Statistical inference of 2-type critical Galton–Watson processes with immigration

Abstract

In this paper the asymptotic behavior of the conditional least squares estimators of the offspring mean matrix for a 2-type critical positively regular Galton–Watson branching process with immigration is described. We also study this question for a natural estimator of the spectral radius of the offspring mean matrix, which we call criticality parameter. We discuss the subcritical case as well.

Appendices

Appendix 1: Estimations of moments

In the proof of Theorem 3.1, good bounds for moments of the random vectors and variables $({\varvec{M}}_k)_{k\in \mathbb {Z}_+}$, $({\varvec{X}}_k)_{k\in \mathbb {Z}_+}$, $(U_k)_{k\in \mathbb {Z}_+}$ and $(V_k)_{k\in \mathbb {Z}_+}$ are extensively used. First note that, for all $k \in \mathbb {N}$, ${\mathbb {E}}( {\varvec{M}}_k \,|\,{\mathcal F}_{k-1} ) = {\varvec{0}}$ and ${\mathbb {E}}({\varvec{M}}_k) = {\varvec{0}}$, since ${\varvec{M}}_k = {\varvec{X}}_k - {\mathbb {E}}({\varvec{X}}_k \,|\,{\mathcal F}_{k-1})$. We present these results without proofs as they can be proven the same way as in Appendix B of Ispány et al. (2014).

Lemma 6.1

Let $({\varvec{X}}_k)_{k\in \mathbb {Z}_+}$ be a 2-type Galton–Watson process with immigration and with ${\varvec{X}}_0 = {\varvec{0}}$. If ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,1}\Vert ^2) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,2}\Vert ^2) < \infty$ and ${\mathbb {E}}(\Vert {\varvec{\varepsilon }}_1\Vert ^2) < \infty$ then

$$\begin{aligned} {\text {Var}}( {\varvec{M}}_k \,|\,{\mathcal F}_{k-1} ) = X_{k-1,1} {\varvec{V}}_{\!\!{\varvec{\xi }}_1} + X_{k-1,2} {\varvec{V}}_{\!\!{\varvec{\xi }}_2} + {\varvec{V}}_{\!\!{\varvec{\varepsilon }}} = U_{k-1} {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}} + V_{k-1} \widetilde{{\varvec{V}}}_{\!\!{\varvec{\xi }}} + {\varvec{V}}_{\!\!{\varvec{\varepsilon }}} \end{aligned}$$

(6.1)

for all $k \in \mathbb {N}$, where

$$\begin{aligned} \widetilde{{\varvec{V}}}_{\!\!{\varvec{\xi }}} := \sum _{i=1}^2 \langle {\varvec{e}}_i, {\varvec{v}}_\mathrm {right}\rangle {\varvec{V}}_{\!\!{\varvec{\xi }}_i} = \frac{\beta {\varvec{V}}_{\!\!{\varvec{\xi }}_1}-(1-\delta ){\varvec{V}}_{\!\!{\varvec{\xi }}_2}}{\beta +1-\delta }. \end{aligned}$$

If ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,1}\Vert ^3) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,2}\Vert ^3) < \infty$ and ${\mathbb {E}}(\Vert {\varvec{\varepsilon }}_1\Vert ^3) < \infty$ then, for all $k \in \mathbb {N}$,

$$\begin{aligned}&{\mathbb {E}}( {\varvec{M}}_k^{\otimes 3} \,|\,{\mathcal F}_{k-1} )= X_{k-1,1} {\mathbb {E}}[({\varvec{\xi }}_{1,1,1} - {\mathbb {E}}({\varvec{\xi }}_{1,1,1})^{\otimes 3}] \nonumber \\&\quad +\, X_{k-1,2} {\mathbb {E}}[({\varvec{\xi }}_{1,1,2} - {\mathbb {E}}({\varvec{\xi }}_{1,1,2})^{\otimes 3}] +\, {\mathbb {E}}[({\varvec{\varepsilon }}_1 - {\mathbb {E}}({\varvec{\varepsilon }}_1)^{\otimes 3}] . \end{aligned}$$

(6.2)

