# Poisson like distribution in $\left(\sum_{k=0}^{n-1} x^{2^k}\right)^{n}$?

Given $y_{n}(x)=\left(\sum_{k=0}^{n-1} x^{2^k}\right)^{n}$. An example ($n=5$) may look like $$y_5(x)={x^{80}}+5 x^{72}+5 x^{68}+5 x^{66}+5 x^{65}+10 x^{64}+20 x^{60}+20 x^{58}+20 x^{57}+20 x^{56}+20 x^{54}+20 x^{53}+40 x^{52}+20 x^{51}+40 x^{50}+30 x^{49}+35 x^{48}+60 x^{46}+60 x^{45}+60 x^{44}+60 x^{43}+80 x^{42}+50 x^{41}+61 x^{40}+60 x^{39}+100 x^{38}+90 x^{37}+85 x^{36}+70 x^{35}+95 x^{34}+65 x^{33}+75 x^{32}+{120 x^{31}}+120 x^{30}+100 x^{29}+110 x^{28}+100 x^{27}+90 x^{26}+90 x^{25}+100 x^{24}+100 x^{23}+90 x^{22}+70 x^{21}+66 x^{20}+70 x^{19}+55 x^{18}+65 x^{17}+75 x^{16}+60 x^{15}+50 x^{14}+50 x^{13}+40 x^{12}+30 x^{11}+31 x^{10}+25 x^{9}+15 x^{8}+10 x^{7}+5 x^{6}+x^{5}$$ If you plot exponents of addends of $y_n$ against prefactors, this looks (for $n=7$) like

$\hskip1.7in$

Can this be described by a kind of Poisson distribution for general $n$?

• Very nice question. Were you able to find out more about this problem? – Yuriy S Mar 10 '16 at 21:43
• I'd consider instead $f(x)^n = \left(\sum_{k=0}^\infty x^{2^k-1}\right)^n = \sum_{m=0}^\infty a_m(n) x^m$ for $|x| < 1$ ? the central limit theorem should tell us that the coefficients $\frac{a_m(n)}{n}$ will tend to the normal distribution – reuns Mar 11 '16 at 6:33
• Are you sure your plot is for $n=7$ and not $n=8$? The average degree should be $2^n-1$ which is only $127$ for $n=7$ and $255$ for $n=8$ matching the plot. – A.S. Mar 11 '16 at 6:41
• Probabilistically speaking, you have a multinomial $M\sim MN(n,\frac 1 n,\dots,\frac 1 n)$ dotted with $V=(2^0,\dots, 2^{n-1})$: $S_M=M\cdot V$. You get $\mu_M=2^n-1$ and can compute variance exactly. Approximate $S_M$ by $S_B=\sum_{i=0}^{n-1} 2^{i}B(n,\frac 1 n)$ or go even further to get $S_P=\sum_{i=0}^{n-1} 2^{i}Pois(1)$. All cumulants of a Poisson are $1$, hence cumulants of $S_P$ are $$\kappa_k=\frac {2^{kn}-1}{2^k-1}$$ yielding variance $\kappa_2\approx \frac{4^n}3$ and $\kappa_{3}\approx\frac {8^n}7$. Quality of approximation of $S_M$ by $S_P$ deteriorate for higher cumulants. – A.S. Mar 11 '16 at 7:46
• The above yields asymptotic skew of $\frac {3^{3/2}}{7}\approx 0.74$ – A.S. Mar 11 '16 at 7:51

This is not a Poisson distribution, but is related. It is a rather interesting distribution, but let me not disclose it at the moment.

First note that, up to the multiple $n^n$, this is the probability generating function for the sum of $n$ iid random variables, which are uniformly distributed on the set $\{1,2,\dots,2^{n-1}\}$.

Dividing by $2^{n-1}$, we get the sum of $n$ iid random variables $\xi_{1,n},\dots,\xi_{n,n}$, uniformly distributed on the set $\{1,2^{-1},2^{-2}\dots,2^{1-n}\}$. Then the characteristic function of the sum $S_n = \xi_{1,n}+\dots+\xi_{n,n}$ is $$\varphi_{S_n}(t) = \varphi_{\xi_{1,n}}(t)^n = \left(\frac1n \sum_{k=0}^{n-1}e^{it2^{-k}}\right)^n = \left(1+ \frac1n \sum_{k=0}^{n-1}\big(e^{it2^{-k}}-1\big)\right)^n\\ \to \exp\left\{\sum_{k=0}^{\infty}\big(e^{it2^{-k}}-1\big)\right\} = \prod_{k=0}^{\infty}\exp\left\{e^{it2^{-k}}-1\right\},\quad n\to\infty.$$ Therefore, $$S_n\overset{w}{\longrightarrow} S = \sum_{k=0}^\infty \frac{\zeta_k}{2^k},\quad n\to\infty,$$ where $\zeta_k$ are iid with Poisson(1) distribution.

Let us study the limit disribution $S$. As @A.S. wrote, its cumulants are $(1-2^{-k})^{-1}$.

Further, let $0.\alpha_1\alpha_2\alpha_3\dots$ be the binary form of fractional part of $S$. Then the sequence $\{\alpha_n,n\ge 1\}$ of digits is stationary (in fact, this is the case for any random variable of the form $\sum_{k=0}^\infty \zeta_k 2^{-k}$ with iid integer-valued variables $\zeta_k$). It is possible to "identify" the marginal distribution of digits, that is, the probability $p = P(\alpha_1 = 1)$. Define for $k\ge 0$ $$E_k = 1 - 2e^{-1}\sum_{n=0}^\infty \frac{1}{(2^k (2n+1))!}.$$ Then $$p = \frac12\left(1-\prod_{k=0}^\infty E_k\right).$$ If $p\neq 1/2$, which seems to be the case, then the distribution of $S$ is singular. (In the unlikely case that $p=1/2$, it is still singular since the digits $\alpha_n$ are dependent.) It is also continuous, but at the moment I don't see a simple way to prove this.

• Didn't think of rescaling by $2^{n-1}$. What else can you say about the limiting distribution $\zeta$ apart from its characteristic function and cumulants $\kappa_k=(1-2^{-k})^{-1}$? – A.S. Mar 12 '16 at 15:03
• @A.S., I tried to scale by variance too, but quickly found that there is no limit with such scaling. Concerning the distribution, I believe it is continuous and singular, but didn't try to prove this yet. – zhoraster Mar 12 '16 at 15:43
• I got some !simulation pictures – A.S. Mar 12 '16 at 15:55
• @A.S., the sequence of binary digits of the fractional part is stationary. I have some problems computing even the marginal distribution of digits (probably I'm just a bit tired), but it is almost obvious that the digits are dependent. So the singularity should be quite easy to prove. – zhoraster Mar 12 '16 at 20:43
• +1 Whoa, I admit that this is not my branch of math, I deal with regularly. I might need more than 3 days to judge all this. Could you point towards a good online reference to provide my the basics I need? – draks ... Mar 14 '16 at 19:35

This is not an answer, but a series of thoughts about the problem.

As a start, the coefficient of $x^k$ in $y_n(x)$ seems to be the number of ways that $k$ can be written as the sum of $n$ powers of $2$ not exceeding $2^{n-1}$.

Another approach might be to use the fact that the coefficient of $x^m$ in $f(x)$ is $\frac{f^{m}(0)}{m!}$, and evaluate $y^{m}(0)$.

Another thought is that having $n$ as both the exponent and limit of the sum seems odd. It might be profitable to consider $y_{n, m}(x) = \left(\sum_{i=0}^n x^{2^i}\right)^m$.

That's all I can think of for now.