1 Introduction
The classical Borel-Cantelli lemma plays an important role in probability theory and its applications. The modern theory of dynamical systems uses a special version of the Borel-Cantelli lemma (see for instance [4], [10]).
Let $\langle \mathbb{M},\hspace{0.1667em}\mathfrak{F},\hspace{0.1667em}\mu ,\hspace{0.1667em}T\rangle $ be a dynamical system with $T-$ invariant probability measure μ. The classical Poincare’ recurrence theorem states that for any fixed subset $A\in \mathfrak{F}$ with $\mu (A)\gt 0$ the equality
\[ \mu (\{x\in A\mid \hspace{0.1667em}\hspace{0.1667em}{T^{n}}(x)\in A\hspace{0.1667em}\hspace{0.1667em}\text{for infinitely many}\hspace{2.5pt}n\in \mathbb{N}\})=\mu (A)\]
holds. We consider the sequence of nontrivial measurable subsets ${A_{n}}$ and can still ask for the μ-measure of the limsup set:
In the case $\Sigma \mu ({T^{-n}}{A_{n}})=\Sigma \mu ({A_{n}})\lt \infty $, the convergence case of the Borel-Cantelli lemma implies that the $\mu -$ measure of the limsup set is zero. If $\Sigma \mu ({A_{n}})=\infty $, and ${T^{-n}}{A_{n}}$ are independent, then for $\mu -$ measure of the limsup set is one (see for instance [11]). The last assertion has a limited value for deterministic dynamical systems, since one rarely deals with purely independent sets. In the case, if $\Sigma \mu ({A_{n}})=\infty $ and the events ${T^{-n}}{A_{n}}$ are dependent, the situation is more complicated and more interesting. A definition, found in [4], applies to this situation.
Definition 1 (see [4]).
The divergence case of the Borel-Cantelli lemma is not helpful for finding BC sequences since this case of the lemma requires independent sets. To obtain a BC sequence, we need impose some restrictions.
If, for a dynamical system, all sequences of subsets ${A_{n}}$ that satisfy (1) and certain additional conditions are BC, we obtain what is called a dynamical Borel-Cantelli lemma.
The first example of such a lemma, in which only sequences of balls centered at a fixed point and with weakly monotonically decreasing radii are allowed, was proved by J. Kurzweil [10]. For a dynamical system with mixing property, the “abundance” of BC sequences can be interpreted as an aspect of strong chaos and stochastic behavior of the system. It is proved (see [4], [12], [7], [1]) that for a wide class of hyperbolic or”fast” mixing systems, various sequences of sets have the BC property. The sets which are to be considered in this kind of problems are usually decreasing sequences of balls with the same center (see [10], [14], [6]) or cylinders. In [4] Chernov proved the dynamical BC lemma for a Gibbs measures. In [7] Kim and Galatolo established that the Borel-Cantelli property and the waiting time problem are in general strictly connected.
Stationary determinantal processes provide a natural and important class of shift-invariant probability measures on symbolic spaces. They arise in probability theory, random matrix theory, mathematical physics, and the theory of point processes. In contrast to Bernoulli or Markov measures, determinantal measures are characterized by determinant formulae for finite-dimensional distributions and typically exhibit negative dependence. In the stationary case on $\mathbb{Z}$, such processes are generated by convolution kernels
where $f:{\mathbb{S}^{1}}\to [0,1]$ is a measurable function. The corresponding measure ${\mu _{f}}$ is invariant under the shift, and therefore it naturally defines a measure-preserving symbolic dynamical system.
A key tool in the present paper is the ψ-mixing property of stationary determinantal processes. Fan, Liao and Qiu obtained necessary and sufficient conditions for the ψ-mixing property in terms of the regularity of the generating function f and the decay of its Fourier coefficients. This makes stationary determinantal processes suitable for studying dynamical Borel–Cantelli type problems for cylinder targets.
The paper is organized as follows. Section 2 contains the necessary preliminaries on Borel–Cantelli sequences, cylinder sets, stationary determinantal measures, and ψ-mixing, and also presents the main results of the paper. In Section 3, we prove the uniform and conditional estimates for determinantal cylinder probabilities. Section 4 is devoted to the proof of the exponential cylinder mixing estimate and the main strong dynamical Borel–Cantelli theorem.
2 Preliminaries and formulation of main results
2.1 Borel–Cantelli sequences and the Sprindzuk criterion
Let $\langle \mathbb{M},\hspace{0.1667em}\mathfrak{F},\hspace{0.1667em}\mu ,\hspace{0.1667em}T\rangle $ be a dynamical system with $T-$ invariant probability measure μ.
Let ${A_{n}}\subset \mathbb{M}$ be a sequence of measurable sets. We set ${B_{n}}:={T^{-n}}{A_{n}}$. Define the set $\limsup {B_{n}}$ as
A classical Borel-Cantelli lemma in probability theory states:
Lemma 1 (Borel-Cantelli).
