For any $a,b\in\mathbb{R}$, we denote $a\vee b$ and $a\wedge b$ as the maximum and minimum of $a$ and $b$, respectively. For any $a\in\mathbb{R}$, we denote $\lfloor a\rfloor$ as the largest integer strictly smaller than $a$. For any sequences $a_{n}$ and $b_{n}$, $a_{n}=o(b_{n})$ denotes $a_{n}/b_{n}\to 0$ as $n\to\infty$. We denote $a_{n}=O(b_{n})$, or equivalently $a_{n}\lesssim b_{n}$, if $a_{n}\leq Cb_{n}$ for some constant $C>0$, where $C$ is an universal constant. We denote the indicator function for a set $A$ as $I(\cdot\in A)$ or $I_{A}(\cdot)$. For a given $p$-dimensional vector $u=(u_{1},\ldots,u_{p})^{T}$ and set $S\subseteq\{1,\ldots,p\}$, we define $u_{S}=(u_{j})^{T}_{j\in S}\in\mathbb{R}^{|S|}$, where $|S|$ is the cardinality of $S$. For given index sets $S$, $S^{\prime}\subseteq\{1,\ldots,p\}$ and real matrix $B\in\mathbb{R}^{p\times p}$, we denote $B_{(S,S^{\prime})}$ as the $|S|\times|S^{\prime}|$ submatrix consisting only of $S$th columns and $S^{\prime}$th rows of $B$, and let $B_{S}=B_{(S,S)}$. For a real matrix $B$, we denote $S_{B}$ as the index set for nonzero elements of $B$ and call $S_{B}$ the support of $B$. We define $\mathcal{C}_{p}$ as the class of all $p\times p$ dimensional positive definite matrices. For any $p\times p$ symmetric matrix $B$, $\lambda_{\min}(B)$ and $\lambda_{\max}(B)$ are the minimum and maximum eigenvalues of $B$, respectively.

For any $p$-dimensional vector $u=(u_{1},\ldots,u_{p})^{T}$, we define vector norms $\|u\|_{1}=\sum_{j=1}^{p}|u_{j}|$, $\|u\|_{2}=(\sum_{j=1}^{p}u_{j}^{2})^{1/2}$ and $\|u\|_{\max}=\max_{1\leq j\leq p}|u_{j}|$. For any $p\times p$ matrix $B=(b_{ij})$, we define the spectral norm, matrix $\ell_{1}$ norm, matrix $\ell_{\infty}$ norm and Frobenius norm by

respectively.

For a given positive integer $m$, we denote $\chi^{2}_{m}$ as the chi-square distribution with degrees of freedom $m$. For any random variables $Y_{1},Y_{2}$ and $Y_{3}$, we denote $Y_{1}\overset{d}{\equiv}Y_{2}\oplus Y_{3}$ if the distribution of $Y_{1}$ is equal to that of $Y_{2}+Y_{3}$, and $Y_{2}$ and $Y_{3}$ are independent. For given positive numbers $a$ and $b$, $Gamma(a,b)$ and $IG(a,b)$ are the gamma distribution and inverse-gamma distribution with shape parameter $a$ and rate parameter $b$, respectively. $Beta(a,b)$ is the beta distribution whose density function at $x\in(0,1)$ is proportional to $x^{a-1}(1-x)^{b-1}$. We denote $N_{p}(X\mid\mu,\Sigma)$ as the density function of $N_{p}(\mu,\Sigma)$ at $X\in\mathbb{R}^{p}$. We denote the inverse-Wishart distribution by $IW_{p}(\nu,\Phi)$, where the degree of freedom and scale matrix are $\nu>p-1$ and $\Phi\in\mathcal{C}_{p}$, respectively.

We consider the model

where $\Omega_{n}=\Sigma_{n}^{-1}$ is a $p\times p$ precision matrix and $X_{i}=(X_{i1},\ldots,X_{ip})^{T}\in\mathbb{R}^{p}$ for all $i=1,\ldots,n$. The MCD guarantees that there exists unique lower triangular matrix $A_{n}=(a_{jl})$ and diagonal matrix $D_{n}=diag(d_{j})$ such that

where $a_{jj}=0$ and $d_{j}>0$ for all $j=1,\ldots,p$. Let $S_{A_{n}}$ be the support of the Cholesky factor $A_{n}$, and $S_{j}$ be the support of the $j$th row of $A_{n}$. Let $\mathbb{P}_{\Omega_{n}}$ and $\mathbb{E}_{\Omega_{n}}$ be the probability measure and expectation corresponding to the model (12), respectively.

