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A mysterious result, probably by Euler himself, goes as follows:

If $n$ is a positive integer and $f:\mathbb R \rightarrow \mathbb C$ is integrable on the open interval $(0, n)$, then $$\int_0^1\int_0^1\ldots\int_0^1 f(\lfloor x_1 + x_2 + \ldots +x_n\rfloor) \, dx_1 \, dx_2\ldots dx_n = \sum_{k=1}^\infty A(n,k)\frac{f(k)}{k!}, $$ where the $A(n,k)$'s are the Eulerian numbers.

Is there an analogous result without the floor function ? Thats is, is there an analogous formula for the integral

$$ \int_0^1\int_0^1\ldots\int_0^1 f(x_1 + x_2 + \ldots + x_n)\,dx_1\,dx_2\ldots dx_n \; ? $$

dohmatob
  • 9,753
  • @GabrielRomon Ya, that is indeed true and I considered it myself. It also provides a way to approximate the integral via statistical sampling. – dohmatob Nov 10 '17 at 09:39

2 Answers2

7

Here is a detailed solution motivated by @Arthur's. Let $\mathbf w \in \mathbb R^n$, non of whose coordinates is $0$. For any real $z$, define

$$G_{\mathbf w,z} := \{\mathbf x \in \mathbb R^n | \mathbf w^T \mathbf x \le z\}, \;H_{\mathbf w,z} := \{\mathbf x \in \mathbb R^n | \mathbf w^T \mathbf x = z\}.$$

The sought-for integral is precisely $$\int_{[0,1]^n} f(\mathbf w^T\mathbf x)d\mathbf x= \mathbb E_{x_1,\ldots, x_n \sim \mathcal U[0, 1]} [f(\mathbf w^T \mathbf x)]$$ with the choice $w_1 = \ldots w_n = 1$.

Now, let $F_{\mathbf w}$ be the cdf of $\mathbf w^T\mathbf x$ for i.i.d $x_1,\ldots,x_n \sim \mathcal U[0,1]$. Then its not hard to see that $F_{\mathbf w}(z) = \operatorname{vol}_n (G_{\mathbf w,z} \cap[0,1]^n)$, with density $f_{\mathbf w}(z) = \|\mathbf w\|_2^{-1} \operatorname{vol}_{n-1} (H_{\mathbf w,z} \cap[0,1]^n)$. By elementary properties of expectations, one has

$$ \begin{split} \int_{[0,1]^n} f(\mathbf w^T\mathbf x)d\mathbf x &= \mathbb E_{x_1,\ldots, x_n \sim \mathcal U[0, 1]}f(\mathbf w^T\mathbf x) = \int_{-\infty}^\infty \mathbb E(f(\mathbf w^T\mathbf x) | \mathbf w^T\mathbf x = z)f_{\mathbf w}(z)dz\\ &= \int_{-(-\mathbf w)_+^T\mathbf 1_n}^{(\mathbf w)_+^T\mathbf 1_n} \|\mathbf w\|_2^{-1}\operatorname{vol}_{n-1} (H_{\mathbf w,z} \cap[0,1]^n)f(z)dz. \end{split} $$

Invoking Theorem 4 of this paper yields $$ \|\mathbf w\|_2^{-1}\operatorname{vol}_{n-1} (H_{\mathbf w,z} \cap[0,1]^n) = \frac{1}{(n-1)! \prod_{k=1}^n w_k}\sum_{K \subseteq [\![n]\!]}(-1)^{\#K}\left(z - \mathbf w^T\mathbf 1_K\right)_+^{n-1}, $$

where $\mathbf w^T \mathbf 1_K := \sum_{k \in K}w_k$. Putting things together gives

$$ \int_{[0,1]^n} f(\mathbf w^T\mathbf x)d\mathbf x= \frac{1}{(n-1)! \prod_{k=1}^n w_k}\sum_{K \subseteq [\![n]\!]}(-1)^{\#K}\sigma_{\mathbf w, K}(f), $$

where $\sigma_{\mathbf w, K}(f):= \Lambda_{-(-\mathbf w)_+^T\mathbf 1_n,(\mathbf w)_+^T\mathbf 1_n,\mathbf w^T\mathbf 1_K}(f)$ and $$\Lambda_{a,b,c}(f) := \int_{a}^b \left(z - c\right)_+^{n-1}f(z)dz. $$

Thus the whole game is about computing the numbers $\Lambda_{a,b,c}(f)$.

