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Coefficient convolution and shape imbalance

Abstract

Strict positivity of the normalized incomplete beta Turanian coefficients.

Theorem 1.1 (Strict decrease of the convolution).

Lean statement: D5/S3/Analytic/IncompleteBeta/Result.convolution_strictAnti_imbalance

Proof. Machine-checked in Lean as D5/S3/Analytic/IncompleteBeta/Result.convolution_strictAnti_imbalance (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. D. B. Karp (2016). Normalized incomplete beta function: log-concavity in parameters and other properties. DOI: 10.1007/s10958-016-2958-z. URL: https://arxiv.org/abs/1509.05120v1.

Commentary.

For positive real a and m and every natural n, the sum over k from zero to n of c(k,a,m+d)c(n-k,a,m-d) is strictly decreasing as d ranges from zero, included, to m, excluded. Here c(j,a,b) is Gamma(a+b+j) divided by Gamma(b)Gamma(a+j+1). Thus the convolution decreases with the absolute difference of the two positive shape parameters when their sum is fixed.

Termwise beta integration expresses the convolution as Gamma(2a+2m+n)/Gamma(2m) times the normalized beta mean of W. The kernel W(u) is the sum over k from zero to n of u to the power a+k times (1-u) to the power a+n-k, divided by Gamma(a+k+1)Gamma(a+n-k+1). Reflection of the finite sum makes W symmetric. Differentiating each summand gives two adjacent terms whose sum telescopes. The derivative equals [u(1-u)] to the power a-1 times [(1-u) to the power n+1 minus u to the power n+1], divided by Gamma(a)Gamma(a+n+1). It is positive between zero and one half. All kernel powers have positive exponents, so W is continuous at both endpoints. The strict beta comparison therefore applies, with a common positive Gamma factor.

Theorem 1.2 (Positive coefficients for all positive shifts).

Lean statement: D5/S3/Analytic/IncompleteBeta/Result.result

Proof. Machine-checked in Lean as D5/S3/Analytic/IncompleteBeta/Result.result (✓ std3). ∎

Resolves. Problems/karp-2016-incomplete-beta-coefficient-log-concavity (proved) by D5/S3/Analytic/IncompleteBeta/Result.result.

Source. Repository-derived.

Acknowledgement. D. B. Karp (2016). Normalized incomplete beta function: log-concavity in parameters and other properties. DOI: 10.1007/s10958-016-2958-z. URL: https://arxiv.org/abs/1509.05120v1.

Commentary.

For every positive real a, b, alpha and beta and every natural n, the sum over k from zero to n of c(k,a,b+alpha)c(n-k,a,b+beta) minus c(k,a,b)c(n-k,a,b+alpha+beta) is strictly positive. These are the coefficients of the normalized incomplete beta Turanian; they are independent of the series variable.

The two convolutions have the common mean shape b+(alpha+beta)/2. Their nonnegative half differences are respectively the absolute value of alpha-beta divided by two and (alpha+beta)/2. Positivity of both shifts makes the first strictly smaller. Convolution symmetry handles either ordering of the shifts, and strict decrease with imbalance gives the asserted positive difference.

References