# Showing $\frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^q \to E((X-EX)^q)$ in probability [closed]

I am trying to show for $X_n$ iid st. $E|X|^q < \infty$ that

$$\frac{1}{n} \sum^n (X_i - \bar{X})^q \to E((X-EX)^q)$$ in probability.

I used the following approach: We note

• $\frac{1}{n} \sum_{i=1}^n X_i \to EX$ in prob. by the WLLN, i.e., $\bar{X}_n \to EX$ in probability
• $Y_i := (X_i - EX)^q$ are iid

i.e. we use Slutsky on $\frac{1}{n} \sum^n (X_i - \bar{X})^q$ to obtain iid random variables to then use WLLN as per below:

$$\frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^q \quad \xrightarrow{\text{Slutsky}} \quad \frac{1}{n} \sum^n (X_i - EX)^q \quad\xrightarrow{\text{WLLN}} \quad E(X-EX)^q \>.$$

Now, I realised that, of course, once I applied Slutsky I have taken $n \to \infty$ so I cant really take the last step via the WLLN. Can somebody help me fix this proof ?

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(+1) Welcome to the site, PeeJay. If this is homework, please kindly add the homework tag. Like on math.SE, we give hints for homework and (usually) full solutions otherwise. If it's not homework, just say so. Thank you very much for showing your work and thought process so far! Cheers. :) – cardinal Jan 31 '12 at 1:05
nope, it s not homework. thanks for the welcome ! – Beltrame Jan 31 '12 at 1:19
Here are some things to think about: (1) What does Slutsky's theorem say? In particular, think about if your first "convergence" statement appealing to Slutsky makes sense. (2) Is $q$ an integer? If not, then do you mean to have $|X_i - \bar X|^q$ and $|X - \mathbb E X|^q$ throughout? (3) What additional requirements on $q$ might you need? (4) Can you reduce it easily to a slightly simpler case which should, at the very least, dispense with some extra notation? When you've had a chance to think about this, you might consider making corresponding edits to the question. – cardinal Jan 31 '12 at 4:10
Please don't crosspost: math.stackexchange.com/questions/104003/… – cardinal Jan 31 '12 at 4:21
Closed because a duplicate on maths mentioned by @cardianl got answered. – mbq Jan 31 '12 at 8:27