Step 1: Apply Fajan's rule.
According to Fajan's rule, covalent character increases with increase in polarization of the anion by the cation.
Polarization depends on size and charge of ions.
Step 2: Analyze cation.
In all compounds, the cation is same \( Ca^{2+} \).
Thus, polarization depends only on the anion.
Step 3: Compare halide ions.
Size of halide ions increases down the group:
\[
F^- < Cl^- < Br^- < I^-
\]
Larger ions are more polarizable.
Step 4: Effect on covalent character.
Greater polarizability leads to higher covalent character.
Thus, covalent character increases in the order:
\[
F^- < Cl^- < Br^- < I^-
\]
Step 5: Write order of compounds.
\[
CaF_2 < CaCl_2 < CaBr_2 < CaI_2
\]
Step 6: Conclusion.
Thus, increasing order of covalent character is:
\[
\boxed{CaF_2 < CaCl_2 < CaBr_2 < CaI_2}
\]