On Rényi Entropy Power Inequalities

This paper gives improved Rényi entropy power inequalities (R-EPIs). Consider a sum S
_n
= Σ
_k=1
ⁿ
X
_k
of n independent continuous random vectors taking values on ℝ
^d
, and let α ∈ [1, ∞]. An R-EPI provides a lower bound on the order-α Rényi entropy power of S
_n
that, up to a multiplicative constant (which may depend in general on n, α, d), is equal to the sum of the order-α Rényi entropy powers of the n random vectors {X
_k
}
_k=1
ⁿ
. For α = 1, the R-EPI coincides with the wellknown entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov, which relies on the sharpened Young's inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix.