Spectral Radii of Large Non-Hermitian Random Matrices

Introduction

The largest eigenvalues of the three Hermitian matrices (Gaussian orthogonal ensemble, Gaussian unitary ensemble and Gaussian symplectic ensemble) are proved to converge to the Tracy-Widom laws by Tracy and Widom (1994, 1996). Since then there have been very active research in this direction. For example, Baik et al. (1999) establish a connection between the longest increasing subsequence problem and the Tracy-Widom law. The relationships among the largest eigenvalues, combinatorics, growth processes, random tilings and the determinantal point processes are found [see, e.g., Tracy-Widom (2002) and Johansson (2007) and the literature therein]. In the studies of the high-dimensional statistics, Johnstone (2001, 2008) and Jiang (2009) prove that the largest eigenvalues of Wishart and Jacobi matrices converge to the Tracy-Widom law. Ramírez et al. (2011) obtain the asymptotic distribution of the largest eigenvalues of beta-Hermite ensemble. Recently, a research interest is the universality of the largest eigenvalues of non-Gaussian matrices; see, for example, Tao and Vu (2011), Erdős et al. (2012) and the references therein.

In this paper we will study the largest absolute values of the eigenvalues of some non-Hermitian matrices. Initiated by Ginibre (1965) for the study of Gaussian random matrices (real, complex and symplectic), the interest has continued and theoretical results are found to have many applications in quantum chromodynamics, chaotic quantum systems and growth processes; see more descriptions from the paper by Akemann, Baik and Francesco (2001). The applications also include dissipative quantum maps [Haake (2010)] and fractional quantum-Hall effect [Di Francesco et al. (1994)]. We refer the readers to Khoruzhenko and Sommers (2001) for more details.

Our analysis of the spectral radius is based on the following result. It is a special case of Theorem 1.2 from Chafaï and Péché (2014) which is another version of Theorem 4.7.1 from Hough et al. (2009).

Chafaï and Péché (2014) also give two general results in their Theorems 1.3 and 1.4 to show the following: if the density function of the eigenvalues of a non-Hermitian random matrix is the same as that in Lemma 1.1, under certain restrictions on $\varphi(x)$, the limiting distribution of the spectral radii is the Gumbel distribution. Their results do not apply to our three ensembles since our models do not meet their restrictions.

Now we present our results on the three ensembles in Subsections 1.1, 1.2 and 1.3, respectively. After this the strategy of the proofs and some comments are given.

if the probability distribution of $X_{n}$ converges weakly to that generated by the cumulative distribution function (cdf) $F(x)$ or $X.$ Now we study the spectral radius.

Let $z_{1},\cdots,z_{n}$ have the density as in (1.1). Define $H_{k}(x)=e^{-x}\sum^{k-1}_{j=0}\frac{x^{j}}{j!}$ for $k\geq 1$. Then $\frac{1}{\sqrt{n}}\max_{1\leq j\leq n}|z_{j}|$ converges weakly to probability distribution function $H(x)=\prod^{\infty}_{k=1}H_{k}(x^{-2})$ for $x>0$ and $H(x)=0$ for $x\leq 0.$

Observe that $H_{k}(x)$ is the cdf $P(\mbox{Poi}(x)\leq k-1)$ for each $k\geq 1$, where $\mbox{Poi}(x)$ is a Poisson random variable with parameter $x>0.$ So $H(x)$ is the product of those cdfs evaluated at $x^{-2}$.

as $x\to+\infty$. So $H(x)$ is heavy-tailed. This property will be verified in Section 2.4.

2 Truncation of Circular Unitary Ensemble

where $C$ is a normalizing constant. Assuming $c=\lim\frac{p}{n},$ Życzkowski and Sommers (2000) show that the empirical distribution of $z_{i}$’s converges to the distribution with density proportional to $\frac{1}{(1-|z|^{2})^{2}}$ for $|z|\leq c$ if $c\in(0,1).$ Dong et al. (2012) prove that the empirical distribution goes to the circular law and the arc law as $c=0$ and $c=1$, respectively.

Trivially, in the above theorem, $\{A_{n};n\geq 3\}$ is bounded and $B_{n}$ has the scale of $(n\log n)^{-1/2}$.

3 Product Ensemble

where $C$ is a normalizing constant and $w_{k}(z)$ is given by the Meijer G-function with

This formula seems not easy to understand at the first sight. However, the function admits an easily recursive formula $w_{1}(z)=\exp(-|z|^{2})$ and

for all integer $k\geq 2$; see, for example, Akemann and Burda (2012).

