Geometric Moment Contraction for Stochastic Nesterov Acceleration

We study geometric moment contraction (GMC) of the constant-parameter stochastic Nesterov recursion [ Yk=Thetak+β(Thetak-Thetak-1),qquad Thetak+1=Yk-γ G(Yk,Xk+1). ] Under mean strong monotonicity and stochastic Lp Lipschitz continuity, an explicit Perron comparison proves synchronous Lp contraction when βγ Lp<(1-β)(1-qγ,p). This direct criterion includes infinite-variance gradients for 1<p<2, but its small-step regime requires β<μ/(μ+Lp). A complementary power-Lyapunov argument establishes a positive, generally much smaller, step-size interval for every fixed β<1 and every p>1, using only a finite pth gradient moment. At p=2, a simpler explicit certificate gives [ 0<γ<frac2μ(1-β)2L22(1-β+2β2). ] 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.