Lemma 6.2

Let $({\varvec{X}}_k)_{k\in \mathbb {Z}_+}$ be a 2-type Galton–Watson process with immigration such that $\alpha , \delta \in [0, 1)$ and $\beta , \gamma \in (0, \infty )$ with $\alpha + \delta > 0$ and $\beta \gamma = (1 - \alpha ) (1 - \delta )$ (hence it is critical and and positively regular). Suppose ${\varvec{X}}_0 = {\varvec{0}}$, and ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,1}\Vert ^\ell ) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,2}\Vert ^\ell ) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\varepsilon }}_1\Vert ^\ell ) < \infty$ with some $\ell \in \mathbb {N}$. Then ${\mathbb {E}}(\Vert {\varvec{X}}_k\Vert ^\ell ) = {\text {O}}(k^\ell )$, i.e., $\sup _{k \in \mathbb {N}} k^{-\ell } {\mathbb {E}}(\Vert {\varvec{X}}_k\Vert ^\ell ) < \infty$.

Corollary 6.3

Let $({\varvec{X}}_k)_{k\in \mathbb {Z}_+}$ be a critical, positively regular 2-type Galton–Watson process. Suppose ${\varvec{X}}_0 = {\varvec{0}}$, and ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,1}\Vert ^\ell ) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,2}\Vert ^\ell ) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\varepsilon }}_1\Vert ^\ell ) < \infty$ with some $\ell \in \mathbb {N}$. Then

$$\begin{aligned} {\mathbb {E}}(\Vert {\varvec{X}}_k\Vert ^i) = {\text {O}}(k^i) , \qquad {\mathbb {E}}({\varvec{M}}_k^{\otimes i}) = {\text {O}}(k^{\lfloor i/2 \rfloor }) , \qquad {\mathbb {E}}(U^i_k ) = {\text {O}}(k^i) , \qquad {\mathbb {E}}(V^{2j}_k ) = {\text {O}}(k^j) \end{aligned}$$

for $i, j \in \mathbb {Z}_+$ with $i \le \ell$ and $2 j \le \ell$.

Corollary 6.4

Let $({\varvec{X}}_k)_{k\in \mathbb {Z}_+}$ be a critical, positively regular 2-type Galton–Watson process. Suppose ${\varvec{X}}_0 = {\varvec{0}}$, and ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,1}\Vert ^\ell ) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\xi }}_{1,1,2}\Vert ^\ell ) < \infty$, ${\mathbb {E}}(\Vert {\varvec{\varepsilon }}_1\Vert ^\ell ) < \infty$ with some $\ell \in \mathbb {N}$. Then

• (i) for all $i,j\in \mathbb {Z}_+$ with $\max \{i,j\} \le \lfloor \ell /2 \rfloor$, and for all $\kappa > i + \frac{j}{2} + 1$, we have

  $$\begin{aligned} n^{-\kappa } \sum _{k=1}^n\vert U_k^i V_k^j\vert \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 \qquad \hbox {as }n\rightarrow \infty , \end{aligned}$$

  (6.3)

• (ii) for all $i,j\in \mathbb {Z}_+$ with $\max \{i,j\} \le \ell$, for all $T>0$, and for all $\kappa > i + \frac{j}{2} + \frac{i+j}{\ell }$, we have

  $$\begin{aligned} n^{-\kappa } \sup _{t\in [0,T]} \vert U_{\lfloor nt\rfloor }^i V_{\lfloor nt\rfloor }^j \vert \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 \qquad \hbox {as }n\rightarrow \infty , \end{aligned}$$

  (6.4)

• (iii) for all $i,j\in \mathbb {Z}_+$ with $\max \{i,j\} \le \lfloor \ell /4 \rfloor$, for all $T>0$, and for all $\kappa > i + \frac{j}{2} + \frac{1}{2}$, we have

  $$\begin{aligned} n^{-\kappa } \sup _{t\in [0,T]} \left| \sum _{k=1}^{\lfloor nt\rfloor }[U_k^i V_k^j - {\mathbb {E}}(U_k^i V_k^j \,|\,{\mathcal F}_{k-1})] \right| \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 \qquad \hbox {as }n\rightarrow \infty . \end{aligned}$$

  (6.5)

Remark 6.5

In some parts of this paper we need the above statements with a smaller $\kappa$ than it is provided by Corollary 6.4. Fortunately we can use a decomposition argument in those cases to sharpen the statements, for example see the proof of (5.9) and Lemma 6.7.

