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Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\hspace{0.3cm},\quad a_{2}=(-1)^2\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\hspace{0.3cm} \quad\ldots\quad a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Update: If we look at the term at the central symmetric random variable $a_{\lfloor n/2 \rfloor}$ $$ > a_{\lfloor n/2 \rfloor}=\text{sum of > the products of $\lfloor n/2 \rfloor$ > of different $z_{i}$'s} $$ it is not hard to see that it has uniform distributed phase in $(-\pi,\pi]$. However, its magnitude is blowing up extremely fast!

Simulations showed that for $n=100$ the mean of $\log |a_{\lfloor n/2 > \rfloor}|\approx 15$ which was surprisingly big.

Does anyone knows how to prove that $$ \lim_{n\to\infty}{|a_{\lfloor n/2 > \rfloor}|}=\infty $$ and estimate the rate of growth?

Thanks!

Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\hspace{0.3cm},\quad a_{2}=(-1)^2\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\hspace{0.3cm} \quad\ldots\quad a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Update: If we look at the term at the central symmetric random variable $a_{\lfloor n/2 \rfloor}$ $$ > a_{\lfloor n/2 \rfloor}=\text{sum of > the products of $\lfloor n/2 \rfloor$ > of different $z_{i}$'s} $$ it is not hard to see that it has uniform distributed phase in $(-\pi,\pi]$. 

However, its magnitude is blowing up extremely fast!

Does anyone knows how to compute the limit distribution of $|a_{\lfloor n/2 \rfloor}|$ under the appropriate normalizations?

Thanks!

Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\hspace{0.3cm},\quad a_{2}=(-1)^2\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\hspace{0.3cm} \quad\ldots\quad a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Thanks!

Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\hspace{0.3cm},\quad a_{2}=(-1)^2\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\hspace{0.3cm} \quad\ldots\quad a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Update: If we look at the term at the central symmetric random variable $a_{\lfloor n/2 \rfloor}$ $$ > a_{\lfloor n/2 \rfloor}=\text{sum of > the products of $\lfloor n/2 \rfloor$ > of different $z_{i}$'s} $$ it is not hard to see that it has uniform distributed phase in $(-\pi,\pi]$. However, its magnitude is blowing up extremely fast!

Simulations showed that for $n=100$ the mean of $\log |a_{\lfloor n/2 > \rfloor}|\approx 15$ which was surprisingly big.

Does anyone knows how to prove that $$ \lim_{n\to\infty}{|a_{\lfloor n/2 > \rfloor}|}=\infty $$ and estimate the rate of growth?

Thanks!

Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\quad,\quad a_{2}=\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\quad,\quad\ldots\quad,a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Thanks!

Let $z_1,z_2,\ldots,z_n$ be i.i.d. random variables in the unit circle. Consider the polynomial $$ p(z)=\prod_{i=1}^{n}{(t-z_i)}=t^n+a_{1}t^{n-1}+\cdots+a_{n-2}t^2+a_{n-1}t+a_n $$ where the $a_i$ are the symmetric functions $$ a_{1}=(-1)\sum_{i=1}^{n}{z_{i}}\hspace{0.3cm},\quad a_{2}=(-1)^2\sum_{1\leq i< j\leq n}{z_{i}z_{j}}\hspace{0.3cm} \quad\ldots\quad a_{n}=(-1)^{n}z_{1}z_{2}\ldots z_{n}. $$

How can we estimate the random variable $Z$ defined as $$ Z=\sum_{j=1}^{n}{|a_{j}|} $$ asymptotically as $n\to\infty$?

It is not very difficult to estimate $|\sum_{j=1}^{n}{a_{j}}|$ by estimating $\log p(1)$ via the CLT. However, $Z$ seems to be much more difficult. Any idea of what can work here?

Thanks!