The model (12) can be interpreted as a Gaussian DAG model depending on the sparsity pattern of $A_{n}$. For a set of vertices $V=\{1,\ldots,p\}$ and a set of directed edges $E$, a graph $\mathcal{D}=(V,E)$ is said to be a DAG if there is no directed cycles. It is also called the Bayesian network or belief network. In this paper, we assume that the variables have a known natural ordering in which no edges exist from larger vertices to smaller vertices. It has been commonly assumed in literature including Shojaie and Michailidis (2010), Ben-David et al. (2015), Khare et al. (2016), Yu and Bien (2017) and Cao, Khare and Ghosh (2017). There are relatively few works on DAG models when the ordering of variables is unknown (Kalisch and Bühlmann, 2007; Rütimann and Bühlmann, 2009; van de Geer and Bühlmann, 2013). As discussed in van de Geer and Bühlmann (2013), when the ordering is unknown, a very different technique is needed relative to the known ordering case.

For $i\in V$, define the set of all $i$’s parents as the subset of $V$ smaller than $i$ and sharing an edge with $i$ and denote it as $pa_{i}(\mathcal{D})$. Any multivariate Gaussian distribution that obeys the directed Markov property with respect to a DAG $\mathcal{D}$ is said to be a Gaussian DAG model over $\mathcal{D}$. To be specific, if $X=(X_{1},\ldots,X_{p})^{T}\sim N_{p}(0,\Omega^{-1})$ and $N_{p}(0,\Omega^{-1})$ belongs to a Gaussian DAG model over $\mathcal{D}$, then

for each $j=1,\ldots,p$. Furthermore, if we adopt the MCD as in (2), with the known ordering of variables, $N_{p}(0,\Omega^{-1})$ belongs to a Gaussian DAG model over $\mathcal{D}$ if and only if $a_{jl}=0$ whenever $l\notin pa_{j}(\mathcal{D})$ (Cao, Khare and Ghosh, 2017). In other words, the support of $A$ uniquely determines a DAG $\mathcal{D}$ under the known ordering assumption.  The model $X=(X_{1},\ldots,X_{p})^{T}\sim N_{p}(0,\Omega^{-1})$ given $S_{A}$ is equivalent to a Gaussian DAG model, and it can be represented as a linear autoregressive model,

where $a_{S_{j}}=a_{j,S_{j}}=(a_{jj^{\prime}})_{j^{\prime}\in S_{j}}^{T}$. For more details on the expression (3), refer to Bickel and Levina (2008) and Ben-David et al. (2015). The autoregressive model interpretation enables us to adopt the priors introduced in the linear regression literature. Since $a_{S_{j}}$ corresponds to nonzero elements among $a_{j}=(a_{j1},\ldots,a_{j,j-1})^{T}$, one can use a prior designed for sparse regression coefficient vectors for $a_{j}$, which is our strategy introduced in section 2.3.

In this paper, we consider the high-dimensional setting where $p=p_{n}$ is a function of $n$ increasing to infinity as $n\to\infty$ and $p\geq n$. We assume that the data were generated from a true precision matrix $\Omega_{0n}$, where $\Sigma_{0n}=\Omega_{0n}^{-1}$ is the true covariance matrix. Denote the MCD (2) of the true precision matrix by $\Omega_{0n}=(I_{p}-A_{0n})^{T}D_{0n}^{-1}(I_{p}-A_{0n})$, where $A_{0n}=(a_{0,jl})$, $a_{0j}=(a_{0,j1},\ldots,a_{0,j,j-1})^{T}$ and $D_{0n}=diag(d_{0j})$. For notational convenience, let $\mathbb{P}_{0}=\mathbb{P}_{\Omega_{0n}}$ and $\mathbb{E}_{0}=\mathbb{E}_{\Omega_{0n}}$.