Appendix 2: CLS estimators

For each $n \in \mathbb {N}$, a CLS estimator $\widehat{{\varvec{m}}}_{\varvec{\xi }}^{(n)}$ of ${\varvec{m}}_{\varvec{\xi }}$ based on a sample ${\varvec{X}}_1, \ldots , {\varvec{X}}_n$ can be obtained by minimizing the sum of squares

$$\begin{aligned} \sum _{k=1}^n \big \Vert {\varvec{X}}_k - {\mathbb {E}}({\varvec{X}}_k \,|\,{\mathcal F}_{k-1}) \big \Vert ^2 = \sum _{k=1}^n \left\| {\varvec{X}}_k - {\varvec{m}}_{\varvec{\xi }}{\varvec{X}}_{k-1} - {\varvec{m}}_{\varvec{\varepsilon }}\right\| ^2 \end{aligned}$$

with respect to ${\varvec{m}}_{\varvec{\xi }}$ over $\mathbb {R}^{2\times 2}$. In what follows, we use the notation ${\varvec{x}}_0 := {\varvec{0}}$. For all $n \in \mathbb {N}$, we define the function $Q_n : (\mathbb {R}^2)^n \times \mathbb {R}^{2\times 2} \rightarrow \mathbb {R}$ by

$$\begin{aligned} Q_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n ; {\varvec{m}}_{\varvec{\xi }}') := \sum _{k=1}^n \left\| {\varvec{x}}_k - {\varvec{m}}_{\varvec{\xi }}' {\varvec{x}}_{k-1} - {\varvec{m}}_{\varvec{\varepsilon }}\right\| ^2 \end{aligned}$$

for all ${\varvec{m}}_{\varvec{\xi }}' \in \mathbb {R}^{2\times 2}$ and ${\varvec{x}}_1, \ldots , {\varvec{x}}_n \in \mathbb {R}^2$. By definition, for all $n \in \mathbb {N}$, a CLS estimator of ${\varvec{m}}_{\varvec{\xi }}$ is a measurable function $F_n : (\mathbb {R}^2)^n \rightarrow \mathbb {R}^{2\times 2}$ such that

$$\begin{aligned} Q_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n; F_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n)) = \inf _{{\varvec{m}}_{\varvec{\xi }}' \in \mathbb {R}^{2\times 2}} Q_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n ; {\varvec{m}}_{\varvec{\xi }}') \end{aligned}$$

for all ${\varvec{x}}_1, \ldots , {\varvec{x}}_n \in \mathbb {R}^2$. Next we give the solutions of this extremum problem.

Lemma 6.6

For each $n \in \mathbb {N}$, any CLS estimator of ${\varvec{m}}_{\varvec{\xi }}$ is a measurable function $F_n : (\mathbb {R}^2)^n \rightarrow \mathbb {R}^{2\times 2}$ for which

$$\begin{aligned} F_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n) = H_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n) G_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n)^{-1} \end{aligned}$$

(6.6)

on the set

$$\begin{aligned} \bigl \{ ({\varvec{x}}_1, \ldots , {\varvec{x}}_n) \in (\mathbb {R}^2)^n : \det (G_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n)) > 0 \bigr \} , \end{aligned}$$

where

$$\begin{aligned} G_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n) := \sum _{k=1}^n {\varvec{x}}_{k-1} {\varvec{x}}_{k-1}^\top , \qquad H_n({\varvec{x}}_1, \ldots , {\varvec{x}}_n) := \sum _{k=1}^n ({\varvec{x}}_k - {\varvec{m}}_{\varvec{\varepsilon }}) {\varvec{x}}_{k-1}^\top . \end{aligned}$$

For the existence of these CLS estimators in case of a critical symmetric 2-type Galton–Watson process, i.e., when $\varrho = 1$, we need the following approximations.

