<p>The comments have the answer. Let <span class="math-container">$\{X_n\}_{n=1}^{\infty}$</span> be a sequence of iid random variables with <span class="math-container">$E(X_1)=\mu$</span> and <span class="math-container">$Var(X_1)=\sigma^2$</span> with <span class="math-container">$\mu,\sigma^2&lt;\infty$</span>. Define <span class="math-container">$Y_n=\frac{1}{n}\sum_{i=1}^n X_i.$</span> By WLLN, <span class="math-container">$\text{plim}Y_n=\mu.$</span> Since <span class="math-container">$f(x)=x^3$</span> is a continuous function, by the continuous mapping theorem, <span class="math-container">$\text{plim}f(Y_n)=\text{plim}Y_n^3=f(\mu)=\mu^3.$</span></p>

CC BY-SA 4.0
2019-02-26T02:11:36.020Z
2019-02-26T05:10:23.730Z
2019-02-26T05:10:23.730Z
user21226
<p>From LLN, we have <span class="math-container">$\text{plim}  Y_n=\mu$</span>. Now you have <span class="math-container">$W_n=Y^3_n$</span>, what is the probability limit of <span class="math-container">$W_n$</span>?</p>

<p>Please help me answer this and thank you! </p>

CC BY-SA 4.0
2019-02-22T23:48:54.207Z
Take a look at the [continuous mapping theorem](https://en.wikipedia.org/wiki/Continuous_mapping_theorem).
Matthew Gunn
CC BY-SA 4.0
2019-02-23T08:35:19.200Z
If you want to only find the probability limit, you can replace "plim" with "lim". Proving it involves probability theory.
chan1142
<econometrics>
Simple Econometric question
Mohuya 
# Simple Econometric question

(Asked by: username_0 on 2019-02-21)

From LLN, we have $\text{plim} Y\_n=\mu$. Now you have $W\_n=Y^3\_n$, what is the probability limit of $W\_n$?

Please help me answer this and thank you!

**Tags:** econometrics

> **Comments:**
> **username_1**: Take a look at the [continuous mapping theorem](https://en.wikipedia.org/wiki/Continuous_mapping_theorem).  
> **username_2**: If you want to only find the probability limit, you can replace "plim" with "lim". Proving it involves probability theory.  

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## 1 answers

### Answer by username_3 on 2019-02-26. Score: 1

The comments have the answer. Let $\{X\_n\}\_{n=1}^{\infty}$ be a sequence of iid random variables with $E(X\_1)=\mu$ and $Var(X\_1)=\sigma^2$ with $\mu,\sigma^2<\infty$. Define $Y\_n=\frac{1}{n}\sum\_{i=1}^n X\_i.$ By WLLN, $\text{plim}Y\_n=\mu.$ Since $f(x)=x^3$ is a continuous function, by the continuous mapping theorem, $\text{plim}f(Y\_n)=\text{plim}Y\_n^3=f(\mu)=\mu^3.$


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