Let $f_a$ be the indicator function of the interval $[-a,a]$ divided by $2a$. So $f_a$ is the PDF of a uniform random variable in the interval $[-a,a]$, and $f_a$ is even. Let $g_n = (f_{1/n})^{*n}$ be the n-fold convolution of $f_{1/n}$. So $g_n$ is the PDF of the mean of $n$ independent uniform random variables in $[-1,1]$ -- this is called the Bates distribution.

My questions are:

  1. Does the sequence of functions { $g_n / (f_{1/n}*g_n)$ } converge uniformly to a function $G(x)$ on the interval $[-1,1]$ (the $*$ in the denominator means convolution) ?
  2. If so, is there a closed form expression for $G(x)$ ?

Note that the sequence can also be written { $(f_{1/n})^{*n} / (f_{1/n})^{*(n+1)}$ }. Also note that the denominator $(f_{1/n})^{*(n+1)}$ is positive on the open interval $(-1-1/n,1+1/n)$, so there is no division by $0$.

Convergence at the endpoints $x=\pm1$ is trivial and $G(-1)=G(1)=0$. Using the central limit theorem (a local limit form) I can show that there is convergence at $x=0$ and that $G(0)=1$. I have numerical evidence (plots) that convergence is uniform on all of $[-1,1]$. The function $G(x)$ (if it exists) appears to be nice and smooth with maximum at $x=0$.

Note that both numerator {$(f_{1/n})^{*n}$} and denominator {$(f_{1/n})^{*(n+1)}$} are Dirac sequences (aka delta sequences), which makes the analysis delicate.

I have looked at variants of the central limit theorem, but they only seem applicable at $x=0$. And I couldn't get anywhere with the Fourier Transform, which turns convolutions into products. Eulerian number asymptotics are relevant, but I made no progress with them either. I think there is a technique applicable to the quotient of PDFs that will answer these questions, but I am very unfamiliar with it.

Thank you for any ideas...


1 Answer 1


The answer to 1. is Yes.

The answer to 2. is "almost". For $x \in$ (-1,1) let $\hat t$ be the unique $t$ where $e^{-tx}sinh(t)/t$ is minimized. $\hat t$ is also the unique root of $coth(t) - 1/t = x$. Then $G(x) = {\hat t}/sinh(\hat t)$.

I revisited the Eulerian number asymptotics and found the technique I was looking for - the saddlepoint approximation for PDFs. There is a wikipedia article on it. In the case of the sum of uniform random variables, the approximation is asymptotic across the entire support, and not just near the center, as in the central limit theorem. The approximation can be applied separately to numerator and denominator of the quotient. The approximation uses the moment generating function, which is $sinh(t)/t$ for the uniform random variable in [-1,1].

Papers that were most helpful were:

  • Daniels, H.E. Saddlepoint Approximations in Statistics. The Annals of Mathematical Statistics. Vol. 25, No. 4. (Dec. 1954) pp. 631-650.
  • Janson, Svante. Euler–Frobenius Numbers and Rounding. arXiv:1305.3512v1. 15 May 2013.

The first one treats sums of uniform RVs in Example 5.3, and the second one has the key asymptotic estimate in equation (6.15), with a proof.

Below is a plot that shows 6 elements of the sequence { $(f_{1/n})^{*n} / (f_{1/(n+1)})^{*(n+1)}$ } converging uniformly to the above $G(x)$ (in red).
This plot was made with the R language and the gmp package for rational number arithmetic.

plot illustrating convergence to G(x)


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