Limiting distributions of two random sequences
Let p (0\(\leq p\leq 1)\) be fixed. We define a sequence of random intervals \(\{[L_ n,R_ n]\), \(n\geq 0\}\) and a sequence of random variables \(\{X_ n\), \(n\geq 1\}\), inductively, as follows: Firstly, let \([L_ 0,R_ 0]=[0,1]\). Secondly, let \([L_{n-1},R_{n-1}]\) be defined. Let \(X_ n\) be a uniform random variable over \([L_{n-1},R_{n-1}]\) and put with probability p \[ [L_ n,R_ n]= \begin{cases} [L_{n-1},X_ n]&\;if\;X_ n\geq frac{1}{2}(L_{n-1}+R_{n-1}), \\ [X_ n,R_{n-1}]&\;if\;X_ n<frac{1}{2}(L_{n-1}+R_{n-1}), \end{cases} \] and with probability \(1-p\) \[ [L_ n,R_ n]= \begin{cases} [L_{n-1},X_ n]&\;if\;X_ n<frac{1}{2}(L_{n-1}+R_{n-1}), \\ [X_ n,R_{n-1}]&\;if\;X_ n\geq frac{1}{2}(L_{n-1}+R_{n- 1}).\end{cases} \] It is easy to see that \(X_ n\) converges to a random variable \(Y_ p\) (say) almost surely as \(n\to \infty\). In this paper, the author considered the distribution of \(Y_ 1\) or \(Y_ 0\) and showed that the distribution of \(Y_ 1\) is the beta (2,2) distribution.
scientific article; zbMATH DE number 3854188