Sum of random decreasing numbers between 0 and 1: does it converge??

Let’s define a sequence of numbers between 0 and 1. The first term, $r_1$ will be chosen uniformly randomly from $(0, 1)$, but now we iterate this process choosing $r_2$ from $(0, r_1)$, and so on, so $r_3\in(0, r_2)$, $r_4\in(0, r_3)$… The set of all possible sequences generated this way contains the sequence of the reciprocals of all natural numbers, which sum diverges; but it also contains all geometric sequences in which all terms are less than 1, and they all have convergent sums. The question is: does $\sum_{n=1}^{\infty} r_n$ converge in general? (I think this is called almost sure convergence?) If so, what is the distribution of the limits of all convergent series from this family?

Answer

Let $(u_i)$ be a sequence of i.i.d. uniform(0,1) random variables. Then the sum you are interested in can be expressed as
$$S_n=u_1+u_1u_2+u_1u_2u_3+\cdots +u_1u_2u_3\cdots u_n.$$
The sequence $(S_n)$ is non-decreasing and certainly converges, possibly to $+\infty$.

On the other hand, taking expectations gives
$$E(S_n)={1\over 2}+{1\over 2^2}+{1\over 2^3}+\cdots +{1\over 2^n},$$
so $\lim_n E(S_n)=1.$ Now by Fatou’s lemma,
$$E(S_\infty)\leq \liminf_n E(S_n)=1,$$
so that $S_\infty$ has finite expectation and so is finite almost surely.

Attribution
Source : Link , Question Author : Carlos Toscano-Ochoa , Answer Author : Community

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