Limiting distributions of two random sequences

AbstractFor fixed p (0 ≤ p ≤ 1), let {L0, R0} = {0, 1} and X1 be a uniform random variable over {L0, R0}. With probability p let {L1, R1} = {L0, X1} or = {X1, R0} according as X1 ≥ 12(L0 + R0) or < 12(L0 + R0); with probability 1 − p let {L1, R1} = {X1, R0} or = {L0, X1} according as X1 ≥ 12(L0 + R0) or < 12(L0 + R0), and let X2 be a uniform random variable over {L1, R1}. For n ≥ 2, with probability p let {Ln, Rn} = {Ln − 1, Xn} or = {Xn, Rn − 1} according as Xn ≥ 12(Ln − 1 + Rn − 1) or < 12(Ln − 1 + Rn − 1), with probability 1 − p let {Ln, Rn} = {Xn, Rn − 1} or = {Ln − 1, Xn} according as Xn ≥ 12(Ln − 1 + Rn − 1) or < 12(Ln − 1 + Rn − 1), and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n ≥ 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n → ∞. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 − y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability)