Why sample statistics converge to the right parameter We know that for sample/empirical distribution function $F_n(x)$ we have that 
a) $F_n(x)\xrightarrow[p]{}F(x)$ (pointwise convergence)
b) $\dfrac{\sqrt{n}(F_n(x)-F(x))}{\sqrt{F(x)(1-F(x))}}\xrightarrow[d]{}N(0,1)$
c) $F_n$ converges uniformly in probability to F.
My question is how do we prove that the sample moments of order $k$, and sample central moments of order $k$ converge to $E(X^k)$ and $E(X-E(X))^k$ respectively? (I think need to use the above empirical distribution function properties, but I do not know how, or which...)
Any help would be appreciated.
If you know how to explain that the sample statistics converge, without using any property of the empirical distribution, I would also be thankful. 
 A: I'm assuming that your sample consists of iid random variables $X_1,\dots, X_n$. The law of large numbers says that if a random variable $X$ has a finite expected value, then then the sample mean $\overline X=\frac1n\sum_{i=1}^n X_i$ converges to $E(X)$ in probability. Replacing $X$ by $X^k$, this implies that the sample moments of order $k$ converge in probability to $E(X^k)$, as long as $E(X^k)$ exists.
For the same reason, $\frac1n\sum_{i=1}^n (X_i-E(X))^k$ converges in probability to the central moment of order $k$ if it exists; however, to get convergence of the sample central moments, we need to prove a modification of this statement where $E(X)$ is replaced by $\overline X$. We can do this by expressing the $k$th sample central moment as a polynomial in terms of the $j$th sample moments, $j=1,\dots,k$ (e.g., see http://mathworld.wolfram.com/SampleCentralMoment.html), and then applying the convergence of sample moments.
If we have an infinite iid sequence $X_1,X_2,\dots$, and if we form a sequence of samples where the $n$th sample consists of the first $n$ variables in this sequence, $X_1,\dots,X_n$, then we can use the strong law of large numbers to strengthen the conclusions above: namely, we will get almost sure convergence (which is stronger than convergence in probability) of both the sample moments and sample central moments.
