A Note on the Structure of a Stochastic Model Considered by V. M. Dandekar

Abstract

Mr. Dandekar starts with the stochastic process x₀, x₁ x₂,…, ad inf. which may be characterised by the following defining postulates:

• π₁: Each x _i can take only the two values 0 (= failure) and 1 (= success) and the probability that x₀ = 1 is p.

• π₂: For any m (or less) consecutive x _i ’s at most one can be 1.

• π₃: If any m−1 (or more) consecutive x _i ’s are known to be zeros then the next x _i is 1 with probability p.

• π₃: If x₀ = 0 then the conditional stochastic process x₁, x₂,… is the same as the original process x₀, x₁, x₂,….

Let P _n = P(x _n = 1), n = 0, 1, 2,… (P₀ = p, q = 1−p) and let φ(t) be the generating function $\sum\limits_0^\infty {P_n t^n } .$

Sankhya 15: 251–252.