Categorical white noise Categorical time series such the genomic DNA, the sleep state of a person $\cdots$ etc, are often treated and analyzed. The first question that came to my head was how can such type of series be a white noise? is there a definition for categorical white noise (I haven't found any definition).
 Edit: $X_{t}$, for $t = 0, \pm 1, \pm 2 ,\cdots$ , is a categorical-valued time series with finite state-space $\mathbb{C}= \{c_{1}, c_{2},\cdots, c_{k}\}$ and $p_{j} = pr (X_{t} = c_{j}) >0$

 As an example, here's the categorical time series that represent EEG sleep stat of an infant.
 A: Great question!
A fairly simple but rather imprecise method
First, you need a theoretical prior about the distribution of the variable you are interested on (e.g. infant EEG sleep states). The example you give is that of a categorical distribution, with $p_j=pr(X=c_j)>0$, and $\sum_{j} p_j=1$, with $j= 1, 2, ... , k$
Second, I would test whether the sample distribution follows the a-priori theoretical distribution you are given. A simple Chi-square goodness-of-fit test would suffice for it. If the test indicates that the sample data could have been drawn from such theoretical distribution with a given confidence level, then the categorical white noise is given by the observed discrepancy between the predicted and observed distribution, at that confidence level.
For instance, in a sample of 100 minutes, your prior is that you should observe 30 $ah$, 40 $qt$, and 30 $tr$. Yet, you observe 27, 41, and 32 respectively. If the two are identical at 5%, then the discrepancy is due to white noise.
On the contrary, if you find that the theoretical and sample distributions are different at your desire confidence level, then you might conclude that (i) your theoretical distribution is wrong, and/or (ii) there is some "non-white (non-random) noise".
Spectral analysis
There seems to be a very recent literature on spectral analysis for time series of categorical variables (e.g. here). The only way to evaluate these series seems to be by means of "indicators" variables. For instance, you assume
$$Y_t = \begin{cases} 1 &\mbox{if } X_t=c_k \\ 
0 & \mbox{if } X_t \neq c_k. \end{cases} $$ 
Applying this to your dataset gives a sequence like 1 0 1 0 0 1 0 1 0 1 1 0 0 1 ...
The idea is to analyze the frequency of all the possible cycles, with the length of the cycle being $k$. Then, a spectral analysis of these cycles will show you the strength of the signal at different frequencies. If these are fairly constant, there is indication of the series being white noise.  
Here for instance in the spectral analysis of a white noise:

Image source: Wikipedia
The paper that I refer to have more examples.
