  # Take the Gamblers ruin on [0, N] with a fair coin, i.e., a simple random walk with absorbing boundaries at 0 and N. Let T be the time which elapses before the SRW is absorbed by one of the boundaries (i.e., the duration of the game. Show that P(T < ∞) = 1 and Epecation (T^k) < ∞ for all k.

Question

Take the Gamblers ruin on [0, N] with a fair coin, i.e., a simple random walk with absorbing boundaries at 0 and N. Let T be the time which elapses before the SRW is absorbed by one of the boundaries (i.e., the duration of the game. Show that P(T < ∞) = 1 and Epecation (T^k) < ∞ for all k.

check_circleExpert Solution
Step 1

Proof of P (T < ∞) = 1:

The probability P (T < ∞) = 1 implies that the duration of the game will be finite with probability 1. In other words, it means that the probability that the game goes on forever (that is, probability that the duration of the game will be infinite) is 0.

Suppose the gambler starts playing with initial amount i, (0 < i < N).

After the nth gamble (n ≥ 0), let the fortune of the gambler be Rn. Thus, R0 = i.

The game stops if Rn = 0 or Rn = N, which happens after time T.

Denote Qi = P (Gambler plays forever | R0 = i). Thus, P (T < ∞) = 1 – Qi.

The game is absorbed (that is, stops) when any of the boundaries, 0 (ruin) or N (desired win) is attained. For any amount of fortune lying between 0 and N, the game is continued.

Using these conditions, Qi can be modelled by the following recursive relation:

Step 2

Calculation:

The characteristic equation of the above linear homogeneous recurrence is solved as follows:

Step 3

Conclusion:

Thus, Qi = P (Gambler plays forever | R0 = i) = 0.

As a result, the probability that the duration of the game will be finite is:

P (T < ∞) = 1 &n...

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