Central limit theorem versus law of large numbers The central limit theorem states that the mean of i.i.d. variables, as $N$ goes to infinity, becomes normally distributed. 
This raises two questions:


*

*Can we deduce from this the law of large numbers? If the law of large numbers says that the mean of a sample of a random variable's values equals the true mean $\mu$ as $N$ goes to infinity, then it seems even stronger to say that (as the central limit says) that the value becomes $\mathcal N(\mu, \sigma)$ where $\sigma$ is the standard deviation. Is it fair then to say that central limit implies the law of large numbers?

*Does the central limit theorem apply to linear combination of variables?
 A: For law of large numbers, you need to have all variables to be defined on the same probability space (as the law of large numbers is a statement about probability of an event determined by $\bar X_n$, for all $n$ simultaneously). For convergence in distribution, you can have different probability spaces, and that simplifies many aspects of the proofs (e.g., increasing nested spaces, very common for various triangular array proofs). But it also means you cannot make any statements concerning the joint distributions of $\bar X_n$ and $\bar X_{n+1}$, say. So no, convergence in distribution does not imply the law of large numbers, unless you have a common probability space for all variables.
A: First, though there are many definitions, one of the standard forms of the central limit theorem says that $\sqrt{n}(\bar{X}_n-EX)$ converges in distribution to $\mathcal N(0, Var(X))$, where $\bar{X}$ is the sample mean of $n$ iid copies of some random variable $X$. 
Secondly, suppose we have two independent random variables $X$ and $Y$. Then 
$$\sqrt{n}(\frac{1}{n}\sum_{j=1}^n(aX_j+Y_j) - E(aX+Y)) \to \mathcal N(0, Var(aX+Y))$$
or
$$\sqrt{n}a(\bar{X}_n- EX)+\sqrt{n}(\bar{Y}_n -EY) \to \mathcal N(0, a^2Var(X)+Var(Y)).$$
In other words, a linear combination of random variables wont converge to a linear combination of normals under the CLT, just one normal. This makes sense because a linear combination of random variables is just a different random variable that CLT can be applied to directly. 
A: The OP says

The central limit theorem states that the mean of i.i.d. variables, as N goes to infinity, becomes normally distributed.

I will take this to mean that it is the OP's belief that for i.i.d. random
variables $X_i$ with mean $\mu$ and standard deviation $\sigma$, the 
cumulative distribution function $F_{Z_n}(a)$ of
$$Z_n = \frac{1}{n} \sum_{i=1}^n X_i$$
converges to the cumulative distribution function of $\mathcal N(\mu,\sigma)$,
a normal random
variable with mean $\mu$ and standard deviation $\sigma$.  Or, the
OP believes minor re-arrangements of this formula, e.g. the distribution
of $Z_n - \mu$ converges to the distribution of $\mathcal N(0,\sigma)$,
or the distribution of $(Z_n - \mu)/\sigma$ converges to the distribution of $\mathcal N(0,1)$, the standard normal random variable.  Note as an example
that these statements imply that 
$$P\{|Z_n - \mu| > \sigma\} = 1 - F_{Z_n}(\mu + \sigma) + F_{Z_n}((\mu + \sigma)^-) \to 1-\Phi(1)+\Phi(-1) \approx 0.32$$
as $n \to \infty$.
The OP goes on to say

This raises two questions:
  
  
*
  
*Can we deduce from this the law of large numbers? If the law of large numbers says that the mean of a sample of a random variable's values equals the true mean μ as N goes to infinity, then it seems even stronger to say that (as the central limit says) that the value becomes N(μ,σ) where σ is the standard deviation.
  

The weak law of large numbers says that for i.i.d. random variables $X_i$
with finite mean $\mu$, given any $\epsilon > 0$,
$$P\{|Z_n - \mu| > \epsilon\} \to 0 ~~ \text{as}~ n \to \infty.$$
Note that it is not necessary to assume that the standard deviation is
finite.
So, to answer the OP's question,


*

*The central limit theorem as stated by the OP does not imply
the weak law of large numbers.  As $n \to \infty$, the OP's 
version of the central limit theorem says that 
$P\{|Z_n-\mu| > \sigma\} \to 0.317\cdots$ while
the weak law says that $P\{|Z_n-\mu| > \sigma\} \to 0$

*From a correct statement of the central limit theorem, one can 
at best deduce only a restricted form of the weak law of large numbers
applying to random variables with finite mean and standard
deviation.  But the weak law of large numbers also holds for random
variables such as Pareto random variables with finite means but
infinite standard deviation.

*I do not understand why saying that the sample mean converges
to a normal random variable with nonzero standard deviation is
a stronger statement than saying that the sample mean converges
to the population mean, which is a constant (or a random variable
with zero standard deviation if you like). 
