Higher order optimal approximation of Csiszarʹs image-divergence Original Research Article

Here are established various sharp and nearly optimal probabilistic inequalities giving the high order approximation of Csiszarʹs f-divergence between two probability measures, which is the most essential and general tool for their comparison. The above are done through Taylorʹs formula, generalized Taylor–Widderʹs formula, an alternative recent expansion formula. Based on these we give many representation formulae of Csiszarʹs distance, then we estimate in all directions their remainders by using either the norms approach or the modulus of continuity way. Most of the last probabilistic estimates are sharp or nearly sharp, attained by basic simple functions.