Many statistical estimators are computed by stochastic optimization, making the statistical properties of the optimization output central to their use. For Adam-type updates, the current gradient enters both the momentum buffer and the second-moment state used to normalize it.  We establish convergence rates and asymptotic normality while retaining this dependence and deriving the required trajectory control. The general result uses a fixed first-moment decay rate $\beta_1\in[0,1)$, a second-moment decay rate $\beta_{2,t}=1-1/t$, and base stepsizes $\alpha_t=\alpha t^{-\gamma}$ with $1/2<\gamma<1$. Under global strong convexity and smoothness of the population risk and a uniform conditional sub-exponential bound on martingale-difference gradient noise, we derive a path wise envelope and unconditional cumulative and pointwise expected excess-risk bounds of orders $T^{1-\gamma}$ and $t^{-\gamma}$, respectively, up to logarithmic factors, without assuming bounded iterates or bounded stochastic gradients. With additional local Hessian and conditional covariance regularity, these estimates yield asymptotic linearity of the full Polyak--Ruppert (PR) average at the population minimizer, with the classical sandwich covariance in fixed dimension.  Independent-sample covariance estimation provides confidence intervals for fixed linear functionals.

 

A separate diagonal quadratic model with zero momentum and bounded i.i.d.\ noise variables shows why current-step normalization matters for centering. For $\beta_{2,t}=1-t^{-\kappa}$, $0<\kappa\le1$, we identify an explicit third-moment correction.  It is negligible at the root-$T$ scale for $\kappa>1/2$; when its coefficient is nonzero, it shifts the Gaussian center at $\kappa=1/2$ and determines a slower convergence scale below the threshold.  Lagged normalization has a centered limit under the sufficient condition $\gamma+\kappa>1$. Fixed second-moment decay rates are outside these results. Numerical experiments illustrate the convergence and distributional conclusions.