Geometric Moment Contraction for Stochastic Nesterov Acceleration

We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion $$Y_k={\mathrm{\Theta }}_k+\beta ({\mathrm{\Theta }}_k-{\mathrm{\Theta }}_{k-1}),{\mathrm{\Theta }}_{k+1}=Y_k-\gamma G(Y_k,X_{k+1})\mathrm{.}$$ Under mean strong monotonicity and stochastic $L^p$ Lipschitz continuity, an explicit Perron comparison proves synchronous $L^p$ contraction when $\beta \gamma L_p<(1-\beta )(1-q_{\gamma ,p})$. This direct criterion includes infinite-variance gradients for $1<p<2$, but its small-step regime requires $\beta <\mu \mathrm{/}(\mu +L_p)$. A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed $\beta <1$ and every $p>1$, using only a finite $p$th gradient moment. At $p=2$, a simpler explicit certificate gives $$0<\gamma <\frac{2\mu (1-\beta {)}^2}{L_2^2(1-\beta +2{\beta }^2)}\mathrm{.}$$ Its quadratic high-momentum scaling is a limitation of the chosen metric, not a sharp stability boundary. We quantify this loss, provide a general mean-only quadratic $S$-procedure, and exploit endpoint Lyapunov inequalities under stronger samplewise sector information. Verified endpoint certificates can be orders of magnitude less conservative than the explicit metric.