Stochastic Modeling

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By Klee V. (ed.)

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Example text

Im as above, then pil il+1 > 0. Now check the recurrence of P: if in the original chain pii = 0 then the return to state i occurs in both chains on the same event, hence the return probability to state i will be the same. If pii > 0 then in the new chain, the return probability is equal to 1 × Pi (return to i after time 1 in the original chain) 1 − pii 1 = (1 − pii ) 1 − pii which is 1. e. the solutions to both equations are the same. Hence, the minimal solution to hP = h with hi = 1 is the same as that to hP = h.

And ⎛ 0 1 ⎜1 − p 1 0 ⎜ (c) ⎜ 0 1 − p2 ⎝ .. . 0 p1 0 .. 0 0 p2 .. ⎞ ... 0 ... . 0 . ⎟ ⎟ . . 0 . ⎟ ⎠ . . .. . These models describe so-called birth-and-death processes, or birth-death processes, where state i represents the size of the population, and during a transition a member of the population may die or a new member may be born. In case (a) only births are allowed, and the chain is deterministic. Here, every state i forms a non-closed class and is non-essential. In model (b) a ‘death’ occurs with the same chance 1 − p and a birth with the same chance p, regardless of the size i of the population at the given time (unless i = 0 of course).

By the strong Markov property, Pi (Xn = i for at least two values of n ≥ 1) = fi2 , and more generally, for all k Pi (Xn = i for at least k values of n ≥ 1) = fik . 25) (i) Denote by Bk the event that Xn = i for at least k values of n ≥ 1. Then, obviously, (i) (i) (i) events Bk are decreasing with k: B1 ⊇ B2 ⊇ . , and the event that Xn = i for (i) infinitely many values of n is the intersection k≥1 Bk . 26) which equals 1 when fi = 1 and 0 when fi < 1. O the heavy change, . . Now thou art gone, and never must return!

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