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Let $(U_n: n\geq 1)$ be a sequence of independent random variable and each distributed uniformly on the interval $[0\;1]$. Let $X_0=0,$ and define $X_n$ for $n\geq 1$ by the following recursion: $X_n=\max\left\{X_{n-1}, \frac{X_{n-1}+U_n}{2}\right \}$

Does $\lim_{n \rightarrow \infty} X_n $ exists in the a.s. sense? Does $\lim_{n \rightarrow \infty} X_n$ exists in the m.s. sense ? If the limit exits identify the limiting random variable.

Suyog Kapsikar
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  • 'Answer the following' is not so welcome here. Can you show what you have done so far? Have you realized that $X_n$ is non-negative and increasing? – Kavi Rama Murthy Oct 02 '19 at 11:53

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