We now define some notations related to the data set. Let ${\bf X}_{n}=(X_{1},\ldots,X_{n})^{T}$ $\in\mathbb{R}^{n\times p}$ be the data of size $n$, and $\tilde{X}_{j}=(X_{1j},\ldots,X_{nj})^{T}\in\mathbb{R}^{n}$ be the $j$th column of ${\bf X}_{n}$. For a given index set $S\subseteq\{1,\ldots,p\}$, let ${\bf X}_{S}=(\tilde{X}_{j})_{j\in S}\in\mathbb{R}^{n\times|S|}$ be the data matrix consisting only of $S$th columns of ${\bf X}_{n}$. Let $Z_{ij}=(X_{i1},\ldots,X_{i,j-1})^{T}\in\mathbb{R}^{j-1}$ and $\tilde{Z}_{j}=(Z_{1j},\ldots,Z_{nj})^{T}\in\mathbb{R}^{n\times(j-1)}$ for all $j=2,\ldots,p$.

For a given positive integer $1\leq s\leq p$, we define $\Psi_{\max}(s)^{2}=\sup_{S:0<|S|\leq s}\lambda_{\max}({\bf X}_{S}^{T}{\bf X}_{S})$ and $\Psi_{\min}(s)^{2}=\inf_{S:0<|S|\leq s}\lambda_{\min}({\bf X}_{S}^{T}{\bf X}_{S})$, where the supremum and infimum are taken over all index sets $S\subseteq\{1,\ldots,p\}$. We say that the restricted eigenvalue condition is met for some integer $s$ if $n^{-1}\Psi_{\min}(s)^{2}$ is bounded away from zero uniformly for all large $n$. This condition has been used in the high-dimensional regression literature to control the behavior of the design matrix. The autoregressive model representation (3) connects the eigenvalues of the precision matrix $\Omega_{0n}$ with those of the design matrix in (3) because the quantity ${\bf X}_{S_{j}}$ corresponds to the design matrix based on the representation. Thus, the bounded eigenvalue assumption (A1) in section 2.5 essentially corresponds to the restricted eigenvalue condition.

In this paper, we suggest the following prior distribution for our model:

for some positive constants $\nu_{0},c_{1},c_{2},R_{2},\ldots,R_{p}$ and $\gamma$, where $f_{nj}$ is a probability mass function on $\{0,1,\ldots,R_{j}\wedge(j-1)\}$ and $\widehat{a}_{S_{j}}=\big{(}{\bf X}_{S_{j}}^{T}{\bf X}_{S_{j}}\big{)}^{-1}{\bf X}_{S_{j}}^{T}\tilde{X}_{j}$. The proposed prior is empirical in the sense that it depends on the data, so we call the prior (4) the empirical sparse Cholesky (ESC) prior. To obtain the desired asymptotic properties, appropriate conditions for hyperparameters in (4) will be introduced in section 3. Note that the prior for $d_{j}$ can be generalized to the proper prior $IG(\nu_{0}/2,\nu_{0}^{\prime})$ for some constant $\nu_{0}^{\prime}>0$ and the results in section 3 also hold for this prior choice. However, for computational convenience, we describe and prove the main results with the improper prior $\pi(d_{j})\propto d_{j}^{-\nu_{0}/2-1}$.

For the conditional prior of $a_{j}$ given $d_{j}$, we first introduce zero components through the prior $\pi_{j}$ and impose the Zellner’s g-prior (Zellner, 1986) on the nonzero components, $a_{S_{j}}$. The use of Zellner’s g-prior simplifies the calculation of the marginal posterior for $S_{j}$. Martin, Mess and Walker (2017) suggested a similar prior in the high-dimensional linear regression model. Also note that the ESC prior has a connection to the DAG-Wishart prior (Ben-David et al., 2015; Cao, Khare and Ghosh, 2017). Theorem 7.3 in Ben-David et al. (2015) shows that the DAG-Wishart prior on $(A_{n},D_{n})$ given a DAG implies the inverse-gamma distribution on $d_{j}$ and multivariate normal distribution on the nonzero elements of $a_{j}$ given $d_{j}$, where $(a_{j},d_{j})$ are mutually independent for all $j=1,\ldots,p$. Thus, the ESC prior (4) is quite close to the DAG-Wishart prior when the support of $A_{n}$ is given.