Now we consider the largest radius and the result is given below. We allow $k$ changes with $n$ in this paper. First, we need some notation. Let $\Phi$ denote the cumulative distribution function of $N(0,1)$. For $\alpha\in(0,\infty)$, define

and $\Phi_{\infty}(x)=\Phi(x)$. The digamma function $\psi$ is defined by

Let $k=k_{n}$ be a sequence of positive integers. The following holds. (a). If $\lim_{n\to\infty}k_{n}/n=0$, particularly for $k_{n}\equiv k$, then \alpha_{n}\big{(}n^{-k_{n}/2}\max_{1\leq j\leq n}|z_{j}|-1\big{)}-\beta_{n} converges weakly to the cdf $\exp(-e^{-x})$, where

(b). If $\lim_{n\to\infty}k_{n}/n=\alpha\in(0,\infty)$, then

(c). If $\lim_{n\to\infty}k_{n}/n=\infty$, then

Taking $k=1$ in (a) of Theorem 3, the corresponding limiting result is obtained by Rider (2003). Here we not only get the result for finite $k$, but for all possible range of $k_{n}$, which leads to the three transition zones: $k_{n}/n\to\alpha$ with $\alpha=0$, $\alpha\in(0,\infty)$ and $\alpha=\infty.$

As mentioned below Lemma 1.1, Theorems 1.3 and 1.4 from Chafaï and Péché (2014) conclude that the limiting distributions are always the Gumbel. The two theorems do not imply any of our results. Although the limiting distributions in Theorem 2 and case (a) of Theorem 3 are the Gumbel, since the density functions in (1.3) and (1.5) have two parameters $k$ and $n$ with $k$ depending on $n$, their assumptions are not satisfied.

Finally, let us look at the tail behavior of the distribution in (b) of Theorem 3. In fact we have

as $x\to+\infty$, where $C=\frac{\sqrt{\alpha}e^{-\alpha/8}}{2\sqrt{2\pi}}$. It is different from that of $e^{N(0,1)}$, the standard logarithmic normal distribution: $P(e^{N(0,1)}\geq x)\sim\frac{1}{\sqrt{2\pi}\,\log x}e^{-(\log x)^{2}/2}$ as $x\to+\infty.$ This will be verified in Section 2.4.

Strategy of the proofs. By using Lemma 1.1, the absolute values of eigenvalues $|z_{i}|$’s are “independent”. So we are dealing with the maxima of independent random variables with different distributions. The first step is to identify the distribution of each random variable. For example, for the product ensemble in Section 1.3, $|z_{j}|$ has the same distribution as the product of some i.i.d. random variables with Gamma distributions (Lemma 2.4). Then we analyze the tail probabilities of the product of random variables carefully through moderate deviations (Proposition 2.1). This step costs the major effort.

1. It is noteworthy to mention that, though the main idea is analyzing the maxima of independent random variables, the proofs are not trivial. In the classical study of the maxima of i.i.d. random variables, the limiting distributions are only of three types: Fréchet distribution, Gumbel distribution and Weibull distribution; see, for example, Resnick (2007). However, the limiting distributions appeared in Theorems 1 and (b) of Theorem 3 are new.

2. The eigenvalues of the three random matrices investigated in this paper are rotation-invariant. This special property gives us the advantage of independence by Lemma 1.1. When the eigenvalues are not of the invariant property, it seems there have no good understanding on the largest radii. For example, if $z_{1},\cdots,z_{n}$ have joint density

where $\tau\in(-1,1)$ is a parameter and $C$ is a normalizing constant [Lemma 4 from Petz and Hiai (1998)]. See also a similar example on page 3403 from Rider (2003) and (1.1) from the Arxiv paper by Kuijlaars and López.

4. Tracy and Widom (1994, 1996) prove that the largest eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles converge to the Tracy-Widom laws. Recently there have been an active research on the universality of the eigenvalues of non-Gaussian matrices; see, for example, Tao and Vu (2011), Erdős et al. (2012) and the references therein. In particular, Erdős et al. generalize the results by Tracy-Widom to the matrices with non-Gaussian entries. Our Theorems 1, 2 and 3 consider the eigenvalues of matrices with Gaussian entries. It will be interesting to study the universality of the three results for the matrices with non-Gaussian entries.

Finally, the organization of the rest of paper is as follows. We will prove Theorems 1, 2 and 3 in Sections 2.1, 2.2 and 2.3, respectively. The verifications of (1.2) and (1.9) are given in Section 2.4.

Proofs

In this section, we will prove Theorems 1, 2 and 3 in each subsection.

