Trend 2.9 of the Beryllium Group (Group IIA) Trends says that the thermal stabilities of the nitrates, carbonates, and peroxides increase down the column. (compare this trend with Trend 1.5 of the Lithium Group (Group IA) Trends).
The thermal stabilities of the these large anion compounds (nitrates, carbonates, sulfates, peroxides, and superoxides) increases down the column because the size of the cation increases down the column. The cations fill the interstitial holes of these anion lattices. Since these large anions have large interstitial holes, larger cations fill the lattice better. Therefore cations farther down the group form better lattices: compounds with greater stability.
Another way to look at this trend is to look at the stabilities of the products of all of these thermal decompositions, which in this case is the metal oxide (M2O).
MgCO3 ----> MgO + CO2
BaCO3 ----> BaO + CO2
Since the metal oxides for the smaller elements such as MgO are relatively stable, the decomposition occurs fairly readily, making the carbonates, nitrates and so on less thermally stable. As you go down the group the metal oxide BaO is unstable because of the bigger cation does not fit in the lattice. This makes the carbonates, nitrates, etc. less likely to decompose to the oxide and therefore more thermally stable.