Problem 697

Randomly Decaying Sequence

Given a fixed real number $c$ , define a random sequence $(X_{n})_{n \geq 0}$ by the following random process:

$X_{0} = c$ (with probability 1).
For $n > 0$ , $X_{n} = U_{n} X_{n - 1}$ where $U_{n}$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U_{m})_{m < n}$ .

If we desire there to be precisely a $25$ probability that $X_{100} < 1$ , then this can be arranged by fixing $c$ such that $\log_{10} c \approx 46.27$ .

Suppose now that $c$ is set to a different value, so that there is precisely a $25$ probability that $X_{10 000 000} < 1$ .

Find $\log_{10} c$ and give your answer rounded to two places after the decimal point.

随机递减序列

给定实数 $c$ ，如下的随机过程可以生成一个随机序列 $(X_{n})_{n \geq 0}$ ：

$X_{0} = c$ （概率为1）。
对于所有 $n > 0$ ， $X_{n} = U_{n} X_{n - 1}$ ，其中 $U_{n}$ 是在 $0$ 到 $1$ 之间均匀随机选择的实数，且与之前所有的 $(U_{m})_{m < n}$ 独立。

如果我们希望 $X_{100} < 1$ 的概率恰好为 $25$ ，我们只需选择满足 $\log_{10} c \approx 46.27$ 的实数 $c$ 。

假设现在我们希望 $X_{10 000 000} < 1$ 的概率恰好为 $25$ ，我们需要给 $c$ 选择一个不同的数值。

求 $\log_{10} c$ 并将你的答案保留小数点后两位。