Let $a_{ni}\in[0,1)$ be constants for $i\geq 1,n\geq 1$ and $\sup_{n\geq 1,i\geq 1}a_{ni}<1$. For each $i\geq 1$, $a_{i}:=\lim_{n\to\infty}a_{ni}$. Assume $c_{n}:=\sum_{i=1}^{\infty}a_{ni}<\infty$ for each $n\geq 1$ and $c:=\sum_{i=1}^{\infty}a_{i}<\infty$, and $\lim_{n\to\infty}c_{n}=c$. Then

Proof. Note that $\prod^{\infty}_{i=1}(1-a_{ni})$ and $\prod^{\infty}_{i=1}(1-a_{i})$ are well defined, and $\prod^{\infty}_{i=1}(1-a_{ni})>0$ for each $n\geq 1$ and $\prod^{\infty}_{i=1}(1-a_{i})>0$. It suffices to show that

which goes to zero as $k\to\infty$. Therefore, we have

Set $a=\sup_{n\geq 1,i\geq 1}a_{ni}$. Then $0\leq a<1.$ It follows that

Proof of Theorem 1. By Lemma 1.1 and (1.1), $\max_{1\leq j\leq n}|z_{j}|$ and $\max_{1\leq j\leq n}Y_{nj}$ have the same distribution, where $Y_{n1},\cdots,Y_{nn}$ are independent such that $Y_{nj}$ has the probability density function (pdf) proportional to $y^{2j-1}(1+y^{2})^{-(n+1)}I(y\geq 0)$ for $1\leq j\leq n$. Thus, to prove the theorem, it suffices to show

Let $X_{i}$, $i\geq 1$ be a sequence of i.i.d. random variables with cumulative distribution function (cdf) $F$. Let $X_{1:n}\leq X_{2:n}\leq\cdots\leq X_{n:n}$ be the order statistics of $X_{1},X_{2},\cdots,X_{n}$ for each $n\geq 1$. Then from page 14 on the book by Balakrishnan and Cohen (1991), we know that the cdf of $X_{i:n}$ is given by

for each $1\leq i\leq n$. If $F$ has a probability density function $f$, then the pdf of $X_{i:n}$ is given by

The monotonicity of the order statistics implies that $F_{i:n}(x)$ is non-increasing in $i$ for each $x$, that is,

Let $\{u_{n},~{}n\geq 1\}$ be a sequence of constants such that $\lim_{n\to\infty}n(1-F(u_{n}))=:\tau\in(0,\infty)$. Write $\tau_{n}=n(1-F(u_{n})).$ Then it follows from the first equality in equation (2.4) that

as $n\to\infty$ for each fixed integer $i\geq 1$.

Now, we take $F(y)=\frac{y^{2}}{1+y^{2}}$ for $y>0$. Fix $x>0$, set $u_{n}=u_{n}(x)=\sqrt{n}x$. Then $\lim_{n\to\infty}n(1-F(u_{n}))=x^{-2}$. Then from (2.7)

for each fixed integer $i\geq 1$. For each $n\geq 1$, define

Then it follows from (2.6) that $\sup_{n\geq 1,i\geq 1}a_{ni}=\sup_{n\geq 1}a_{n1}$. By the first identity in (2.4),

From (2.8) we have $\lim_{n\to\infty}a_{ni}=1-H_{i}(x^{-2})=:a_{i}$ for each $i\geq 1$. Moreover, we have

By exchanging the ordering of the sums, we know $\sum^{\infty}_{i=1}\sum^{\infty}_{k=i}\frac{(x^{-2})^{k}}{k!}=\sum^{\infty}_{k=1}\sum^{k}_{i=1}\frac{(x^{-2})^{k}}{k!}=\sum^{\infty}_{k=1}\frac{(x^{-2})^{k}}{(k-1)!}=x^{-2}\exp\{x^{-2}\}.$ It follows that $\sum^{\infty}_{i=1}a_{i}=x^{-2}.$ By Lemma 2.1, we have

From (2.5) we obtain the pdf of $X_{j:n}$ given by

which is also the pdf of $Y_{nj}$. Therefore, we have

for $x>0.$ This completes the proof of Theorem 1. $\blacksquare$

2 The Proof of Theorem 2

Notation: $C_{n}\sim D_{n}$ as $n\to\infty$ implies $\lim_{n\to\infty}\frac{C_{n}}{D_{n}}=1$; $C_{n}(t)\sim D_{n}(t)$ uniformly over $t\in T_{n}$ implies $\lim_{n\to\infty}\sup_{t\in T_{n}}|\frac{C_{n}(t)}{D_{n}(t)}-1|=0$; $C_{n}(t)=O(D_{n}(t))$ uniformly over $t\in T_{n}$ implies $\sup_{t\in T_{n}}|\frac{C_{n}(t)}{D_{n}(t)}|$ is bounded; $C_{n}(t)=o(D_{n}(t))$ uniformly over $t\in T_{n}$ implies $\sup_{t\in T_{n}}|\frac{C_{n}(t)}{D_{n}(t)}|$ converges to zero as $n\to\infty$.

For random variables $\{X_{n};\,n\geq 1\}$ and constants $\{a_{n};\,n\geq 1\}$, we write $X_{n}=O_{P}(a_{n})$ if $\lim_{x\to+\infty}\lim_{n\to\infty}P(|\frac{X_{n}}{a_{n}}|\geq x)=0$. In particular, if $X_{n}=O_{P}(a_{n})$ and $\{b_{n};\,n\geq 1\}$ is a sequence of constants with $\lim_{n\to\infty}b_{n}=\infty$, then $\frac{X_{n}}{a_{n}b_{n}}\to 0$ in probability as $n\to\infty$.

