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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?

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    $\begingroup$ Hint: the logarithm was invented as a way to convert multiplication into addition. $\endgroup$
    – whuber
    May 17, 2019 at 16:10
  • $\begingroup$ 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. $\endgroup$ May 17, 2019 at 18:45
  • $\begingroup$ @whuber ok thx i will look into that. $\endgroup$
    – TheDoctor
    May 18, 2019 at 0:26
  • $\begingroup$ The question is much more interesting when "$1/2$" is replaced by "$1/n$" :-). $\endgroup$
    – whuber
    May 18, 2019 at 1:18
  • $\begingroup$ @whuber yeah sorry you were correct. There was a mistake. Unfortunately I overlooked it. It is 1/n not 1/2. $\endgroup$
    – TheDoctor
    May 18, 2019 at 10:43

1 Answer 1

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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: $$ lim_{n→\infty} \frac{1}{n}\sum_{k=1}^{n} \log_{2}(X_k) \Rightarrow \sum_{x\in X} \log_{2}(x)v(x) $$

$$ \log_{2}(1)*\frac{1}{6} + \log_{2}(4)*\frac{1}{4} + \log_{2}(8)*\frac{1}{3} +\log_{2}(16)*\frac{1}{4} = \color{blue}{\frac{5}{2}} $$

Applying Inverse function:

$$ 2^{\color{blue}{\frac{5}{2}}} = \color{red}{5.659} $$

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