Lemma 6.7

Suppose that the assumptions of Theorem 3.1 hold. Then for each $T > 0$,

$$\begin{aligned} n^{-5/2} \sup _{t\in [0,T]} \biggl | \sum _{k=1}^{{\lfloor nt\rfloor }} U_{k-1} V_{k-1} \biggr | \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 \qquad \hbox {as }n \rightarrow \infty . \end{aligned}$$

Proof

The aim of the following discussion is to decompose $\sum _{k=1}^{{\lfloor nt\rfloor }} U_{k-1} V_{k-1}$ as a sum of a martingale and some other terms. Using the recursions (4.7), (4.4) and Lemma 6.1, we obtain

$$\begin{aligned} {\mathbb {E}}(U_{k-1}V_{k-1} \,|\,{\mathcal F}_{k-2})&= {\mathbb {E}}\Bigl ((U_{k-2} + \langle {\varvec{u}}_{\mathrm {left}}, {\varvec{M}}_{k-1} + {\varvec{m}}_{\varvec{\varepsilon }}\rangle ) \bigl (\lambda V_{k-2} + \langle {\varvec{v}}_{\mathrm {left}}, {\varvec{M}}_{k-1} + {\varvec{m}}_{\varvec{\varepsilon }}\rangle \bigr ) \,\Big |\, {\mathcal F}_{k-2}\Bigr ) \\&= \lambda U_{k-2} V_{k-2} + \langle {\varvec{v}}_{\mathrm {left}}, {\varvec{m}}_{\varvec{\varepsilon }}\rangle U_{k-2} + \lambda \langle {\varvec{u}}_{\mathrm {left}}, {\varvec{m}}_{\varvec{\varepsilon }}\rangle V_{k-2} + {\varvec{u}}_{\mathrm {left}}^\top {\varvec{m}}_{\varvec{\varepsilon }}{\varvec{m}}_{\varvec{\varepsilon }}^\top {\varvec{v}}_{\mathrm {left}}\\&\quad + {\varvec{u}}^\top {\mathbb {E}}({\varvec{M}}_{k-1} {\varvec{M}}_{k-1}^\top \,|\,{\mathcal F}_{k-2}) {\varvec{v}}\\&= \lambda U_{k-2} V_{k-2} + \text {constant} + \hbox {linear combination of }U_{k-2}\hbox { and }V_{k-2}. \end{aligned}$$

Thus

$$\begin{aligned} \sum _{k=1}^{{\lfloor nt\rfloor }} U_{k-1} V_{k-1}&= \sum _{k=2}^{{\lfloor nt\rfloor }} \big [ U_{k-1} V_{k-1} - {\mathbb {E}}(U_{k-1}V_{k-1} \,|\,{\mathcal F}_{k-2}) \big ] + \sum _{k=2}^{{\lfloor nt\rfloor }} {\mathbb {E}}(U_{k-1}V_{k-1} \,|\,{\mathcal F}_{k-2}) \\&= \sum _{k=2}^{{\lfloor nt\rfloor }} \big [ U_{k-1} V_{k-1} - {\mathbb {E}}(U_{k-1}V_{k-1} \,|\,{\mathcal F}_{k-2}) \big ] + \lambda \sum _{k=2}^{{\lfloor nt\rfloor }} U_{k-2} V_{k-2} \\&\quad + {\text {O}}(n) + \hbox {linear combination of }\sum _{k=2}^{{\lfloor nt\rfloor }} U_{k-2} \quad \hbox { and } \quad \sum _{k=2}^{{\lfloor nt\rfloor }} V_{k-2}. \end{aligned}$$

Consequently

$$\begin{aligned}&\sum \limits _{k=2}^{{\lfloor nt\rfloor }} U_{k-1} V_{k-1} = \frac{1}{1-\lambda } \sum \limits _{k=2}^{{\lfloor nt\rfloor }} \big [ U_{k-1} V_{k-1} - {\mathbb {E}}(U_{k-1}V_{k-1} \,|\,{\mathcal F}_{k-2}) \big ] \\&- \frac{\lambda }{1-\lambda } U_{{\lfloor nt\rfloor }-1} V_{{\lfloor nt\rfloor }-1} + {\text {O}}(n) + \hbox {linear combination of }\sum \limits _{k=2}^{{\lfloor nt\rfloor }} U_{k-2} \quad \hbox { and }\quad \sum \limits _{k=2}^{{\lfloor nt\rfloor }} V_{k-2}. \end{aligned}$$

