Splitting intervals (Q5903823)

Let X be a random variable (r.v.) with density k of bounded variation on [0,1]; let H be a r.v. taking values \(\pm 1\) with probability 1/2. Take independently two independent sequences of copies \(\{X_ n\}\) and \(\{H_ n\}\). Starting with \(M_ 0=0\), \(L_ 0=1\) define midpoints \(M_ n\) and lengths \(2L_ n\) recursively by \[ L_{n+1}=L_ n\cdot X_{n+1}\quad and\quad M_{n+1}=M_ n+H_{n+1}(L_ n-L_{n+1}); \] hence the new interval has one endpoint in common with the old interval while the intervals get progressively smaller. The author finds the general form of the limiting density of \(\{M_ n\}\). For the case where k is uniform on (1/2,1] see \textit{L. Devroye}, \textit{G. Letac} and \textit{V. Seshadri} [Stat. Probab. Lett. 4, 183-186 (1986; Zbl 0599.60030)].