Proof. From definition, it is easy to see that $\lim_{n\to\infty}a(x_{n})=+\infty$ and $\lim_{n\to\infty}b(x_{n})=0$ and $\min_{1\leq j\leq j_{n}}c_{n,j}>0$ as $n$ is large enough. Thus, $\min_{1\leq j\leq j_{n}}[(j-1)c_{n,j}+a(x_{n})+b(x_{n})y]\to+\infty$ as $n\to\infty$. It is well known that $1-\Phi(x)\sim\frac{\phi(x)}{x}$ as $x\to\infty.$ Therefore, (2.9) follows from (2.10). Now let us prove (2.10).

Fact 1: Uniformly over $1\leq j\leq l_{n}$,

by using the third assertion in (2.12) and

Fact 2: Uniformly over $l_{n}<j\leq j_{n}$, which is different from the assumption on (2.13) and (2.14),

It then follows from (2.13), (2.14) and (2.11) that

where the middle limit in (2.12) is used in the second step. Similarly, it follows from (2.15), (2.16) and (2.11) that

by using (2.17) and (2.18) in the equality and the middle assertion in (2.12) in the last step. By adding up the above eqaution and (2.18), we obtain (2.10). $\blacksquare$

There exists a constant $C>0$ such that for all $r>k\geq 1$,

where for $i=1,2$, $l_{i}(t)$ is a polynomial in $t$ of degree $\leq 3i$, depending on $r$ and $k$, and all of its coefficients are of order O(\big{(}\frac{r}{(r-k)k}\big{)}^{i/2}).

Proof of Theorem 2. Review the density formula in (1.3). Set $m_{n}=n-p.$ For ease of notation, we sometimes write $m$ for $m_{n}.$ By assumption, $h_{2}^{\prime}<\frac{m_{n}}{n}<h_{1}^{\prime}$ for all $n\geq 2$ where $h_{i}^{\prime}=1-h_{i}\in(0,1)$ for $i=1,2.$ Then we need to prove $(\max_{1\leq j\leq p}|z_{j}|-A_{n})/B_{n}$ converges weakly to the cdf $\exp(-e^{-x})$, where $A_{n}=c_{n}+\frac{1}{2}(1-c_{n}^{2})^{1/2}(n-1)^{-1/2}a_{n}$, $B_{n}=\frac{1}{2}(1-c_{n}^{2})^{1/2}(n-1)^{-1/2}b_{n}$,

for $x>3$. We proceed this through several steps.

Step 1: Reduction to an easy formulation. Let $U_{i}$, $i\geq 1$ be a sequence of i.i.d. random variables uniformly distributed over $(0,1)$, and $U_{1:n}\leq U_{2:n}\leq\cdots\leq U_{n:n}$ be the order statistics of $U_{1},U_{2},\cdots,U_{n}$ for each $n\geq 1$. From (2.5), the density function of $U_{j:m_{n}+j-1}$ is

Denote the corresponding cdf as $F_{j:m_{n}+j-1}(x)$. Notice the pdf of $(U_{j:m_{n}+j-1})^{1/2}$ is proportional to $x^{2j-1}(1-x^{2})^{m_{n}-1}$. For each $n\geq 2$, let $\{Y_{nj};\,1\leq j\leq p\}$ be independent random variables such that $Y_{nj}$ and $(U_{j:m_{n}+j-1})^{1/2}$ have the same distribution. By Lemma 1.1 and (1.1), $\max_{1\leq j\leq p}|z_{j}|$ and $\max_{1\leq j\leq p}Y_{nj}$ have the same distribution. We claim that, to prove the theorem, it suffices to show

where we use the facts $a_{n}\to\infty$, $b_{n}\to 0$ and $c_{n}\in(0,1)$ in the above. Since $B_{n}$ has the scale of $(n\log n)^{-1/2}$, by (2.20),

weakly, which leads to the desired conclusion. Now we proceed to show (2.19).

for $x\in(0,1).$ In fact, since for each $1<j\leq p$,

which implies that $U_{j-1:m_{n}+j-2}\leq U_{j:m_{n}+j-1}$ for $1<j\leq p$. This yields (2.21).

