1. (necessary condition) Each state with at least one unknown exiting transi-
tion probability must be visited infinitely often with probability 1 as either
T or N or both go to infinity.
2. (sufficient condition) Infinitely many of the individuals who have reached
each state with at least one unknown exiting transition probability must
be observed for at least one time period in that state.
In the following we will enunciate some results about the conditions on P
under which both the necessary and sufficient conditions are fulfilled. Let us
first state the condition when T —> ∞ with N fixed.
Proposition 5 Assume that P is such that each state with at least one unknown
exiting transition probability is persistent. Then T —> ∞ ensures the fulfillment
of condition 1 for any N > l.11 Condition 2 is obviously fulfilled for T —> ∞.
The proof of the first part is in Billingsley (1986), Theorem 8.2. Notice that
each state of an irreducible Markov chain with finite state space is persistent
(Billingsley, 1986, Example 8.7). Therefore in our case assuming that the chain
is irreducible ensures that it is persistent, which in turn ensures that ML esti-
mates show the usual asymptotic properties when T —> ∞ for any N > 1, finite
or infinite.
A similar result can be obtained when N —> ∞, but it depends on the initial
conditions of the process. These are defined by a vector of initial probabilities
P= ∣∣P⅛∣∣, representing, Vh ∈ S, the probability of being in state h in T = 0. Of
course, Ph > 0 Vh ∈ S, and ɪ) Ph = 1.
ħεs
Proposition 6 Assume that P and p are such that Phk P^k > θ for eac^1 state
к & S with at least one unknown exiting transition probability, for at least one
h⅛ ∈ 5 and for some finite integer r⅛. Then, TV —> ∞ ensures the fulfill-
ment of condition 1 for any T > max{¾, к ∈ 5}, where 7⅛ is, for each k,
the minimum r⅛ such that Phk Phk k > θ∙ Moreover, condition 2 is fulfilled if
T > max {rik, ⅛ ∈5} + l = T*.
The condition on p and P ensures that there is at least one initial state and
one route which allows each state with at least one unknown exiting transition
probability to be reached in a finite number of steps. Then, if T is large enough,
and N —> ∞, each state with at least one unknown exiting transition probability
will be reached infinitely many times. One additional time period is needed to
estimate the transition probabilities. Notice that if Ph ≠ 0 Vh ∈ S, and the
chain is irreducible, then the conditions on p and P are met, and moreover
T* = 2, so that the ML estimates show the usual asymptotic properties when
N —> ∞ for T > 2, finite or infinite.
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11For definitions of irreducibility and persistence, see Billingsley (1986), Section 8.
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