You could always use a rectangular distribution as an conventional way tondepitct random error (additive error).

The problem of defining the notion transition matrix does not exist, since it is a widely well understood element of the theory of Markov chains. Hence, probably, the question is about designing appropriate transition matrix to a given random process. And here we have two cases to consider - when the process is determined by some theoretical conditions on random variables, or when the process desribes real changes of real quantities somehow modelled by a theoretical random process (the mathematical stochastic structure).

Just for clarity, these are some typical examples:

For the first case: we have already a sequence of random variables N(t), t=0,1,2,..., say with Poisson distributions P(L(t)), where L(t), is an increasing sequence of positive real numbers. Q1: Is this a Markov process; Q2: if yes - what is its transition matrix?

For the second case: we have some data indicationg that the observed values n(t) of some process with natural values, observed at, say 10 places in a form 10 sequences:

n1(0), n1(1), n1(2), n1(3), ....

n2(0), n2(1), n2(2), n2(3), ...

. .. . . . .

n10(0), n10(1), n10(2), n10(3), .....

Q1: Can we assume that a Markov process is well fitting these data?

Q2: If yes - what is the suitable transition matrix?

= = = = = = = == = = = = =

Now, I am almost sure that the problem is related to the second case. Next advices assume this as a fact. Then the first step in solving problems with choosing the appropriate stochastic model in the form of the Markov process is to

1. establish/choose/determine the quantities or classes supposed to vary as such a process, called possible STATES (e.g. for the mentioned consumer brand, these could be names of a brand);

2. determine, what means the probability of a class (in the above branch - the portion of money spent by consumers during a trading week; if so, then the nr of the consecutive week is the "instant" of the Markov chain)

3. try to formulate (qualitatively at least), for own clarity of the subject, causes of changes of the states (e.g. of changes popularity of some brands into favoring another brand (usually these should imply a need to introduce some subsets of the states[=brands] since not many people will change preference e.g. for buying cars into trains:) )

This would be the first stage of developing the model; as a result one should get at least some graph of possible passages between states, initially - without concrete values of probabilities. Lack of an arrow from state B to state C would denote then, that we do not expect any transition/switching of the preference from brand B onto the preference of brand C. This is a very important stage of the construction, since we are winning then a great simplification of the sought transition matrix: lack of an arrow means 0 value at the entry of the indexed site - as in the example above named entry posed in the row "B" column "C".

Hopefully this will enough for first suggestion. Possibly this is nothing knew, then we shall go further after additional questions/comments/. . . .