[proofplan] The estimator error is decomposed as the inverse sample second-moment matrix multiplied by the sample average of the orthogonality moments $X_i u_i$. We first prove the needed [weak law of large numbers](/theorems/1127) for integrable finite-dimensional random vectors, then apply it to $X_iX_i^\top$ and $X_i u_i$. Since the population matrix $Q$ is positive definite, matrices sufficiently close to $Q$ are invertible and their inverses are uniformly bounded. The product representation then implies that $\hat{\beta}_n-\beta_0$ converges to $0$ in probability, while the event on which the sample matrix is singular has probability tending to $0$. [/proofplan]

[step:Prove the weak law for integrable finite-dimensional random vectors] Let $d \in \mathbb{N}$ and let $Z_1,Z_2,\dots : \Omega \to \mathbb{R}^d$ be i.i.d. random vectors with $\mathbb{E}[|Z_1|] < \infty$. We claim that \begin{align*} \frac{1}{n}\sum_{i=1}^n Z_i \xrightarrow{\mathbb{P}} \mathbb{E}[Z_1]. \end{align*} For $M > 0$, define the truncated random vector $Z_i^{(M)} : \Omega \to \mathbb{R}^d$ by \begin{align*} Z_i^{(M)} := Z_i\,\mathbb{1}_{\{|Z_i|\le M\}}. \end{align*} Then $|Z_i^{(M)}| \le M$ and \begin{align*} \mathbb{E}\left[\left|Z_1 - Z_1^{(M)}\right|\right] = \mathbb{E}\left[|Z_1|\mathbb{1}_{\{|Z_1|>M\}}\right] \to 0 \end{align*} as $M \to \infty$, by integrability of $|Z_1|$. Fix $\varepsilon > 0$. Choose $M > 0$ such that \begin{align*} \mathbb{E}\left[\left|Z_1 - Z_1^{(M)}\right|\right] < \frac{\varepsilon^2}{12}. \end{align*} By Markov's inequality applied to the non-negative random variable \begin{align*} \left|\frac{1}{n}\sum_{i=1}^n (Z_i-Z_i^{(M)})\right|, \end{align*} we get \begin{align*} \mathbb{P}\left( \left|\frac{1}{n}\sum_{i=1}^n (Z_i-Z_i^{(M)})\right|>\frac{\varepsilon}{3} \right) \le \frac{3}{\varepsilon}\mathbb{E}\left[\left|Z_1-Z_1^{(M)}\right|\right]. \end{align*} Also \begin{align*} \left|\mathbb{E}[Z_1-Z_1^{(M)}]\right| \le \mathbb{E}\left[\left|Z_1-Z_1^{(M)}\right|\right]. \end{align*} For the bounded centered average, independence gives \begin{align*} \mathbb{E}\left[ \left| \frac{1}{n}\sum_{i=1}^n \left(Z_i^{(M)}-\mathbb{E}[Z_1^{(M)}]\right) \right|^2 \right] = \frac{1}{n}\mathbb{E}\left[\left|Z_1^{(M)}-\mathbb{E}[Z_1^{(M)}]\right|^2\right] \le \frac{4M^2}{n}. \end{align*} Therefore [Chebyshev's inequality](/theorems/1126) gives \begin{align*} \mathbb{P}\left( \left| \frac{1}{n}\sum_{i=1}^n \left(Z_i^{(M)}-\mathbb{E}[Z_1^{(M)}]\right) \right|>\frac{\varepsilon}{3} \right) \le \frac{36M^2}{n\varepsilon^2}. \end{align*} Combining the three estimates by the triangle inequality, we obtain \begin{align*} \limsup_{n\to\infty} \mathbb{P}\left( \left| \frac{1}{n}\sum_{i=1}^n Z_i-\mathbb{E}[Z_1] \right|>\varepsilon \right) \le \frac{3}{\varepsilon}\mathbb{E}\left[\left|Z_1-Z_1^{(M)}\right|\right]. \end{align*} The chosen $M$ makes the right-hand side arbitrarily small by sending $M \to \infty$. Hence the claimed convergence in probability holds. [/step]