For each $n\geq 2$, set $a_{nj}=1-F_{p+1-j:m_{n}+p-j}(\beta_{n}(x))=1-F_{p+1-j:n-j}(\beta_{n}(x))$ for $1\leq j\leq p$. From (2.21), for each $n$, $a_{ni}$ is non-increasing in $i$. Since $Y_{nj}^{2}$ and $U_{j:m_{n}+j-1}$ are identically distributed, we have

It is easy to check the following holds: suppose $\{l_{n};\,n\geq 1\}$ is sequence of positive integers. Let $z_{ni}\in[0,1)$ be constants for all $1\leq i\leq l_{n}$ with $\max_{1\leq i\leq l_{n}}z_{ni}\to 0$ and $\sum^{l_{n}}_{i=1}z_{ni}\to z\in[0,\infty)$. Then

Next we will use (2.22) and (2.23) to prove (2.19). In fact, we only need to verify that

Step 3: The analysis of dominated terms. Fix $\delta\in(\frac{1}{2},\frac{2}{3})$. Let $j_{n}=[n^{\delta}]$, the integer part of $n^{\delta}$. For $1\leq j\leq j_{n}$, define

Then we see that uniformly over $1\leq j\leq j_{n}$,

In Lemma 2.3, take $r=n-j$ and $k=n-p$ to have

uniformly over $1\leq j\leq j_{n}$ as $n\to\infty$, where

and where, for $i=1,2$, $l_{i}(t)$ is a polynomial in $t$ of degree $\leq 3i$, depending on $n$, and all of its coefficients are of order $O(n^{-i/2})$ by the assumption $h_{1}<\frac{p}{n}<h_{2}$ for all $n\geq 2.$ Now, by taking $B=(u_{nj},\infty)$ we obtain

uniformly for $1\leq j\leq j_{n}$ as $n\to\infty$. From L’Hospital’s rule, we have that for any $r\geq 0$

Since $\min_{1\leq j\leq j_{n}}u_{nj}\to\infty$ as $n\to\infty$ by (2.26), it follows from (2.27) that

holds uniformly over $1\leq j\leq j_{n}$. Furthermore, since the coefficients of $l_{i}(t)$ are uniformly bounded by $O(n^{-i/2})$ for $i=1,2$, we have

uniformly over $1\leq j\leq j_{n}$, and thus obtain that

uniformly over $1\leq j\leq j_{n}$. Therefore, we have

In Lemma 2.2, by taking $x_{n}=nc_{n}^{2}/(1-c_{n}^{2})$ and $c_{nj}=x_{n}^{-1/2}(1+o(1))$ where “$o(1)$” is as indicated in (2.26), we then get

Step 4: Non-dominated terms are negligible. From (2.26) again, we see

for all large $n$. Then it follows from (2.28) that $a_{nj_{n}}=O(n^{-3/2})$, and hence

This together with (2.30) yields (2.24). The proof is then completed. $\blacksquare$

3 The Proof of Theorem 3

We begin with some preparation. The following result characterizes the structure of the radius of the eigenvalues from the product ensemble.

Let $k$ and $z_{1},\cdots,z_{n}$ be as in (1.5). Let $\{s_{j,r},\,1\leq r\leq k,j\geq 1\}$ be independent random variables and $s_{j,r}$ have the Gamma density $y^{j-1}e^{-y}I(y>0)/(j-1)!$ for each $j$ and $r.$ Then $\max_{1\leq j\leq n}|z_{j}|^{2}$ and $\max_{1\leq j\leq n}\prod_{r=1}^{k}s_{j,r}$ have the same distribution.

Proof. Let $\{s_{j,r};\,1\leq r\leq k,j\geq 1\}$ be independent random variables and $s_{j,r}$ follow a Gamma($j$) distribution with density function $y^{j-1}e^{-y}I(y\geq 0)/\Gamma(j)$ for all $1\leq r\leq k$ and $j\geq 1.$ Define $v_{1}(y)=\exp(-y)$, $y>0$, and set for $j\geq 2$

One can easily verify that for each $j\geq 1$, $v_{j}(y)$ is proportional to $w_{j}(y^{1/2})$, i.e., for some constants $d_{j}>0$,

Let $z$ be any complex number with $\textit{Re}(z)>0$, and define for $j\geq 1$

Note that $\gamma_{1}(z)=\Gamma(z)=\int_{0}^{\infty}y^{z-1}e^{-y}dy$. For $j\geq 2$, by using (2.31),

Assume $Y_{nj}$, $1\leq j\leq n$ are independent random variables, and for each $1\leq j\leq n$, the density of $Y_{nj}$ is proportional to $y^{2j-1}w_{k}(y)$. By Lemma 1.1 and (1.1), $\max_{1\leq j\leq n}|z_{j}|^{2}$ and $\max_{1\leq j\leq n}Y_{nj}^{2}$ are identically distributed. Furthermore, since the density function of $Y_{nj}^{2}$, denoted by $f_{j}(y)$, is proportional to $y^{j-1}w_{k}(y^{1/2})$, and thus proportional to $y^{j-1}v_{k}(y)$ from (2.32), we have from (2.33) that

for $1\leq j\leq n$. Let the characteristic function of $\log Y_{nj}^{2}$ be denoted by $g_{j}(t)$. Then we have

from (2.33). Since $\Gamma(j+it)/\Gamma(j)$ is the characteristic function of $\log s_{j,r}$, it follows that $\log Y_{nj}^{2}$ has the same distribution as that of $\sum^{k}_{r=1}\log s_{j,r}$, or equivalently, $Y_{nj}^{2}$ has the same distribution as that of $\prod^{k}_{r=1}s_{j,r}$ for $j\geq 1$. This implies the desired conclusion. $\blacksquare$

