Poisson Binomial Distribution with Evenly Distributed Bernoulli Trial Probabilities Consider the Poisson binomial distribution with $n$ coins and coin probabilities ${1 \over n}, {2\over n}, \dots, {n-1 \over n}, 1$. Do we know an asymptotic for this distribution?
Le Cam's theorem says when the $p_i$ are small the distribution approaches Poisson($\mu)$, but the $p_i$ are not small here. We can even say it's not Poisson: by the Stein-Chen method applied by Barbour and Hall a lower bound on total variation distance between the Poisson($\mu$) and Poisson binomial distribution with coin probabilities $p_i$ is
$$\frac{1}{32} \min (\frac{1}{\mu}, 1) \displaystyle\sum_{i=1}^n p_i^2$$
In our case $\mu$ and $\sum p_i^2$ are $\Theta(n)$ so this is $\Omega(1)$.
 A: Intuitively, there's no problem because for each $k=1, 2, \ldots, n$ we may pair the variable of mean $k/n$ with that of mean $(n-k)/n$ (except possibly when $2k=n,$ which will not matter asymptotically).  The sum of these variables has mean $1/2$ and (obviously) has no skewness.  The variances of these sums are
$$\frac{k}{n}\left(1 - \frac{k}{n}\right) + \left(1 - \frac{k}{n}\right)\frac{k}{n} = 2\frac{k}{n}\left(1 - \frac{k}{n}\right)$$
which range from $2/n^2$ to $1/2.$  This range might give us pause, but asymptotically it causes no problems: when we standardize the sum (its mean is $\mu(n)=(1+n)/2$ and its variance is $\sigma^2(n)=(n^2-1)/(6n),$ both elementary computations) it converges fairly rapidly to a standard Normal distribution.  The excess kurtosis is always negative (the tails are obviously limited in extent) but shrinks to zero.
A rigorous demonstration computes that the cumulant generating function of the standardized sum is $t^2/2 + O(n^{-1})O(t^4)$ which converges as $n$ grows large to the c.g.f. of the standard Normal distribution.
As a check, this R simulation generates 10,000 such sums for $n=20$ and visually compares their distribution to the Normal distribution with parameters $\mu(20),\sigma^2(20):$

The red curve plots the corresponding theoretical probabilities.  It is (already) an excellent fit.
n <- 20
n.sim <- 1e4
set.seed(17)

p <- (1:n)/n
x <- colSums(matrix(rbinom(n*n.sim, 1, p), nrow=n))

hist(x, freq=FALSE, breaks=min(x):max(x))
m <- sum(p)
v <- sum(p * (1-p))
curve(pnorm(x+1, m, sqrt(v)) - pnorm(x, m, sqrt(v)), col="Red", add=TRUE, lwd=2, n=301)