Examples

  • Volume of hypercube $[0,1]^n$. This is a pathologically simple example and is only here to act as a sanity check. Take $w_1 = \ldots = w_n = f = 1$, and for any $K \subseteq [\![n]\!]$ with $\#K = k$, one has

$$\sigma_{\mathbf w, K}(f) = \int_{k}^n (z-k)^{n-1} dz = \frac{1}{n}(n-k)^n. $$

Thus $$\int_{[0,1]^n} dx_1 \ldots dx_n = \frac{1}{n!}\sum_{k=0}^n C^n_k(-1)^k(n-k)^n = 1 $$

  • Expected value of sum of $n$ i.i.d uniform random variables. Take $w_1 = \ldots = w_n = 1$ and $f = \operatorname{id}$. For any $K \subseteq [\![n]\!]$ with $\#K = k$, one has

$$\sigma_{\mathbf w, K}(f) = \int_0^n(z-k)^{n-1}zdz = \frac{(n - k)^n(n^2 + k)}{n(n+1)} $$

$$ \begin{split} \int_{[0,1]^n}(x_1+ \ldots + x_n)dx_1 \ldots dx_n &= \frac{1}{(n+1)!}\sum_{k=0}^n (-1)^kC^n_k (n-k)^{n}(n^2 + k)\\ &= \frac{1}{(n+1)!}\frac{n}{2}(n+1)! = \frac{n}{2} \end{split} $$

dohmatob
  • 9,753
4

I propose the following:

If $A(c)$ is the hyperarea of the region of the hyperplane $x_1+\cdots + x_n = c$ bounded by the hypercube $0\leq x_1, \ldots,x_n\leq 1$, the integral is equal to $\frac 1{\sqrt n} \int_0^{n}A(c)f(c)dc$.

I might have missed something, though, as I haven't done any general checking. It works for $n = 1,2$ and $f(x) = 1$, so I don't think I've missed any constant factors, at least. Comments and corrections are welcome.

Arthur
  • 204,511
  • Nice! I can derive your formula upto withing the $\frac{1}{\sqrt{n}}$ factor (mistake ?). Indeed, let $H_z := {(x_1,\ldots,x_n) | \sum_i x_i = z}$. Then, $$ \begin{split} \int_{[0,1]^n} f(\sum_i x_i)dx_1 \ldots dx_n &= \int_{z=0}^n \left(\int_{[0,1]^n \cap H_z}f(z)dx_1\ldots dx_n\right)dz\ &= \int_{z=0}^n \left(\int_{[0,1]^n \cap H_z}dx_1\ldots dx_n\right)f(z)dz\ = \int_{z=0}^n \operatorname{vol}([0,1]^n \cap H_z)f(z)dz \end{split} $$ – dohmatob Nov 10 '17 at 10:22
  • @dohmatob I needed the $\frac1{\sqrt n}$ factor to make the $n = 2, f(x) = 1$ case work, since the graph of $A(c)$ in that case is an isosceles triangle with height $\sqrt2$ and base $2$. It might not be the correct factor in the general case, but for the two simplest cases it works, so that's why I guessed it. However, it seems reasonable, since it represents the distance between the hyperplanes $x_1+\cdots +x_n = c$ and $x_1+\cdots x_n = c+1$ (in other words, the velocity at which the hyperplane "sweeps over" the hypercube) – Arthur Nov 10 '17 at 10:26
  • you're right about the strange $\sqrt{n}$ term in the denominator. Actually, this term is just the $\ell_2$ norm of the weighting vector $w = (1,\ldots,1)$ required to pass from volume of the polytope to the hyper-area of its boundary, via derivatives w.r.t $z$. See my post below. That is $\partial_z \operatorname{vol}(z) = |w|_2 \operatorname{surface}(z)$. – dohmatob Nov 13 '17 at 10:58