  We study the asymptotic behavior, uniform-in-time, of a nonlinear dynamical
system under the combined effects of fast periodic sampling with period
$\delta$ and small white noise of size $\varepsilon,\thinspace
0<\varepsilon,\delta \ll 1$. The dynamics depend on both the current and recent
measurements of the state, and as such it is not Markovian. Our main results
can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem
(CLT) type results. LLN type result shows that the resulting stochastic process
is close to an ordinary differential equation (ODE) uniformly in time as
$\varepsilon,\delta \searrow 0.$ Further, in regards to CLT, we provide
quantitative and uniform-in-time control of the fluctuations process. The
interaction of the small parameters provides an additional drift term in the
limiting fluctuations, which captures both the sampling and noise effects. As a
consequence, we obtain a first-order perturbation expansion of the stochastic
process along with time-independent estimates on the remainder. The zeroth- and
first-order terms in the expansion are given by an ODE and SDE, respectively.
Simulation studies that illustrate and supplement the theoretical results are
also provided.