Let $k$ be as in (1.5) and $\{s_{j,r},\,1\leq r\leq k,j\geq 1\}$ be independent r.v.’s such that $s_{j,r}$ has density $y^{j-1}e^{-y}I(y>0)/(j-1)!$ for all $j,r$. Set $\eta(x)=x-1-\log x$ and

Set $\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}$ for $x>0.$ Then for $1\leq i\leq n$

Proof. Set $Y_{j}=\prod^{k}_{r=1}s_{j,r}$ for $j\geq 1.$ Then,

for $j\geq 1$. The moment generating functions of $\log s_{j,r}$ is

by (1.8). Note that $\eta(x)=x-1-\log x$ for $x>0$. Since $\eta(x)=\int^{x}_{1}\frac{s-1}{s}ds$, it is easy to verify that

By using the expression $\log x=x-1-\eta(x)$ we can rewrite $\log Y_{j}$ as

Since $E(\log Y_{j})=k\psi(j)$, we see that

Note that for any two sequences of reals numbers $\{x_{n}\}$ and $\{y_{n}\}$,

Let $k$ be as in (1.5) and $M_{n}(i)$ be defined as in Lemma 2.5. Assume $\{j_{n};\,n\geq 1\}$ is a sequence of numbers satisfying $1\leq j_{n}\leq\frac{1}{2}n$ for all $n$. Then, for any sequence of positive integers $\{k_{n}\}$, M_{n}(j_{n})=O_{P}\Big{(}\frac{j_{n}k_{n}^{1/2}}{n}\Big{)}. Further, if $\lim_{n\to\infty}k_{n}/n=0$, then M_{n}(j_{n})=O_{P}\Big{(}\frac{k_{n}\log n}{n}\Big{)}.

Proof. By using the Minkowski inequality and (2.36) we get

by (2.36). Since $s_{j,1}$ has density $y^{j-1}e^{-y}I(y>0)/(j-1)!$, we see that E\big{(}s_{j,1}^{-4}\big{)}=\frac{\Gamma(j-4)}{\Gamma(j)}. By the Marcinkiewicz-Zygmund inequality (see, for example, Corollary 2 from Chow and Teicher, 2003), we obtain $E(s_{j,1}-j)^{8}\leq Kj^{4}$ for any $j\geq 1$ where $K$ is a constant not depending on $j$. Then, it follows from Hölder’s inequality that

for any $j\geq 4$ where $C$ is a constant. Combining the last two assertions, we get $E(M_{n}(j_{n}))\leq O(\frac{j_{n}k_{n}^{1/2}}{n}).$ This implies the first conclusion.

Now we prove the second one. Recall $\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}$ for $x>0$ as in (1.8). By Formulas 6.3.18 and 6.4.12 from Abramowitz and Stegun (1972),

as $x\to+\infty$. It is easy to check $E\log s_{j,1}=\frac{1}{\Gamma(j)}\int_{0}^{\infty}(\log y)y^{j-1}e^{-y}\,dy=\psi(j)$. Thus, from the first expression, we have

By Theorem 1 on page 217 from Petrov (1975), we have that

uniformly for $x\in(0,a_{n})$ and $n/2\leq j\leq n$ as $n\to\infty$, where $\{a_{n};\,n\geq 1\}$ is an arbitrarily given sequence of positive numbers with $a_{n}=o(n^{1/6})$. By taking $r=0$ in (2.27), we see that $1-\Phi(x)\sim\frac{1}{\sqrt{2\pi}\,x}e^{-x^{2}/2}$ as $x\to+\infty.$ Now select $x=2(\log n)^{1/2}$ in (2.40) to have

uniformly for $n/2\leq j\leq n$ as $n\to\infty$. Similarly we have

uniformly for $n/2\leq j\leq n$ as $n\to\infty$. This implies

proving the second conclusion. $\blacksquare$

Review the notation we use before: $\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}$ for $x>0$ as in (1.8) and

for $j\geq 1$, where $\{s_{j,r},\,1\leq r\leq k,j\geq 1\}$ are independent random variables such that $s_{j,r}$ has density $y^{j-1}e^{-y}I(y>0)/(j-1)!$ for all $j,r$.