Cao, Khare and Ghosh (2017) used the DAG-Wishart prior to recover the sparse DAG and estimate the precision matrix in high-dimensional settings. Thus, their prior on $(A_{n},D_{n})$ is quite close to ours, and can be viewed as a set of priors for autoregressive model (3) as discussed in the previous paragraph. For the support of DAGs, they imposed the element-wise sparsity using independent Bernoulli distributions with the hyperparameter $q_{n}$, which has a nice interpretation as the individual edge probability. Based on the hierarchical DAG-Wishart prior, they obtained the strong model selection consistency for the DAG and the posterior convergence rate for the precision matrix with respect to the spectral norm. However, they did not directly adopt the autoregressive model interpretation as in (3), which is different from our approach. By using the ESC prior, we can adopt state-of-the-art techniques on selection consistency for the regression coefficient (Martin, Mess and Walker, 2017) and achieve the strong model selection consistency under much weaker conditions than those in Cao, Khare and Ghosh (2017). Furthermore, compared to the existing literature, we obtain faster posterior convergence rates for precision matrices and Cholesky factors under weaker conditions using the techniques introduced by Lee and Lee (2018, 2017) and Martin, Mess and Walker (2017). Indeed, the posterior convergence rates for Cholesky factors are nearly or exactly optimal depending on the relative size of true sparseness for the entire dimension.

We suggest adopting the fractional likelihood with power $\alpha\in(0,1)$,

The use of fractional likelihood has received increased attention in recent years (Martin and Walker, 2014; Syring and Martin, 2016; Miller and Dunson, 2018). In this paper, we use the fractional likelihood mainly because of its appealing theoretical properties under relatively weaker conditions compared to the actual posterior (Bhattacharya, Pati and Yang, 2018). In the proof of the main results of this paper, the use of the fractional likelihood enables us to effectively deal with the ratio of estimated residual variances $\widehat{d}_{S_{j}}$ (the proof of Theorem 3.1) and the ratio of likelihood $L_{nj}(a_{j},d_{j})$ (the proof of Lemma 10.2), where $\widehat{d}_{S_{j}}$ and $L_{nj}(a_{j},d_{j})$ will be defined later.

Here we give a more detailed justification of using the fractional likelihood. The proposed conditional prior for $a_{S_{j}}$ in (4) tracks the data closely because it is centered at the least square estimate. It may cause the unexpected inconsistency (Walker and Hjort, 2001). The fractional likelihood approach can prohibit it by preventing the posterior from following the data too closely. To be more specific, the use of fractional likelihood can be interpreted as an empirical Bayes procedure by considering

Hence, the resulting posterior consists of an ordinary likelihood function and a data-dependent prior $\pi(A_{n},D_{n})/L_{n}(A_{n},D_{n})^{1-\alpha}$. Note that the power $\alpha$ only appears in the prior. From this point of view, the prior is rescaled by a fractional likelihood which has an effect of penalizing parameter values that track the data too closely, while the penalty effect is controlled by the hyperparameter $\alpha$.

The choice of $\alpha$ can be important from a practitioner’s point of view even though theoretical results in this paper hold for any choice of $0<\alpha<1$. In practice, we suggest using $\alpha$ close to 1 to mimic the usual likelihood in finite sample scenario if there is no suspect of model failure, i.e. misspecification. As long as one chooses $\alpha$ sufficiently close to 1, e.g. $\alpha=0.999$ or $\alpha=0.9999$, our experience confirms that the $\alpha$-posterior can be hardly distinguishable from the “usual” posterior even in a finite sample scenario.

The prior (4) and fractional likelihood (5) lead to the following joint posterior distribution,

where $\widehat{d}_{S_{j}}=n^{-1}\tilde{X}_{j}^{T}(I_{n}-\tilde{P}_{S_{j}})\tilde{X}_{j}$ and $\tilde{P}_{S_{j}}={\bf X}_{S_{j}}({\bf X}_{S_{j}}^{T}{\bf X}_{S_{j}})^{-1}{\bf X}_{S_{j}}^{T}$. We refer to the posterior (6) as the $\alpha$-posterior and denote it by $\pi_{\alpha}(\cdot\mid{\bf X}_{n})$ to clarify that we consider the $\alpha$-fractional likelihood. Throughout the paper, $\alpha\in(0,1)$ is a fixed constant.

For given positive constants $0<\alpha<1$, $0<\epsilon_{0}<1/2$, $C_{\rm bm}$ and a sequence of positive integers $s_{0}$, we introduce conditions (A1)-(A4) for the true precision matrix:

Condition (A1) ensures that the eigenvalues of $\Omega_{0n}$ are bounded by fixed constants, which has been commonly used for the estimation of the high-dimensional precision matrices (Ren et al., 2015; Cai, Liu and Zhou, 2016; Banerjee and Ghosal, 2015) as well as the high-dimensional DAGs (Yu and Bien, 2017; Khare et al., 2016). In this paper, condition (A1) is mainly used to get upper bounds of $d_{0j}$, $d_{0j}^{-1}$ and $\|A_{0n}\|$.