[step:Apply the weak law to the sample second moment and orthogonality moment]Define matrix-valued random variables $A_i : \Omega \to \mathbb{R}^{p\times p}$ and vector-valued random variables $b_i : \Omega \to \mathbb{R}^p$ by \begin{align*} A_i := X_iX_i^\top,\qquad b_i := X_i u_i. \end{align*} Equip $\mathbb{R}^{p\times p}$ with the Frobenius norm $|A|_F := \left(\sum_{j,k=1}^p A_{jk}^2\right)^{1/2}$. Since \begin{align*} |A_i|_F = |X_i|^2, \end{align*} the hypothesis $\mathbb{E}[|X_i|^2]<\infty$ gives $\mathbb{E}[|A_i|_F]<\infty$. Also \begin{align*} u_i = Y_i-X_i^\top\beta_0, \end{align*} so \begin{align*} \mathbb{E}[|u_i|^2] \le 2\mathbb{E}[|Y_i|^2]+2|\beta_0|^2\mathbb{E}[|X_i|^2] <\infty. \end{align*} By the [Cauchy-Schwarz inequality](/theorems/432) applied to the real random variables $|X_i|$ and $|u_i|$, \begin{align*} \mathbb{E}[|b_i|] = \mathbb{E}[|X_i||u_i|] \le \left(\mathbb{E}[|X_i|^2]\right)^{1/2} \left(\mathbb{E}[|u_i|^2]\right)^{1/2} <\infty. \end{align*} The weak law from the previous step therefore applies to $(A_i)_{i=1}^\infty$ and $(b_i)_{i=1}^\infty$, yielding \begin{align*} S_n=\frac{1}{n}\sum_{i=1}^n A_i \xrightarrow{\mathbb{P}} Q, \qquad m_n:=\frac{1}{n}\sum_{i=1}^n b_i \xrightarrow{\mathbb{P}} \mathbb{E}[X_i u_i]=0. \end{align*}[/step]

[guided]The sample quantities we need are averages of i.i.d. finite-dimensional random vectors, so the previous step is designed exactly for them. For the matrix average, we view a $p\times p$ matrix as a vector in $\mathbb{R}^{p^2}$ and use the Frobenius norm \begin{align*} |A|_F := \left(\sum_{j,k=1}^p A_{jk}^2\right)^{1/2}. \end{align*} For $A_i := X_iX_i^\top$, the entries are $(A_i)_{jk}=(X_i)_j(X_i)_k$, and a direct computation gives \begin{align*} |A_i|_F^2 = \sum_{j,k=1}^p (X_i)_j^2(X_i)_k^2 = \left(\sum_{j=1}^p (X_i)_j^2\right)^2 = |X_i|^4. \end{align*} Thus $|A_i|_F=|X_i|^2$, and the assumption $\mathbb{E}[|X_i|^2]<\infty$ is exactly the integrability hypothesis needed for the weak law. For the moment vector $b_i := X_i u_i$, we must first check integrability. Since $u_i=Y_i-X_i^\top\beta_0$, the elementary inequality $|a-b|^2\le 2|a|^2+2|b|^2$ gives \begin{align*} \mathbb{E}[|u_i|^2] \le 2\mathbb{E}[|Y_i|^2]+2\mathbb{E}[|X_i^\top\beta_0|^2] \le 2\mathbb{E}[|Y_i|^2]+2|\beta_0|^2\mathbb{E}[|X_i|^2] <\infty. \end{align*} Applying Cauchy-Schwarz to $|X_i|$ and $|u_i|$ gives \begin{align*} \mathbb{E}[|X_i u_i|] = \mathbb{E}[|X_i||u_i|] \le \left(\mathbb{E}[|X_i|^2]\right)^{1/2} \left(\mathbb{E}[|u_i|^2]\right)^{1/2} <\infty. \end{align*} Therefore the weak law applies and yields \begin{align*} \frac{1}{n}\sum_{i=1}^n X_iX_i^\top \xrightarrow{\mathbb{P}} \mathbb{E}[X_iX_i^\top]=Q, \qquad \frac{1}{n}\sum_{i=1}^n X_i u_i \xrightarrow{\mathbb{P}} \mathbb{E}[X_i u_i]=0. \end{align*}[/guided]

