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Part c gives the integral representationThe integrand vanishes on the diagonal $y=x$. Thus the assumed off-diagonal Peskun ordering impliesfor every $f\in L^2(\pi )$.

On the mean-zero subspace, the variational characterization of the spectral gap of a positive reversible kernel isIt follows immediately thatEquivalently, the energy inequality says $K⪯Q$ in the Löwner order on $L_0^2(\pi )$. Positivity permits the operator monotonicity of the square root and hence $K^{1/2}⪯Q^{1/2}$; the spectral representations in the question identify the top spectral values and give the same gap inequality.

P. H. Peskun authored the 1973 paper underlying Peskun ordering, a comparison of the efficiency of Markov chain Monte Carlo algorithms with the same stationary target.^[ref]

For finite reversible Markov chains, increasing every off-diagonal transition probability while preserving the stationary law cannot increase the asymptotic variance of time averages. The comparison formalizes why spending less time repeating the current state can help statistical estimation, provided the ordering's hypotheses hold. It compares the variance of estimators, not merely their one-step acceptance rates.^[ref]