The stochastic limit $X$ in the mean square sense is given the definition: For a row (sequence?) of stochastic variables $X_n$ if $\displaystyle\lim_{n\to\infty}E\{(X_n-X)^2\}$ = 0 and we write $\text{l.i.m.} X_n = X$. This I understand as the limit of a sequence argument akin to usual analysis.
When we talk about the derivative of a stochastic process $X_t^\prime$, we define
$X_t' = \frac{dX_t}{dt} = \text{l.i.m} \frac{X_{t+h} - X_t}{h}$
where $X_t, t\geq 0$ is defined to be stochastic process. When I see the $\text{l.i.m}$ operator, I expect to see a row (sequence?) of stochastic processes. However, in the reference I have, only the index $t$ is written.
Shouldn't an $'n'$ be also written to depict that the derivative is the square mean limit of a row of stochastic processes? Or have I got the idea wrong that the limit is in the sequence index $n$ and the derivative is defined for the stochastic process index $t$? To me, the following seems the clearer definition of the derivative
$X_t' = \frac{dX_t}{dt} = \text{l.i.m} \frac{X_{n,t+h} - X_{n,t}}{h}$
which is sort of a pointwise argument for the index $t$ of the stochastic process. I guess my confusion is mainly from the notation used to express the two concepts where one index seems to be omitted for the clarity sake.
It is like the notation for limit of a sequence and that in continuous time. Specifically, the derivative is defined by a process $\{X_t', t \ge 0\}$ that satisfies \begin{align*} \lim_{h\rightarrow0}E\left(\left(\frac{X_{t+h}-X_t}{h} -X_t'\right)^2 \right) = 0. \end{align*}