Are there fat-tailed distributions with FINITE variance and defined moments? Reading here seems like there shouldn't be.
The only example I studied was the Cauchy distribution, which indeed has fat tails, but also undefined moments, and hence variance. Is there any distribution with tails fatter than normal distribution (but still the same "bell shape")?
 A: A fat-tailed distribution is a distribution for which the tail behaves like a power law, i.e.
$$F(x) \sim 1- x^{-\alpha}$$
for $x\to \infty $ and some $\alpha>0$. Equivalently 
$$
f(x) \sim x^{-\alpha-1}, \quad x\to \infty, \quad \alpha>0,
$$
where $f$ is the pdf.

The density of the student $t$-distribution can be written as 
$$
f(x) = C_1(1+x^2/\nu)^{-(\nu+1)/2}, \quad \nu > 0,
$$
where $C_1$ is the normalization constant in the pdf.
Asymptotically $(1+cx^a)^b \sim c^bx^{ab}$, $x\to \infty$ for some constants $a>0,b,c\in \mathbb R$ so
$$
f(x) \sim C_1(x^2/\nu)^{-(\nu+1)/2} = C_2 \lvert x\rvert ^{-\nu-1}, \quad x\to \pm \infty
$$
for some other constant $C_2$ ($C_2$ absorbed $\nu^{(\nu+1)/2}$). That means that the $t$-distribution is fat-tailed.
As we know, the student $t$-distribution has finite variance if $\nu>2$ and has moments up to order $\nu$. Hence:
Yes, fat-tailed bell-shaped distributions with finite variance exist.
A: Often, "fat-tailed" is synonymous with "outlier-prone." You can have outlier-prone distributions with finite tails, hence all moments finite.
For example, mix a U(-1,1) with a U(-10000, 10000), with .0001 mixing probability on the latter.  Here is some R code that simulates 1000000 such observations, calculates the extreme Z values, and the kurtosis.
set.seed(12345)
N = 1000000
p = .0001
U0 = runif(N)
U1 = runif(N,-1,1)
U2 = runif(N, -10000, 10000)
X = ifelse(U0 > p, U1, U2)
sort.Z =sort(scale(X))
head(sort.Z)  
tail(sort.Z)
kurt <- function(y) {
n = length(y)
p = rep(1/n,n)
ey = sum(y*p)
vy = sum((y-ey)^2*p)
sdy = sqrt(vy)
kurt = sum( ((y-ey)/sdy)^4 *p) -3
return(kurt)
}
kurt(X)
Kurtosis is the average of the Z-values, each taken to the fourth power. Extreme Z-values, like here, indicate outliers. The extreme Z-values printed by the code above are 
-163.9421 -158.7251 -155.8845 -154.8858 -154.3153 -153.1298
and 
148.0770 148.1702 159.8075 161.1619 163.0165 164.8768
Observations greater than 100 standard deviations from the mean, as in this example, are commonly called "outliers."
The kurtosis printed from the code above is of course high as well: 16094.69.
Notice that the distribution is perfectly flat at its "peak," providing a good counterexample to the incorrect notion that high kurtosis corresponds to a "peaked", or "pointy" distribution.
