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Thank you. I get it better now. However, the oxidation state of Ba in both compounds is still +2. So there has been no change in the oxidation state in my opinion.35.
Barium is reduced check it either by o.s or just by the fact that there is loss of oxygen from BaNO3 to form BaO. (Removal of oxygen is reduction)
As the fuel is 'lit up' it means heat is provided and when heat is given to nitrates, they decompose to metal oxide, NO2 and O2....
Mg(NO3)2 cannot be formed due to the same reason: The nitrate decomposes due to heat so there cannot be a displacement reaction of Mg to form magnesium nitrate...