What you have to verify is your model, not your time series (though sample time series from your model can be used for this).

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For example, you are assuming that your data is described by a first-order Markov chain, i.e., the probability of observing a given state only depends on the past state or, with other words, the process has a memory of 1. You can substantiate this assumption by showing that the process does not have a longer memory. 

As an example, consider a Markov chain with only two states, called A and B. Let’s say that you have counted the subchains of length 3 with A in the middle and got:

- A→A→A: 723 times
- A→A→B: 387 times
- B→A→A: 233 times
- B→A→B: 130 times

If the Markov chain is of order 1, the proportions of subchains ending with A should not depend on whether the chain started with A or B. This is a general statistical task, which is often represented in form of a [contingency table](http://en.wikipedia.org/wiki/Contingency_table). The corresponding contigency table for this case would be:

	               prior state
	                 A    B   total
	––––––––––––––––––––––––––––––––
	next state A |  723  233   956
	next state B |  387  130   517
	total        | 1110  363  1473

This can, e.g., be tested with Fisher’s exact test, which for this data should yield that the next state and the prior state are independent and thus the state A in our example Markov chain does not preserve any memory of the preceding state (state B hower might do this, so we have to test it as well).

As you have six states, you would have to perform six such tests with 6×6 contingency tables. Be aware that this is [multiple testing](http://en.wikipedia.org/wiki/Multiple_testing). Also, in theory your data can pass this test but have an even longer memory, but given your data, I consider this unlikely.

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Another way would be comparing some statistical properties of your original data with your simulated ones (which should be the same), for example the distribution of state durations or waiting times between two instances of a state. Should these differ, there is something wrong about your model. (However, your model can still be wrong if the distributions comply.)