Mixing convergence of LSE for supercritical AR(2) processes with Gaussian innovations using random scaling

Abstract

We prove mixing convergence of the least squares estimator of autoregressive parameters for supercritical autoregressive processes of order 2 with Gaussian innovations having real characteristic roots with different absolute values. We use an appropriate random scaling such that the limit distribution is a two-dimensional normal distribution concentrated on a one-dimensional ray determined by the characteristic root having the larger absolute value.

Appendices

Appendices

Stable convergence

We recall the notions of stable and mixing convergence.

Definition A.1

Let $(\Omega ,{\mathcal F},{{\mathbb {P}}})$ be a probability space and ${\mathcal G}\subset {\mathcal F}$ be a sub-$\sigma$-field. Let $({\varvec{X}}_n)_{n\in \mathbb {N}}$ and ${\varvec{X}}$ be $\mathbb {R}^d$-valued random variables defined on $(\Omega ,{\mathcal F},{{\mathbb {P}}})$, where $d\in \mathbb {N}$.

(i) We say that ${\varvec{X}}_n$ converges ${\mathcal G}$-stably to ${\varvec{X}}$ as $n\rightarrow \infty$, if the conditional distribution ${{\mathbb {P}}}^{{\varvec{X}}_n\,|\,{\mathcal G}}$ of ${\varvec{X}}_n$ given ${\mathcal G}$ converges weakly to the conditional distribution ${{\mathbb {P}}}^{{\varvec{X}}\,|\,{\mathcal G}}$ of ${\varvec{X}}$ given ${\mathcal G}$ as $n\rightarrow \infty$ in the sense of weak convergence of Markov kernels. It equivalently means that

$$\begin{aligned} \lim _{n\rightarrow \infty } {\mathbb {E}}_{{\mathbb {P}}}(\xi {\mathbb {E}}_{{\mathbb {P}}}(h({\varvec{X}}_n) \,|\,{\mathcal G}) ) = {\mathbb {E}}_{{\mathbb {P}}}( \xi {\mathbb {E}}_{{\mathbb {P}}}(h({\varvec{X}}) \,|\,{\mathcal G}) ) \end{aligned}$$

for all random variables $\xi :\Omega \rightarrow \mathbb {R}$ with ${\mathbb {E}}_{{\mathbb {P}}}(\vert \xi \vert )<\infty$ and for all bounded and continuous functions $h:\mathbb {R}^d\rightarrow \mathbb {R}$.

(ii) We say that ${\varvec{X}}_n$ converges ${\mathcal G}$-mixing to ${\varvec{X}}$ as $n\rightarrow \infty$, if ${\varvec{X}}_n$ converges ${\mathcal G}$-stably to ${\varvec{X}}$ as $n\rightarrow \infty$, and ${{\mathbb {P}}}^{{\varvec{X}}\,|\,{\mathcal G}} = {{\mathbb {P}}}^{\varvec{X}}$${{\mathbb {P}}}$-almost surely, where ${{\mathbb {P}}}^{\varvec{X}}$ denotes the distribution of ${\varvec{X}}$ on $(\mathbb {R}^d,{\mathcal B}(\mathbb {R}^d))$ under ${{\mathbb {P}}}$. Equivalently, we can say that ${\varvec{X}}_n$ converges ${\mathcal G}$-mixing to ${\varvec{X}}$ as $n\rightarrow \infty$, if ${\varvec{X}}_n$ converges ${\mathcal G}$-stably to ${\varvec{X}}$ as $n\rightarrow \infty$, and $\sigma ({\varvec{X}})$ and ${\mathcal G}$ are independent, which equivalently means that

$$\begin{aligned} \lim _{n\rightarrow \infty } {\mathbb {E}}_{{\mathbb {P}}}(\xi {\mathbb {E}}_{{\mathbb {P}}}(h({\varvec{X}}_n) \,|\,{\mathcal G}) ) = {\mathbb {E}}_{{\mathbb {P}}}(\xi ){\mathbb {E}}_{{\mathbb {P}}}(h({\varvec{X}})) \end{aligned}$$

for all random variables $\xi :\Omega \rightarrow \mathbb {R}$ with ${\mathbb {E}}_{{\mathbb {P}}}(\vert \xi \vert )<\infty$ and for all bounded and continuous functions $h:\mathbb {R}^d\rightarrow \mathbb {R}$.