Using (6.5) with $(\ell , i , j) = (4, 1, 1)$ we have

$$\begin{aligned} n^{-5/2}\sup _{t \in [0,T]}\, \Biggl \vert \sum _{k=2}^{\lfloor nt\rfloor }\big [U_{k-1} V_{k-1} - {\mathbb {E}}(U_{k-1} V_{k-1} \,|\,{\mathcal F}_{k-2}) \big ] \Biggr \vert \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 \qquad \hbox {as }n\rightarrow \infty . \end{aligned}$$

Thus, in order to show the statement, it suffices to prove

$$\begin{aligned} n^{-5/2} \sum _{k=1}^{{\lfloor nT\rfloor }} U_k \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 , \qquad n^{-5/2} \sum _{k=1}^{{\lfloor nT\rfloor }} |V_k| \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 , \qquad n^{-5/2} \sup _{t \in [0,T]} | U_{\lfloor nt\rfloor }V_{\lfloor nt\rfloor }| \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0\nonumber \\ \end{aligned}$$

(6.7)

as $n \rightarrow \infty$. Using (6.3) with $(\ell , i , j) = (2, 1, 0)$ and $(\ell , i , j) = (2, 0, 1)$, and (6.4) with $(\ell , i , j) = (3, 1, 1)$ we have (6.7), thus we conclude the statement. $\square$

Using the same ideas as above one can prove the following.

Lemma 6.8

Suppose that the assumptions of Theorem 3.1 hold. For each $T > 0$, we have

$$\begin{aligned} n^{-2} \sup _{t\in [0,T]} \left| \sum _{k=1}^{\lfloor nt\rfloor }V_k^2 - \frac{\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}} \, {\varvec{v}}_{\mathrm {left}}, {\varvec{v}}_{\mathrm {left}}\rangle }{1-\lambda ^2} \sum _{k=1}^{\lfloor nt\rfloor }U_{k-1} \right| \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}0 \qquad \hbox {as }n \rightarrow \infty . \end{aligned}$$

Now we can prove asymptotic existence and uniqueness of CLS estimators of the offspring mean matrix and of the criticality parameter.

Proposition 6.9

Suppose that the assumptions of Theorem 3.1 hold, and $\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}} {\varvec{v}}_{\mathrm {left}}, {\varvec{v}}_{\mathrm {left}}\rangle + \langle {\varvec{V}}_{\!\!{\varvec{\varepsilon }}} {\varvec{v}}_{\mathrm {left}}, {\varvec{v}}_{\mathrm {left}}\rangle + \langle {\varvec{v}}_{\mathrm {left}}, {\varvec{m}}_{\varvec{\varepsilon }}\rangle ^2 > 0$. Then $\lim _{n \rightarrow \infty } {\mathbb {P}}(\Omega _n) = 1$, where $\Omega _n$ is defined in (3.2), and hence the probability of the existence of a unique CLS estimator $\widehat{{\varvec{m}}}_{\varvec{\xi }}^{(n)}$ converges to 1 as $n \rightarrow \infty$, and this CLS estimator has the form given in (3.1) on the set $\Omega _n$. If $\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}} {\varvec{v}}_{\mathrm {left}}, {\varvec{v}}_{\mathrm {left}}\rangle > 0$ then $\lim _{n \rightarrow \infty } {\mathbb {P}}(\widetilde{\Omega }_n) = 1$, where $\widetilde{\Omega }_n$ is defined in (3.5), and hence the probability of the existence of the estimator $\widehat{\varrho }_n$ converges to 1 as $n \rightarrow \infty$.

Proof

Recall convergence $(n^{-1} U_{{\lfloor nt\rfloor }})_{t\in \mathbb {R}_+} \mathop {\longrightarrow }\limits ^{{\mathcal D}}({\mathcal Y}_t)_{t\in \mathbb {R}_+}$ from (4.6). Using Lemmas 6.8, 6.7, and a version of the continuous mapping theorem (see Ispány and Pap (2010, Lemma 3.1)), one can show

$$\begin{aligned} \sum _{k=1}^n \left[ \begin{array}{l} n^{-3} U_{k-1}^2 \\ n^{-5/2} U_{k-1} V_{k-1} \\ n^{-2} V_{k-1}^2 \end{array}\right] \mathop {\longrightarrow }\limits ^{{\mathcal D}}\left[ \begin{array}{l} \int _0^1 {\mathcal Y}_t^2 \, \mathrm {d}t \\ 0 \\ \frac{\langle {\varvec{v}}_{\mathrm {left}},{\varvec{m}}_{\varvec{\varepsilon }}\rangle }{1-\lambda } \, \int _0^1 {\mathcal Y}_t \, \mathrm {d}t \end{array}\right] \qquad \hbox {as }n \rightarrow \infty . \end{aligned}$$