From (2.38), there exist an integer $j_{0}$ such that for all $j_{0}\leq j\leq n-j_{n}$

By the first inequality above, for all large $n$,

Hence, by assumption $(\frac{k_{n}}{n})^{1/2}=o(\frac{j_{n}k_{n}}{n})$ we see that

for all large $n$. Therefore we have for $j_{0}\leq j\leq n-j_{n}$

for all $t>0$ and large $n$ which does not depend on $t.$ By selecting $t=0.99j(\log n-\log j)$ we have

where the last three minima are taken over all real numbers satisfying the corresponding constraints. It is easily seen that the minimum of $s(\log n-2\log s)$ for $j_{0}^{1/2}\leq s\leq(n-j_{n})^{1/2}$ is achieved at the two end points of the interval, $t=j_{0}^{1/2}$ or $s=(n-j_{n})^{1/2}$. Thus, for all large $n$,

From the given condition $(\frac{k_{n}}{n})^{1/2}\frac{j_{n}}{(\log n)^{1/2}}=\infty$, we obtain

for all large $n$. Therefore, combining all of the inequalities from (2.43) to the above, we have

Finally, observe that, for each $1\leq j<j_{0}$, $\log Y_{j}$ is a sum of $k_{n}$’s many i.i.d. random variables with $Ee^{t\log Y_{j}}<\infty$ for all $|t|<\frac{1}{2}$. Then, by the Chernoff bound (see, for instance, p. 27 from Dembo and Zeitouni, 1998),

The last two assertions imply the desired result. $\blacksquare$

Let $\psi(x)$ be as in (1.8), $a(x)$ and $b(x)$ be as in Theorem 2.1, and $z_{j}$’s and $k_{n}$ be as in Theorem 3. Define $\Phi_{0}(y)=\Lambda(y)$, $a_{n}=a(n/k_{n})$, $b_{n}=b(n/k_{n})$ if $\alpha=0$, and $a_{n}=0$, $b_{n}=1$ if $\alpha\in(0,\infty]$. Then

Proof. For each of the three cases: $\alpha=0$, $\alpha\in(0,\infty)$, and $\alpha=\infty$ we will show that there exists a sequence of positive integers $\{j_{n}\}$ with $1\leq j_{n}\leq n/2$ such that

where $M_{n}(\cdot)$ is defined as in Lemma 2.5, and

Review the definition of $Y_{j}$ in (2.41). The above result together with (2.46), Lemmas 2.4 and 2.5 implies that

the two limits above imply (2.44) due to the fact that $\max_{1\leq j\leq n}\log|z_{j}|$ and $\frac{1}{2}\max_{1\leq j\leq n}\log Y_{j}$ are identically distributed by Lemma 2.4.

Now we start to verify equations (2.45)-(2.47) with a choice of $j_{n}$ given by

Proof of (2.45). It is easy to verify that the conditions in Lemma 2.7 are satisfied, and thus (2.42) holds. In case $\alpha\in(0,\infty]$, $a_{n}=0$ and $b_{n}=1$, and (2.45) holds in this case. When $\alpha=0$, $a_{n}+b_{n}y>0$ for all large $n$, by applying (2.42) with $x=0$ we have

that is, (2.45) holds. This completes the proof of (2.45) for all three cases.

Proof of (2.46). To prove (2.46), it suffices to show $M_{n}(j_{n})=O_{P}((\frac{k_{n}}{n})^{1/2}(\log n)^{-1})$ since since $b_{n}\geq(\log n)^{-1/2}$ for all large $n.$ We use Lemma 2.6 this time. When $\alpha\in(0,\infty]$, $j_{n}=O_{P}(n^{1/8})$ from (2.49), and then we have from the first conclusion in Lemma 2.6 that

When $\alpha=0$, we have from the two conclusions in Lemma 2.6 that

since $\frac{n^{1/8}}{k_{n}^{1/2}}\leq n^{-1/8}$ if $k_{n}\geq n^{1/2}$ and $\frac{k_{n}^{1/2}\log n}{n^{1/2}}\leq n^{-1/8}$ if $k_{n}<n^{1/2}$.

Proof of (2.47). Set T_{n}(j_{n})=\max\limits_{n-j_{n}+1\leq j\leq n}\big{(}\frac{1}{j}\sum^{k_{n}}_{r=1}(s_{j,r}-j)+k_{n}\psi(j)\big{)}. Then

Notice $\sum^{k_{n}}_{r=1}s_{j,r}$ is a sum of $jk_{n}$ i.i.d. random variables with distribution $\mbox{Exp}(1)$, that is, it has density $e^{-x}I(x\geq 0).$ Since the mean and the variance of $\mbox{Exp}(1)$ are both equal to $1$, we normalize the sum by

By Theorem 1 on page 217 from Petrov (1975), for any sequence of positive numbers $\delta_{n}$ such that $\delta_{n}=o((nk_{n})^{1/6})$,

uniformly over $x\in[0,\delta_{n}]$ and $n/2\leq j\leq n$ as $n\to\infty$. Now reorganize the index in (2.51) to obtain

where a_{ni}=P\big{(}W_{n-i+1}>x_{n,i}\big{)} and

Recalling (2.49), we know $j_{n}=o(n)$. From the second expression in (2.38) we have

uniformly over $1\leq i\leq j_{n}$ as $n\to\infty$. It follows that

uniformly over $1\leq i\leq j_{n}$ as $n\to\infty$. Since $a_{n}+b_{n}y=O((\log n)^{1/2})$,

uniformly over $1\leq i\leq j_{n}$. Therefore, by combining the above two expansions we get

uniformly over $1\leq i\leq j_{n}$. We emphasize the above is true when $i=j_{n}=1$, which can be seen directly from (2.54). This fact will be used later.