In Definition A.1, ${{\mathbb {P}}}^{{\varvec{X}}_n\,|\,{\mathcal G}}$, $n\in \mathbb {N}$, and ${{\mathbb {P}}}^{{\varvec{X}}\,|\,{\mathcal G}}$ are the ${{\mathbb {P}}}$-almost surely unique ${\mathcal G}$-measurable Markov kernels from $(\Omega ,{\mathcal F})$ to $(\mathbb {R}^d,{\mathcal B}(\mathbb {R}^d))$ such that for each $n\in \mathbb {N}$,

$$\begin{aligned} \int _G {{\mathbb {P}}}^{{\varvec{X}}_n\,|\,{\mathcal G}}(\omega ,B)\,{{\mathbb {P}}}(\textrm{d}\omega ) = {{\mathbb {P}}}({\varvec{X}}_n^{-1}(B) \cap G) \qquad \text {for every} \ G\in {\mathcal G}, \ B\in {\mathcal B}(\mathbb {R}^d). \end{aligned}$$

and

$$\begin{aligned} \int _G {{\mathbb {P}}}^{{\varvec{X}}\,|\,{\mathcal G}}(\omega ,B)\,{{\mathbb {P}}}(\textrm{d}\omega ) = {{\mathbb {P}}}({\varvec{X}}^{-1}(B) \cap G) \qquad \text {for every} \ G\in {\mathcal G}, \ B\in {\mathcal B}(\mathbb {R}^d), \end{aligned}$$

respectively. For the notion of weak convergence of Markov kernels towards a Markov kernel, see Häusler and Luschgy (2015, Definition 2.2). For more details on stable convergence, see Häusler and Luschgy (2015, Chapter 3 and Appendix A).

A multidimensional stable limit theorem

Recall that $\log ^+(x):= \log (x)\mathbb {1}_{\{x\geqslant 1\}} + 0\cdot \mathbb {1}_{\{ 0\leqslant x < 1\}}$ for $x\in \mathbb {R}_+$, and for an event A with ${{\mathbb {P}}}(A)>0$, ${{\mathbb {P}}}_A$ denotes the conditional probability measure given A. For an $\mathbb {R}^d$-valued stochastic process $({\varvec{U}}_n)_{n\in \mathbb {Z}_+}$, the increments $\Delta {\varvec{U}}_n$, $n \in \mathbb {Z}_+$, are defined by $\Delta {\varvec{U}}_0:= {\varvec{0}}$ and $\Delta {\varvec{U}}_n:= {\varvec{U}}_n - {\varvec{U}}_{n-1}$ for $n \in \mathbb {N}$.

We recall a multidimensional analogue of Theorem 8.2 in Häusler and Luschgy (2015) which was proved in Barczy and Pap (2023, Theorem 1.4).

Theorem B.1

(Barczy and Pap (2023, Theorem 1.4)) Let $({\varvec{U}}_n)_{n\in \mathbb {Z}_+}$ and $({\varvec{B}}_n)_{n\in \mathbb {Z}_+}$ be $\mathbb {R}^d$-valued and $\mathbb {R}^{d\times d}$-valued stochastic processes, respectively, defined on a probability space $(\Omega ,{\mathcal F},{{\mathbb {P}}})$ and adapted to a filtration $({\mathcal F}_n)_{n\in \mathbb {Z}_+}$. Suppose that ${\varvec{B}}_n$ is invertible for sufficiently large $n \in \mathbb {N}$. Let $({\varvec{Q}}_n)_{n\in \mathbb {N}}$ be a sequence in $\mathbb {R}^{d\times d}$ such that ${\varvec{Q}}_n \rightarrow {\varvec{0}}$ as $n \rightarrow \infty$ and ${\varvec{Q}}_n$ is invertible for sufficiently large $n \in \mathbb {N}$. Let $G \in {\mathcal F}_\infty := \sigma (\bigcup _{k=0}^\infty {\mathcal F}_k)$ with ${{\mathbb {P}}}(G) > 0$. Assume that the following conditions are satisfied:

• (i) there exists an $\mathbb {R}^{d\times d}$-valued, ${\mathcal F}_\infty$-measurable random matrix ${\varvec{\eta }}:\Omega \rightarrow \mathbb {R}^{d\times d}$ such that ${{\mathbb {P}}}(G \cap \{\exists \,{\varvec{\eta }}^{-1}\}) > 0$ and

  $$\begin{aligned} {\varvec{Q}}_n {\varvec{B}}_n^{-1} {\mathop {\longrightarrow }\limits ^{{{\mathbb {P}}}_G}}{\varvec{\eta }}\qquad \text {as} \quad n \rightarrow \infty , \end{aligned}$$

