I have a discrete random variable, $X \in \{0,1,2,3\}$. Define the indicator function:
$$ 1_{k}\left(x\right) = \begin{cases} 1, & \text{if $x=k$} \\ 0, & \text{otherwise} \\ \end{cases}$$
where, $x$, is a realization of $X$. Consider the following random variables:
$$N_k = \sum_{i=1}^{N} 1_{k}\left(x_i\right)$$
where $N$ is the total number of trials; $x_i$, is the realization of $X$ from the $i$-th trial, and $N_k$ is the number of trials which resulted in a value of $x_i=k$. I then define:
$$J_k=\frac{N_k}{N}$$
In this case $J_k$ is the fraction of trials which resulted in $x_i=k$. I would like to find the sample standard deviation of $J_k$. How can I do this?
I assume that it ought to be a function of the sample standard deviation of $X$, but I am unsure how the standard deviation propagates from $X$ to $J_k$ since we have $J_k=J_k\left(X\right)$.
Also, note that $\sum_{k=0}^3 J_k=1$. Essentially $J_k$ is an estimate of $\mathbb{P}\left(X=k\right)$, and I want to know how to get error bounds for this estimate.
One other important piece of information that I failed to mention earlier, is that the $x_i$ are in fact arranged on a hexagonal lattice, and there are local correlations, so that $x_i$ is correlated with its nearest neighbors. These correlations die out quickly though, so I think a Markov property assumption would be valid.