Which of the following statements is not correct?

Basic strength of alkaline earth metal hydroxides increases down the group because ionization energy of metal decreases and electropositive character increases down the group

Other alkali metal carbonates are thermally stable. When you move down the group, the alkaline earth metal sulphates show an increase in the electropositive character and the thermal stability. The alkali metal sulphates (with exception of lithium sulphate) are thermally stable

The carbonates of all alkaline earth metals decompose on heating to form the corresponding metal oxide and carbon dioxide.

We know that hydration energy and lattice energy are inversely proportional to the radii of the ions. Now, for sulphates and carbonates, the decrease in hydration energy is much more than the decrease in lattice energy and so, their solubility decreases down the group.
The solubility of the sulphates and carbonates follows the order:  

Be>Mg>Ca>Sr>Ba

The solubilities of sulphates and carbonates decrease down the group.