• (ii) $({\varvec{Q}}_n {\varvec{U}}_n)_{n\in \mathbb {N}}$ is stochastically bounded in ${{\mathbb {P}}}_{G\cap \{\exists \,{\varvec{\eta }}^{-1}\}}$-probability, i.e.,

  $$\begin{aligned} \lim _{K\rightarrow \infty } \sup _{n\in \mathbb {N}} {{\mathbb {P}}}_{G\cap \{\exists \,{\varvec{\eta }}^{-1}\}}(\Vert {\varvec{Q}}_n {\varvec{U}}_n\Vert > K) = 0. \end{aligned}$$

• (iii) there exists an invertible matrix ${\varvec{P}}\in \mathbb {R}^{d\times d}$ with $\varrho ({\varvec{P}}) < 1$ such that

  $$\begin{aligned} {\varvec{B}}_n {\varvec{B}}_{n-r}^{-1} {\mathop {\longrightarrow }\limits ^{{{\mathbb {P}}}_G}}{\varvec{P}}^r \qquad \text {as}\quad n \rightarrow \infty \text { for every} \ r \in \mathbb {N}, \end{aligned}$$

• (iv) there exists a probability measure $\mu$ on $(\mathbb {R}^d, {\mathcal B}(\mathbb {R}^d))$ with $\int _{\mathbb {R}^d} \log ^+(\Vert {\varvec{x}}\Vert ) \, \mu (\textrm{d}{\varvec{x}}) < \infty$ such that

  $$\begin{aligned} {\mathbb {E}}_{{\mathbb {P}}}\bigl (\textrm{e}^{\textrm{i}\langle {\varvec{\theta }},{\varvec{B}}_n\Delta {\varvec{U}}_n\rangle } \,|\,{\mathcal F}_{n-1}\bigr ) {\mathop {\longrightarrow }\limits ^{{{\mathbb {P}}}_{G\cap \{\exists {\varvec{\eta }}^{-1}\}}}}\int _{\mathbb {R}^d} \textrm{e}^{\textrm{i}\langle {\varvec{\theta }},{\varvec{x}}\rangle } \, \mu (\textrm{d}{\varvec{x}}) \qquad \text {as } \ n \rightarrow \infty \end{aligned}$$

  for every ${\varvec{\theta }}\in \mathbb {R}^d$.

Then

$$\begin{aligned} {\varvec{B}}_n {\varvec{U}}_n \rightarrow \sum _{j=0}^\infty {\varvec{P}}^j {\varvec{Z}}_j \qquad {\mathcal F}_\infty \text {-mixing under } {{\mathbb {P}}}_{G \cap \{\exists \,{\varvec{\eta }}^{-1}\}} \text { as } \ n \rightarrow \infty , \end{aligned}$$

(B.1)

and

$$\begin{aligned} {\varvec{Q}}_n {\varvec{U}}_n \rightarrow {\varvec{\eta }}\sum _{j=0}^\infty {\varvec{P}}^j {\varvec{Z}}_j \qquad {\mathcal F}_\infty \text {-stably under }{{\mathbb {P}}}_{G \cap \{\exists \,{\varvec{\eta }}^{-1}\}}\text { as } \ n \rightarrow \infty , \end{aligned}$$

(B.2)

where $({\varvec{Z}}_j)_{j\in \mathbb {Z}_+}$ denotes a sequence of ${{\mathbb {P}}}$-independent and identically distributed $\mathbb {R}^d$-valued random vectors being ${{\mathbb {P}}}$-independent of ${\mathcal F}_\infty$ with ${{\mathbb {P}}}({\varvec{Z}}_0 \in B) = \mu (B)$ for all $B \in {\mathcal B}(\mathbb {R}^d)$.

The series $\sum _{j=0}^\infty {\varvec{P}}^j {\varvec{Z}}_j$ in (B.1) and in (B.2) is absolutely convergent ${{\mathbb {P}}}$-almost surely, since, by condition (iv) of Theorem B.1, ${\mathbb {E}}_{{\mathbb {P}}}(\log ^+(\Vert {\varvec{Z}}_0\Vert ))<\infty$ and one can apply Lemma 1.3 in Barczy and Pap (2023). Further, the random variable ${\varvec{\eta }}$ and the sequence $({\varvec{Z}}_j)_{j\in \mathbb {Z}_+}$ are ${{\mathbb {P}}}$-independent in Theorem B.1, since ${\varvec{\eta }}$ is ${\mathcal F}_\infty$-measurable and the sequence $({\varvec{Z}}_j)_{j\in \mathbb {Z}_+}$ is ${{\mathbb {P}}}$-independent of ${\mathcal F}_\infty$.