Condition (A2) restricts the number of nonzero elements in each row of $A_{0n}$ to be smaller than $s_{0}$. Note that $s_{0}$ may increase to infinity as $n$ gets larger. In our setting, it is equivalent to say that the cardinality of $pa_{j}(\mathcal{D}_{0})$ is less than $s_{0}$ for any $j=2,\ldots,p$, where $\mathcal{D}_{0}$ is the DAG corresponding to $A_{0n}$.

Condition (A3) is the well-known beta-min condition, which determines the lower bound for the nonzero signals. The beta-min condition has been used for the exact support recovery of the high-dimensional linear regression coefficients (Wainwright, 2009a; Bühlmann and van de Geer, 2011; Castillo, Schmidt-Hieber and van der Vaart, 2015; Yang, Wainwright and Jordan, 2016; Martin, Mess and Walker, 2017) as well as the high-dimensional DAGs (Khare et al., 2016; Yu and Bien, 2017; Cao, Khare and Ghosh, 2017).

Condition (A4) restricts the number of nonzero elements in each column of $A_{0n}$ to be smaller than $s_{0}$. In other words, the number of edges directed from any vertex is less than $s_{0}$. This assumption is required to deal with the posterior probability of $\|A_{n}-A_{0n}\|_{1}$. Note that if we consider only the banded structure for the Cholesky factor as in Yu and Bien (2017), conditions (A2) and (A4) automatically hold for some $s_{0}$.

Now, we define a class of precision matrices

In section 3, we show that one can achieve the strong model selection consistency for any $\Omega_{0n}\in\mathcal{U}_{p}$. Furthermore, we derive the posterior convergence rates for $A_{0n}$ and show that these are optimal or nearly optimal for the class $\mathcal{U}_{p}$ (or $\mathcal{U}_{p}$ without condition (A3)).

When the recovery of the DAG is of interest, it is desirable to use a Bayesian procedure that guarantees the strong model selection consistency. We show that the $\alpha$-posterior warrants this property under mild conditions. As mentioned earlier, the Gaussian DAG model has an interpretation as a sequence of autoregressive model (3), which enables us to adopt the state-of-the-art techniques for the selection consistency of the regression coefficient in Martin, Mess and Walker (2017).

To use the results in Martin, Mess and Walker (2017), there are two main issues that need to be addressed. The first is the restricted eigenvalue condition for the design matrix. In our setting, the design matrices consist of columns of data matrix ${\bf X}_{n}$, thus each row follows a multivariate normal distribution. We show that under the bounded eigenvalue condition (A1), the restricted eigenvalue condition for any integer $R=o(n)$ automatically holds on some large set $N^{c}$ having $\mathbb{P}_{0}$-probability tending to 1 (Lemma 9.1 in Supplementary Material). A similar result appears in Narisetty and He (2014). The second issue is more challenging than the first. Martin, Mess and Walker (2017) considered only the known (fixed) residual variance case, which corresponds to the known $d_{0j}$ case in our setting. The assumption on the known residual variance results in a relatively straightforward proof for selection consistency. We extended their techniques to the unknown residual variance case by applying (non-central) chi-square concentration inequalities for the estimated residual variances $\widehat{d}_{S_{j}}$ for some index set $S_{j}$, which is motivated by Shin, Bhattacharya and Johnson (2015). It reveals that the ratio of the marginal posteriors $\pi_{\alpha}(S_{j}\mid{\bf X}_{n})/\pi_{\alpha}(S_{0j}\mid{\bf X}_{n})$ actually behaves like the ratio of the conditional posteriors given $d_{0j}$, $\pi_{\alpha}(S_{j}\mid d_{0j},{\bf X}_{n})/\pi_{\alpha}(S_{0j}\mid d_{0j},{\bf X}_{n})$, with $\mathbb{P}_{0}$-probability tending to 1, where $S_{0j}$ is the index set for the nonzero elements in the $j$th row of $A_{0n}$.

