I've been learning a bit about stochastic approximation via the ODE method and have gotten stuck on a point that arises when dealing with projected/truncated SA algorithms. To be concrete let me ask about Kushner and Yin Theorems 5.2.1 and 5.2.3. I'll take a stab at describing the context but there is so much notation involved in these results it's hopeless to state it all here. In Kushner and Yin's notation the truncated SA algorithm is $\theta_{m+1} = \Pi[\theta_m + \epsilon_m Y_m]$ in which $\Pi$ represents the the $L^2$ projection onto a closed constraint set. They define the projection terms $\epsilon_m Z_{m} = \Pi[\theta_{m} + \epsilon_m Y_m] - \theta_{m} -\epsilon_m Y_m$ and create a piecewise constant interpolation $$Z^0(t) = \sum_{m=0}^N \epsilon_m Z_m \text{ for }\sum_{m=0}^N \epsilon_m \leq t < \sum_{m=0}^{N+1} \epsilon_m$$ with centered translates $Z^n(t) = Z^0(t + \sum_{m=0}^{n-1} \epsilon_m) - Z^0(\sum_{m=0}^{n-1} \epsilon_m)$. One of the things to be proven is that the family of functions $\{ Z^n(t)\}$ is equicontinuous almost surely. They argue by contradiction and thereby assume the existence of a constant $\rho > 0$, a sequence of integers $\mu_k \to \infty$ and a sequence of real numbers $\delta_k \to 0$ with the property that $|Z^{\mu_k}(\delta_k)| \geq \rho$. At this point they start to get pretty heuristic and discuss properties of an "asymptotic jump" in terms of properties that are asserted about a single projection term almost surely (such as the fact that a projection term lives in the normal cone of a point on the boundary and points into the interior of the region we're projecting into or that the size of each individual projection term goes to zero). I can't even begin to see how to make precise mathematical sense of this for a variety of reasons but among them is the fact that $Z^{μ_k}(\delta_k)$ may contain multiple projection terms. In fact if the sequence $\delta_k$ goes to 0 much more slowly than the step size sequence $\epsilon_m$ it seems like the number of projection terms may grow as $k \to \infty$.
Can anyone help me understand the equicontinuity proof or provide a reference that is gentler. Note that I also find the corresponding treatment in Kushner and Clark to be pretty opaque; I'd be happy if someone who understands pg. 193-4 of Kushner and Clark can help me with that instead of the Kushner and Yin account described above.