By (4.11) and continuous mapping theorem,

$$\begin{aligned} n^{-5} \det ({\varvec{A}}_n) \mathop {\longrightarrow }\limits ^{{\mathcal D}}\frac{\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}}{\varvec{v}}_{\mathrm {left}},{\varvec{v}}_{\mathrm {left}}\rangle }{1-\lambda ^2} \int _0^1 {\mathcal Y}_t^2 \, \mathrm {d}t \int _0^1 {\mathcal Y}_t \, \mathrm {d}t \qquad \hbox {as }n \rightarrow \infty . \end{aligned}$$

(6.8)

Since ${\varvec{m}}_{\varvec{\varepsilon }}\ne {\varvec{0}}$, by the SDE (2.8), we have ${\mathbb {P}}({\mathcal Y}_t = 0\hbox { for all }t \in [0,1]) = 0$, which implies that ${\mathbb {P}}\bigl ( \int _0^1 {\mathcal Y}_t^2 \, \mathrm {d}t \int _0^1 {\mathcal Y}_t \, \mathrm {d}t > 0 \bigr ) = 1$. Consequently, the distribution function of $\int _0^1 {\mathcal Y}_t^2 \, \mathrm {d}t \int _0^1 {\mathcal Y}_t \, \mathrm {d}t$ is continuous at 0.

If $\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}} {\varvec{v}}_{\mathrm {left}}, {\varvec{v}}_{\mathrm {left}}\rangle > 0$ then, by (6.8),

$$\begin{aligned} {\mathbb {P}}(\Omega _n)= & {} {\mathbb {P}}\left( \det ({\varvec{A}}_n)> 0 \right) = {\mathbb {P}}\left( n^{-5} \det ({\varvec{A}}_n)> 0 \right) \\\rightarrow & {} {\mathbb {P}}\left( \frac{\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}}{\varvec{v}}_{\mathrm {left}},{\varvec{v}}_{\mathrm {left}}\rangle }{1-\lambda ^2} \int _0^1 {\mathcal Y}_t^2 \, \mathrm {d}t \int _0^1 {\mathcal Y}_t \, \mathrm {d}t> 0 \right) = {\mathbb {P}}\left( \int _0^1 {\mathcal Y}_t^2 \, \mathrm {d}t \int _0^1 {\mathcal Y}_t \, \mathrm {d}t > 0 \right) = 1 \end{aligned}$$

as $n \rightarrow \infty$.

If $\langle {\overline{{\varvec{V}}}}_{\!\!{\varvec{\xi }}} {\varvec{v}}_{\mathrm {left}}, {\varvec{v}}_{\mathrm {left}}\rangle > 0$, then (3.6) yields $\widehat{{\varvec{m}}}_{\varvec{\xi }}^{(n)} \mathop {\longrightarrow }\limits ^{{\mathcal D}}{\varvec{m}}_{\varvec{\xi }}$ as $n \rightarrow \infty$, and hence $\widehat{{\varvec{m}}}_{\varvec{\xi }}^{(n)} \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}{\varvec{m}}_{\varvec{\xi }}$ as $n \rightarrow \infty$, thus $(\widehat{\alpha }_n - \widehat{\delta }_n)^2 + 4 \widehat{\beta }_n \widehat{\gamma }_n \mathop {\longrightarrow }\limits ^{{\mathbb {P}}}(\alpha - \delta )^2 + 4 \beta \gamma = (1 - \lambda )^2 > 0$, implying

$$\begin{aligned} {\mathbb {P}}(\widetilde{\Omega }_n) = {\mathbb {P}}\bigl ((\widehat{\alpha }_n - \widehat{\delta }_n)^2 + 4 \widehat{\beta }_n \widehat{\gamma }_n \ge 0\bigr ) \rightarrow 1 \qquad \hbox {as }n \rightarrow \infty , \end{aligned}$$

hence we obtain the statement. $\square$