We also note here that unlike the Lasso type (or its variants) of results with the random design matrix (Wainwright, 2009b), our theory does not require the irrepresentable condition on the true covariance matrix. For example, Yu and Bien (2017) and Khare et al. (2016) require the irrepresentable condition for the asymptotic properties of estimators in DAG models. See section IV of Wainwright (2009b) for more details on the irrepresentable condition.

The assumption $s_{0}\log p=o(n)$ or $s_{0}\log p\leq cn$ for some constant $c>0$ is widely used in the high-dimensional sparse covariance or precision matrix estimation literature. In Theorem 3.1, we assume less restrictive condition $s_{0}\log p\leq n\,c_{3}/2$, which automatically guarantees $s_{0}\leq R_{j}$ for all $j=2,\ldots,p$. Note that the constant $c_{3}$ is defined in Condition (P).

It is worthwhile to compare our result to those of Cao, Khare and Ghosh (2017), Yu and Bien (2017) and Khare et al. (2016). Note that in these works it is also assumed that the ordering of variables is known. Cao, Khare and Ghosh (2017) showed the strong model selection consistency using the hierarchical DAG-Wishart prior. They assumed variants of conditions (A1), (A2) and (A3). First, they relaxed condition (A1) by letting $\epsilon_{0,n}\to 0$ such that $(\log p/n)^{1/2-1/(2+k)}=o(\epsilon_{0,n}^{4})$ for some $k>0$, instead of a fixed $\epsilon_{0}>0$. Second, they assumed the same condition (A2) but further assumed $s_{0}^{2+k}\sqrt{\log p/n}=o(1)$ and $(\log p/n)^{k/(4k+8)}\log n=o(1)$ and considered only the DAGs with the total number of edges at most $8^{-1}s_{0}(n/\log p)^{(1+k)/(2+k)}$, which can be restrictive. Note that, when $p\geq n$, it does not include the banded Cholesky factor having $s_{0}$ nonzero elements for each row. Third, they assumed somewhat strong beta-min condition compared with (A3), which requires $\min_{j,l:a_{0,jl}\neq 0}|a_{0,jl}|^{2}\geq M_{n}^{2}s_{0}^{2}\epsilon_{0,n}^{-1}\,(\log p/n)^{1/(2+k)}$ for some $k>0$ and some sequence $M_{n}\to\infty$. Thus, their assumptions on the tuple $(n,p,s_{0})$ as well as the parameter class are much more restrictive than ours, except for the bounded eigenvalue condition. Furthermore, the choice of hyperparameter in the hierarchical DAG-Wishart prior depends on the unknown sparsity parameter $s_{0}$, thus it is not adaptive to the unknown parameter. More specifically, the hyperparameter $q_{n}$ in the hierarchical DAG-Wishart prior should be set at $q_{n}=s_{0}(\log p/n)^{1/(2+k)}$ for some $k>0$ to achieve the strong model selection consistency.

Yu and Bien (2017) suggested a penalized maximum likelihood estimation for the Cholesky factor of the precision matrix and proved the exact signed support recovery under the condition $\rho^{-2}\|D_{0n}\|\epsilon_{0}^{-1}(12\pi^{2}s_{0}+32)\log p<n$. They considered the class of precision matrices satisfying condition (A1) and having a banded structure with the row-specific bandwidths $s_{0j}=|S_{0j}|$ such that $a_{0,jl}=0$ for all $1\leq l<j-s_{0j}$ and $2\leq j\leq p$. Thus, by taking $s_{0}=\max_{j}s_{0j}$, their class satisfies conditions (A2) and (A4). They also assumed the beta-min condition, $\min_{j,l:a_{0,jl}\neq 0}|d_{0j}^{-1/2}a_{0,jl}|\geq 8\rho^{-1}\sqrt{2\|D_{0n}\|\log p/n}\big{(}4\max_{j}\|\Sigma_{0n,S_{0j}}^{-1}\|_{\infty}+5\epsilon_{0}^{-1}\big{)}$. In general, it holds that $\|\Sigma_{0n,S_{0j}}^{-1}\|_{\infty}=O(s_{0j}^{1/2})$ without further assumption, thus the above condition implies that the minimum nonzero $|d_{0j}^{-1/2}a_{0,jl}|$ is bounded below by $\sqrt{s_{0}\log p/n}$ with respect to a constant multiple, thus stronger than condition (A3). Furthermore, they assumed the irrepresentable condition