Statement I :  Li₂CO₃ decomposes on heating to give CO₂ gas
Statement II : Carbonates of alkali metals give weakly alkaline solutions due to the hydrolysis of carbonate ion

I) 

stability of I_A group carbonates : 

Stability of carbonates increases
Reason: A larger size cation will stabilize the large size anion and hence Cs₂CO₃ is more stable and Li₂CO₃ is least stable
→ Li₂CO₃ → 2Li⁺ + CO₃^-2

II) Anion of strong acid  and cation of strong base can't undergo hydrolysis, but anion of W.A & cation of W.B can undergo hydrolysis  .CO₃^-2 is the anion of weak acid [H₂CO₃], therefore it can undergo hydrolysis.
 

During the hydrolysis of CO₃^-2  ion, OH^- ion releases into the solution
→ Therefore the solution exhibits alkaline nature