<p>This paper studies a specific class of statistical divergences for spectral densities of time series: the spectral <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Rényi divergences, which include the Itakura–Saito divergence as a limiting case. The aim of this paper is to highlight both information-theoretic and statistical properties of spectral <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Rényi divergences. We reveal the connection between the spectral <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Rényi divergence and the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-divergence in robust statistics, and a variational representation of the spectral <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Rényi divergence. Inspired by these results suggesting “robustness" of spectral <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Rényi divergence, we show that the minimum spectral Rényi divergence estimate has a stable optimization path with respect to outliers in the frequency domain, unlike the minimum Itakura–Saito divergence estimator, and thus it delivers more stable estimates, reducing the need for intricate pre-processing.</p>