# Stochastic Sequence; Compute $\lim_{n\to\infty}$

Let $$(X_n)n≥1$$ be a stochastic sequence of i.i.d. random variables, each $$X_n$$ with values in the set {1, 4, 8, 16} and probability distribution:

$$P[X_n = 1] = 1/6, P[X_n = 4] = 1/4, P[X_n = 8] = 1/3, P[X_n = 16] = 1/4$$.

Compute $$lim_{n→∞}(X_1 ·X_2 · · · X_n)^{\frac{1}{n}}$$. [ Hint: Transform the expression whose limit is to be computed such that the ergodic theorem or the law of large numbers can be applied ]

How can i compute $$lim_{n→∞}(X_1 ·X_2 · · · X_n)^{\frac{1}{n}}$$?

Or how can i transform the expression so that the ergodic theorem or the law of large numbers can be applied?

• Hint: the logarithm was invented as a way to convert multiplication into addition.
– whuber
May 17, 2019 at 16:10
• What does $\lim_{n\to\infty}(X_1X_2\ldots X_n)^{\frac{1}{2}}$ mean in this context? Are you quoting the question correctly? If this is some sort of homework, please add the self-study tag and read the tag wiki. May 17, 2019 at 18:45
• @whuber ok thx i will look into that. May 18, 2019 at 0:26
• The question is much more interesting when "$1/2$" is replaced by "$1/n$" :-).
– whuber
May 18, 2019 at 1:18
• @whuber yeah sorry you were correct. There was a mistake. Unfortunately I overlooked it. It is 1/n not 1/2. May 18, 2019 at 10:43

Using $$\log_{2}$$:
$$lim_{n→\infty}\log_{2}\Big((X_1 ·X_2 · · · X_n)\Big)^{\frac{1}{n}} \Rightarrow lim_{n→\infty} \frac{1}{n} \sum_{k=1}^{n} \log_{2}(X_k)$$
Using Ergodic Theorem: $$lim_{n→\infty} \frac{1}{n}\sum_{k=1}^{n} \log_{2}(X_k) \Rightarrow \sum_{x\in X} \log_{2}(x)v(x)$$
$$\log_{2}(1)*\frac{1}{6} + \log_{2}(4)*\frac{1}{4} + \log_{2}(8)*\frac{1}{3} +\log_{2}(16)*\frac{1}{4} = \color{blue}{\frac{5}{2}}$$
$$2^{\color{blue}{\frac{5}{2}}} = \color{red}{5.659}$$