A COMMENTARY ON THE CORRELATED RANDOM WALK: SOME ADDITIONAL RESULTS AND A TWO-STEP CORRELATION

The generally accepted definition of the correlated random walk in one dimension is stated; in which the probabilities of taking a unit step in the same positive or negative direction as in the previous step are given by p(,1) and p(,2), respectively. The probabilities of taking a step in the opposite direction as in the previous step are given by q(,1) and q(,2) for reversals from a previous right or left step, respectively. The relatively brief history of the correlated random walk is highlighted in citing the major contributions of various authors.

By defining a state space of ordered pairs associated with the adjacent steps of the correlated random walk, a new method is presented for obtaining the occupation probability. Observations are then made concerning the variableness of the coefficient of correlation between successive steps in the non-symmetric walk. Also derived are the initial conditions which prevent this variableness, along with the intuitively expected form of the limiting coefficient as the number of steps increases towards infinity.

Given next is an alternate approach for finding the expected duration of play in terms of the expected durations of play conditional upon the point of absorption. The resulting unconditional expected duration of play is then demonstrated to be consistent with that obtained by using the more conventional expectations conditional upon the direction of the first step only.

The final chapter considers a unique extension of the correlated gambler's ruin problem to a problem involving a two-step correlation. Here, the probabilities of winning or losing a given game are dependent upon the outcomes of the previous two games. Conditional and unconditional probabilities of ultimate ruin for this two-step correlated problem are derived.