ is uncountable. What I need is a special case where $P=P_{XY}$ and $Q=P_X\times P_Y$, for two random variables with joint and product distributions are $P_{XY}$ and $P_X\times P_Y$, respectively. Noticing that in this case KL divergence is equal to the mutual information , I need an upper bound of chi-square divergence in terms of mutual information. 
An upper bound for chi-square divergence in terms of KL divergence for general alphabets
pr.probability;it.information theory;st.statistics