[step:Control inversion near the positive definite population matrix]Let $|B|_{\mathrm{op}}$ denote the operator norm of a matrix $B \in \mathbb{R}^{p\times p}$ acting as a [linear map](/page/Linear%20Map) from $(\mathbb{R}^p,|\cdot|)$ to itself, namely \begin{align*} |B|_{\mathrm{op}} := \sup\{|Bv| : v \in \mathbb{R}^p,\ |v|=1\}. \end{align*} For an invertible matrix $B \in \mathbb{R}^{p\times p}$, write $\|B^{-1}\|_{\mathrm{op}}$ for the same operator norm applied to $B^{-1}$. Let $\lambda_{\min}(Q)>0$ denote the smallest eigenvalue of the symmetric positive definite matrix $Q$. Define \begin{align*} \delta := \frac{\lambda_{\min}(Q)}{2}. \end{align*} If $A \in \mathbb{R}^{p\times p}$ is symmetric and $|A-Q|_{\mathrm{op}}<\delta$, then for every $v\in\mathbb{R}^p$ with $|v|=1$, \begin{align*} v^\top A v = v^\top Qv+v^\top(A-Q)v \ge \lambda_{\min}(Q)-|A-Q|_{\mathrm{op}} > \delta. \end{align*} Thus $A$ is positive definite and invertible. Moreover, \begin{align*} \|A^{-1}\|_{\mathrm{op}} = \frac{1}{\lambda_{\min}(A)} \le \frac{1}{\delta} = \frac{2}{\lambda_{\min}(Q)}. \end{align*} Since $|B|_{\mathrm{op}}\le |B|_F$ for every $B\in\mathbb{R}^{p\times p}$ and $S_n \xrightarrow{\mathbb{P}} Q$ in Frobenius norm, we have \begin{align*} \mathbb{P}\left(|S_n-Q|_{\mathrm{op}}<\delta\right)\to 1. \end{align*} On this event, $S_n$ is invertible and \begin{align*} \|S_n^{-1}\|_{\mathrm{op}}\le \frac{2}{\lambda_{\min}(Q)}. \end{align*}[/step]

[guided]The only delicate point in the OLS formula is the inverse $S_n^{-1}$. We use the operator norm to measure how close a random matrix is to the population matrix. For a matrix $B \in \mathbb{R}^{p\times p}$, define \begin{align*} |B|_{\mathrm{op}} := \sup\{|Bv| : v \in \mathbb{R}^p,\ |v|=1\}, \end{align*} which is the operator norm of $B$ as a linear map from $(\mathbb{R}^p,|\cdot|)$ to itself. For an invertible matrix $B \in \mathbb{R}^{p\times p}$, the notation $\|B^{-1}\|_{\mathrm{op}}$ means the same operator norm applied to the inverse linear map $B^{-1}:\mathbb{R}^p\to\mathbb{R}^p$. The population matrix $Q$ is positive definite, so it is bounded away from singularity. Let $\lambda_{\min}(Q)>0$ be its smallest eigenvalue and set \begin{align*} \delta := \frac{\lambda_{\min}(Q)}{2}. \end{align*} Suppose $A$ is a symmetric matrix satisfying $|A-Q|_{\mathrm{op}}<\delta$. For any unit vector $v\in\mathbb{R}^p$, the quadratic form of $A$ satisfies \begin{align*} v^\top A v = v^\top Qv+v^\top(A-Q)v \ge \lambda_{\min}(Q)-|A-Q|_{\mathrm{op}} > \delta. \end{align*} This proves that $A$ is positive definite. A positive definite matrix is invertible, and its inverse has operator norm equal to the reciprocal of its smallest eigenvalue, so \begin{align*} \|A^{-1}\|_{\mathrm{op}} = \frac{1}{\lambda_{\min}(A)} \le \frac{1}{\delta} = \frac{2}{\lambda_{\min}(Q)}. \end{align*} We apply this deterministic fact to $A=S_n$. The convergence $S_n\to Q$ in probability was obtained in Frobenius norm, and the operator norm is bounded by the Frobenius norm: \begin{align*} |B|_{\mathrm{op}}\le |B|_F \end{align*} for every $B\in\mathbb{R}^{p\times p}$. Hence \begin{align*} \mathbb{P}\left(|S_n-Q|_{\mathrm{op}}<\delta\right)\to 1. \end{align*} On this high-probability event, $S_n$ is invertible and its inverse is uniformly controlled: \begin{align*} \|S_n^{-1}\|_{\mathrm{op}}\le \frac{2}{\lambda_{\min}(Q)}. \end{align*}[/guided]

