Statistical modeling multidimensional discrete system I have a system which its state is described by a vector $v=(a, b, c)$, where $a$, $b$ and $c$ can take any value  between $0$ and $100$ and where $a+b+c <= 100$.
I have observations of the state of the system for about 10 years (one per year, so 10 observations).
How can I model the states of the system?
Can I accurately model the state of the system using Markov chain? Or shall I use another technique?
The first approach that I used was to cluster the possible states of the system  (for example a cluster contains all the observations where: $a<33$ & $b<33$ & $c<33$).
Then I fitted a transition matrix and made the predictions (considering a homogeneous transition matrix) 
This works fine but the predictions are of course not accurate because of the clustering. 
 A: If $(A+B+C)<=100$ all three are non-negative, then you can pretend there's fourth non-negative variable $D$ whose value you don't include in the output, so that $(A+B+C+D)=100$. 
Next, we can divide by 100 to rescale, so now $(a+b+c+d) = 1$, where $a = A/100;b=B/100$ etc.
This gives us something that looks a lot like a Dirichlet distribution with $K=4$. 
Now that you can definitely fit; you can throw it into a Gibbs sampler or some variational approach at very least. 
If you find a stationary distribution, all that's left is to remember to transform the 'lowercase' probabilities back into 'uppercase' values of the state by multiplying them by 100 again.

Re: comment:
Bayesian updates are asymmetric by design; if you're conditioning on time, it's time-asymmetric.
For a time-homogenous chain, by simple application of Bayes:
$$p(V_{T+dT}|V_T) = p(V_T|V_{T+dT})*p(V_{T+dT})/p(V_T)$$ 
where $V_T$ is your pick of $(a,b,c,d)$ at a point in time $T$, with $dT$ as a delay. 
Until my time machine is fixed, past is independent of the present, so $p(V_t|V_{T+dT}) = 1$. This leaves us with the problem of finding the ratio $p(V_{T+dT})/p(V_T)$, which corresponds to a transition matrix for some time skip between two states.
For $T_0$, you'd use your best guess, e.g. $0.50$ for two variables and $0.25$ for four if you have no good reason to favor the odds for any one of them. Then find a transition matrix $M \sim Dir$ , satisfying $V_{T_1} = V_{T_0} * M_V$, plug it into $V_{T_2} = V_{T_1} * M_V = V_{T_0} * {M_V}^2$, etc.
You'll want to use whatever time step you can get based on your data. E.g. if you have daily aggregate data, this will give you a change over 1 day; however, as per the $T_2$ case above, you can trivially deal with missing datapoints.
The same procedure works for higher-order chains, including chains with conditional dependencies between pairs of nodes, but I'm not going to write it all up here for now.
