1

Produce a sequence of random variables $\{X_n \}_{n≥0}$ as follows: Let $X_0 = q$ with probability 1, where $q \in (0, 1)$ is some constant. For $n ≥ 1$, let $X_n = X^2_{n−1}$ with probability 1/2 and $X_n = 2X_{n−1} − X^2_{n−1}$ with probability 1/2. Prove that $ \{X_n \}_{n≥0}$ converges almost surely, and find the distribution of the limit random variable.

The convergence result can be proved by Martingale Convergence Theorem. Besides, by the law of total expectation, $\mathbb{E}[X_n]=\mathbb{E}[\mathbb{E}[X_n|X_{n-1}]] = \mathbb{E}[X_{n-1}]=\mathbb{E}[X_0]=q$. By the dominated convergence theorem, $\mathbb{E}[X]=\lim_{n\to \infty} \mathbb{E}[X_n]=q$. Since $0\leq X \leq 1$, $\mathbb{E}[X^k] \leq \mathbb{E}[X]=q$. If we can prove that $\mathbb{E}[X^k]\geq q$, then $\mathbb{E}[X^k]=q$ for any $k\geq 1$, and the distribution of $X$ is determined to be Bernoulli distribution with parameter $q$.

I encounter difficulties when trying to show $\mathbb{E}[X^k]\geq q$.

zhaofeng-shu33
  • 418
  • 1
    Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Kavi Rama Murthy Jun 08 '21 at 07:53
  • 1
    By numerical experiment I find the limit is either 0 or 1. The expectation of $X_n$ is not changed. So the limit should be a Bernoulli distribution with parameter $p$. But I do not know how to prove it. – zhaofeng-shu33 Jun 08 '21 at 07:53

0 Answers0