Finally, we prove (2.47) by considering the three cases: $\alpha=0$, $\alpha\in(0,\infty)$ and $\alpha=\infty$.

uniformly over $1\leq i\leq j_{n}$. In Lemma 2.2, choose $x_{n}=n/k_{n}$, $j_{n}$ as in (2.49) and c_{nj}=(1+O(n^{-3/8}))\big{(}\frac{k_{n}}{n}\big{)}^{1/2} as in (2.55) to obtain

Further, it is easily seen that $\max_{1\leq i\leq j_{n}}a_{ni}\to 0$. Applying (2.23) to (2.53), we arrive at

Case 2: We see that $j_{n}\sim\alpha^{-1/2}n^{1/8}$ from (2.49). By definition, $a_{n}=0$ and $b_{n}=1$. Then it follows from (2.55) that

holds uniformly over $1\leq i\leq j_{n}$ as $n\to\infty$. We claim that

uniformly over $1\leq i\leq j_{n}$. In fact, review that (2.52) holds if $0<x=o(n^{1/3})$. Evidently, $\max_{1\leq i\leq j_{n}}|x_{n,i}|=O(n^{1/8}).$ But there is a possibility that $x_{n,i}<0$ for small values of $i$. Let $j_{0}>1$ be an integer such that $\min_{j_{0}\leq i\leq j_{n}}x_{n,i}>0$. Then we have from (2.52) that (2.56) holds uniformly over $j_{0}\leq i\leq j_{n}$. By using the standard central limit theorem, we know (2.52) holds as well for each $i=1,\cdots,j_{0}-1$. Therefore, for each $i\geq 1$,

by the fact $1-\Phi(x)\sim\frac{1}{\sqrt{2\pi}x}e^{-x^{2}/2}$ as $x\to+\infty$. We now apply Lemma 2.1 to show (2.47). By defining $a_{ni}=0$ for all $i>j_{n}$, with (2.57), we only need to verify the following two conditions: $\sup_{n\geq n_{0},1\leq i\leq j_{n}}a_{ni}<1$ for some integer $n_{0}$ and $\lim_{n\to\infty}\sum^{j_{n}}_{i=1}a_{ni}=\sum^{\infty}_{i=1}(1-\Phi(\alpha^{-1/2}(i-1)+y))$. The first one follows from (2.56) and the fact that $x_{n,i}\geq\frac{1}{2}\alpha^{1/2}(i-1)+y\geq y$ for $1\leq i\leq j_{n}$ for all large $n$. The second condition can be easily verified by the dominated convergence theorem since $a_{ni}\leq 2(1-\Phi(\frac{1}{2}\alpha^{1/2}(i-1)+y))$ for all $1\leq i\leq j_{n}$ as $n$ is sufficiently large and $\sum^{\infty}_{i=1}2(1-\Phi(\frac{1}{2}\alpha^{1/2}(i-1)+y)<\infty$.

Case 3: $\alpha=\infty$. From (2.49), $0\leq(\frac{k_{n}}{n})(j_{n}-1)\leq n^{1/8}$ and thus $x_{n,i}=O(n^{1/8})$ by (2.55). In particular, we have $x_{n,1}=y$, and for all large $n$, $x_{n,i}>0$ if $2\leq i\leq j_{n}$ and $j_{n}\geq 2$. Therefore,

uniformly over $1\leq i\leq j_{n}$ from (2.52). From (2.55), $a_{n1}\to 1-\Phi(y)$ as $n\to\infty$. Obviously, $x_{n,i}\geq\frac{i}{3}(\frac{k_{n}}{n})^{1/2}$ if $2\leq i\leq j_{n}$ and $j_{n}\geq 2$. Thus, use the fact $1-\Phi(x)\sim\frac{1}{\sqrt{2\pi}\,x}e^{-x^{2}/2}$ as $x\to+\infty$ to see that, for large $n$,

since \exp\big{\{}-\frac{k_{n}}{18n}i^{2}\big{\}}\leq\int_{i-1}^{i}\exp\big{\{}-\frac{k_{n}}{18n}x^{2}\big{\}}\,dx for all $i\geq 2.$ Thus, $I(j_{n}\geq 2)\sum^{j_{n}}_{i=2}a_{ni}\to 0$. This and the fact $I(j_{n}\geq 2)\cdot\max_{2\leq i\leq j_{n}}a_{ni}\to 0$ imply that I(j_{n}\geq 2)\big{(}1-\prod^{j_{n}}_{i=2}(1-a_{ni})\big{)}\to 0 as $n\to\infty$. So we have from (2.53) that