I will add an answer showing how bad the approximation from the central limit theorem (CLT) can be for the Pareto distribution, even in a case where the assumptions for CLT is fulfilled. The assumption is that there must be a finite variance, which for the Pareto means that $\alpha > 2$.  For a more theoretical discussion of why this is so, see my answer here:  https://stats.stackexchange.com/questions/94402/what-is-the-difference-between-finite-and-infinite-variance/100161#100161

I will simulate data from the Pareto distribution with parameter $\alpha=2.1$, so that the variance "just barely exists". Redo my simulations with $\alpha=3.1$ to see the difference! Here is some R code:
```r  
    ###  Pareto dist and the central limit theorem
    ###
    require(actuar) # for (dpqr)pareto1()
    require(MASS) #  for Scott()
    require(scales) # for alpha()
    # We use (dpqr)pareto1(x,alpha,1)
    #
    alpha <- 2.1  #  variance just barely exist
    E <-  function(alpha) ifelse(alpha <= 1,Inf,alpha/(alpha-1))
    VAR <- function(alpha) ifelse(alpha <= 2, Inf, 
            alpha/((alpha-1)^2 * (alpha-2)))
    
    R <- 10000
    e <-  E(alpha)
    sigma  <-  sqrt(VAR(alpha))
    sim <-  function(n) {
        replicate(R, {x <- rpareto1(n,alpha,1)
            x <- x-e
            mean(x)*sqrt(n)/sigma },simplify=TRUE)
    }
    sim1 <- sim(10)
    sim2 <- sim(100)
    sim3 <- sim(1000)
    sim4 <- sim(10000) # do take some time ...
    
    ### These are standardized so have all 
    ### theoretically variance 1.
    ### But due to the long tail, the empirical variances are 
    ### (surprisingly!) much lower:
    
    sd(sim1)
    sd(sim2)
    sd(sim3)
    sd(sim4)
    
    ### Now we plot the histograms:
    hist(sim1, prob=TRUE, breaks="Scott", col=alpha("grey05", 
    0.95), main="simulated Pareto means", xlim=c(-1.8,16))
    hist(sim2, prob=TRUE, breaks="Scott", col=alpha("grey30", 
          0.5), add=TRUE)
    hist(sim3, prob=TRUE, breaks="Scott", col=alpha("grey60", 
          0.5), add=TRUE)
    hist(sim4, prob=TRUE, breaks="Scott", col=alpha("grey90", 
          0.5), add=TRUE)
    plot(dnorm, from=-1.8, to=5, col=alpha("red", 0.5), add=TRUE)
```     
And here is the plot:

[![simulated Pareto means, histogram][1]][1]

  [1]: https://i.sstatic.net/CuQqV.png

One can see that even at sample size $n=10000$ we are far away from the normal approximation. That the empirical variances are so much lower than the true theoretical variance $\sigma^2=1$ is due to the fact that we have a very large contribution to the variance from parts of the distribution in the extreme right tail that do not show up in most samples. **This is to be expected always, when the variance "just barely exists"**.  A practical way to think about that is the following. Pareto distributions is often proposed to model distributions of income (or wealth). The expectation of income (or wealth) will have a very large contribution from the very few billionaires. Sampling with practical sample sizes will have a very small probability of including any billionaires in the sample!