[step:Decompose the OLS error and prove convergence in probability]On the event that $S_n$ is invertible, we have \begin{align*} r_n = \frac{1}{n}\sum_{i=1}^n X_iY_i = \frac{1}{n}\sum_{i=1}^n X_i(X_i^\top\beta_0+u_i) = S_n\beta_0+m_n, \end{align*} where \begin{align*} m_n := \frac{1}{n}\sum_{i=1}^n X_i u_i. \end{align*} Therefore \begin{align*} \hat{\beta}_n-\beta_0 = S_n^{-1}m_n \end{align*} on the event that $S_n$ is invertible. Fix $\varepsilon>0$ and define the event \begin{align*} E_n := \left\{|S_n-Q|_{\mathrm{op}}<\delta\right\}. \end{align*} On $E_n$, \begin{align*} |\hat{\beta}_n-\beta_0| \le \|S_n^{-1}\|_{\mathrm{op}}|m_n| \le \frac{2}{\lambda_{\min}(Q)}|m_n|. \end{align*} Hence \begin{align*} \mathbb{P}\left(|\hat{\beta}_n-\beta_0|>\varepsilon\right) \le \mathbb{P}(E_n^c) + \mathbb{P}\left( |m_n|>\frac{\varepsilon\lambda_{\min}(Q)}{2} \right). \end{align*} The first term tends to $0$ by the previous step, and the second term tends to $0$ because $m_n\xrightarrow{\mathbb{P}}0$. Therefore \begin{align*} \hat{\beta}_n \xrightarrow{\mathbb{P}} \beta_0. \end{align*}[/step]

[guided]We now use the regression equation to expose the error term. On the event that $S_n$ is invertible, the OLS estimator is $\hat{\beta}_n=S_n^{-1}r_n$. Since $Y_i=X_i^\top\beta_0+u_i$, we compute \begin{align*} r_n = \frac{1}{n}\sum_{i=1}^n X_iY_i = \frac{1}{n}\sum_{i=1}^n X_i(X_i^\top\beta_0+u_i) = \left(\frac{1}{n}\sum_{i=1}^n X_iX_i^\top\right)\beta_0 + \frac{1}{n}\sum_{i=1}^n X_i u_i. \end{align*} Using the definitions \begin{align*} S_n := \frac{1}{n}\sum_{i=1}^n X_iX_i^\top,\qquad m_n := \frac{1}{n}\sum_{i=1}^n X_i u_i, \end{align*} this becomes \begin{align*} r_n=S_n\beta_0+m_n. \end{align*} Multiplying by $S_n^{-1}$ on the event where $S_n$ is invertible gives \begin{align*} \hat{\beta}_n-\beta_0 = S_n^{-1}r_n-\beta_0 = S_n^{-1}(S_n\beta_0+m_n)-\beta_0 = S_n^{-1}m_n. \end{align*} Now fix $\varepsilon>0$ and let \begin{align*} E_n := \left\{|S_n-Q|_{\mathrm{op}}<\delta\right\}, \end{align*} where $\delta=\lambda_{\min}(Q)/2$. The previous step showed that $\mathbb{P}(E_n)\to 1$, and that on $E_n$ the matrix $S_n$ is invertible with \begin{align*} \|S_n^{-1}\|_{\mathrm{op}}\le \frac{2}{\lambda_{\min}(Q)}. \end{align*} Therefore, on $E_n$, \begin{align*} |\hat{\beta}_n-\beta_0| = |S_n^{-1}m_n| \le \|S_n^{-1}\|_{\mathrm{op}}|m_n| \le \frac{2}{\lambda_{\min}(Q)}|m_n|. \end{align*} Consequently, \begin{align*} \mathbb{P}\left(|\hat{\beta}_n-\beta_0|>\varepsilon\right) \le \mathbb{P}(E_n^c) + \mathbb{P}\left( \frac{2}{\lambda_{\min}(Q)}|m_n|>\varepsilon \right), \end{align*} that is, \begin{align*} \mathbb{P}\left(|\hat{\beta}_n-\beta_0|>\varepsilon\right) \le \mathbb{P}(E_n^c) + \mathbb{P}\left( |m_n|>\frac{\varepsilon\lambda_{\min}(Q)}{2} \right). \end{align*} The first probability tends to $0$ because $S_n$ is close to $Q$ with probability tending to $1$. The second tends to $0$ because the sample orthogonality moment $m_n$ converges to $0$ in probability. Since this holds for every $\varepsilon>0$, we conclude \begin{align*} \hat{\beta}_n \xrightarrow{\mathbb{P}} \beta_0. \end{align*}[/guided]