# Alternative way of showing convergence of central moments

I woud like to show for a random variable $X$

$${1\over n} \sum_{i=1}^n (X_i-\bar{X}_n)^q\to E(X-EX)^q \quad (\text{convergence in probability)}$$

My approach was to define $Y_i:= (X_i-\bar{X}_n)$ and noting that by the Weak Law of Large numbers and as $Y_i$ are iid we have

$${\frac{1}{n}} \sum_{i=1}^n Y_i \to EY = X_i-EX\quad (\text{convergence in probability)}$$

so using Slutsky's Theorem for a continuous function (here the $q$th power) I concluded that the first line is in fact true.

Provided what I have said above is correct I would now like to take an alternative approach via Characteristic Functions ( i.e. $\phi(t) := E(e^{itX} )$ ) But unfortunately I m not sure what needs to be done.

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Convergence in law to a constant implies convergence in probability to this constant. –  Davide Giraudo Jan 25 '12 at 22:34
so I would try to show pointwise convergence of the respective characteristic functions and this would give me the required result via Levy's Continuity Theorem ? –  Beltrame Jan 25 '12 at 23:00