-
(i) Suppose that $\textstyle\sum \mu ({B_{n}})\lt \infty $, then $\mu (\limsup {B_{n}})=0$, i.e., almost every point $x\in \mathbb{M}$ belongs to finitely many ${B_{n}}$.
-
(ii) In the case, $\textstyle\sum \mu ({B_{n}})=\infty $ and the events of the sequence $\{{B_{n}},\hspace{0.1667em}\hspace{0.1667em}n\in \mathbb{N}\}$ are pairwise independent, then $\mu (\limsup {B_{n}})=1$, i.e., almost every point $x\in \mathbb{M}$ belongs to infinitely many ${B_{n}}$.
The last lemma can be restated as follows:
Lemma 2.
-
(i) Suppose that the sum of all $\mu ({A_{n}})$ converges, then for almost every point $x\in \mathbb{M}$, there are only finitely many values of $n\in \mathbb{N}$ such that ${T^{n}}x\in {A_{n}}$.
-
(ii) In the case, $\textstyle\sum \mu ({B_{n}})=\infty $ and the events of the sequence $\{{B_{n}},\hspace{0.1667em}\hspace{0.1667em}n\in \mathbb{N}\}$ are pairwise independent, then for almost every point $x\in \mathbb{M}$, there are infinitely many values of $n\in \mathbb{N}$ such that ${T^{n}}x\in {A_{n}}$.
Next we introduce several necessary definitions.
A sequence of subsets ${A_{n}}\subset \mathbb{M}$ is called a Borel-Cantelli (BC) sequence if for $\mu -$ almost every $x\in \mathbb{M}$ there are infinitely many values of $n\in \mathbb{N}$, such that, ${T^{n}}x\in {A_{n}}$.
Denote by ${\chi _{n}}(x)$ the indicator function of the set ${B_{n}}:={T^{-n}}{A_{n}}$. For every $N\gt 0$ we define the following two sums:
We define the quantities ${R_{mn}}$ which characterizes the dependence of two events ${B_{m}}$ and ${B_{n}}$:
A sufficient condition for a sequence $\{{A_{n}}\}$ to be an sBC sequence, in terms of ${R_{mn}}$, was first found by W. Schmidt, and the proof was later provided by Sprindzuk [13] in the context of Diophantine approximations. This condition was recently adapted to dynamical systems by D. Kleinbock and G. Margulis [9].
We suppose that the numbers ${R_{mn}}$ satifies the following condition:
\[ {\sum \limits_{n=M}^{N}}{\sum \limits_{m=M}^{N}}|{R_{mn}}|\le C\cdot {\sum \limits_{n=M}^{N}}\mu ({A_{n}}),\]
for some constant $C\gt 0$ and for all $N\gt M\gt 1$. The last condition is called $(SP)-$ condition. Theorem 1 ([13], Chapter I, Lemma 10).
2.2 Cylinder sets and geometry of supports
We now introduce cylinder sets in symbolic spaces. An alphabet $\mathcal{A}$ is a finite set of symbols; for example, $\mathcal{A}=\{{a_{1}},{a_{2}},\dots ,{a_{k}}\}$.
A cylinder $C\subset {\{0,1\}^{\mathbb{Z}}}$ is obtained by fixing symbols on a finite interval $L=[m,k]\subset \mathbb{Z}$, i.e., for some $\omega =\{{\omega _{m}},\dots ,{\omega _{k}}\}\in {\{0,1\}^{L}}$, we set
Each cylinder is open and closed in ${\{0,1\}^{\mathbb{Z}}}$. We call m and k the left and right endpoints of an interval L, respectively, and $(m+k)/2$ the center of L. Following [4], we introduce the relevant geometry explicitly. If $L=[m,k]\subset \mathbb{Z}$ is a finite lattice interval and $D\in {\mathbb{N}_{0}}$, its D-neighborhood is
We say that an interval ${L_{2}}$ lies in the D-neighborhood of ${L_{1}}$ if ${L_{2}}\subset {N_{D}}({L_{1}})$.
Definition 3 (see [4]).
Two lattice intervals ${L_{1}}$ and ${L_{2}}$ are called D-nested if either ${L_{1}}\subset {N_{D}}({L_{2}})$ or ${L_{2}}\subset {N_{D}}({L_{1}})$.
Definition 4 (see [4]).
For two lattice intervals ${L_{1}}=[{m_{1}},{k_{1}}]$ and ${L_{2}}=[{m_{2}},{k_{2}}]$, not necessarily disjoint, the asymmetric distance from ${L_{1}}$ to ${L_{2}}$ is
Equivalently,
2.3 Stationary determinantal measures
Let ${\mathbb{S}^{1}}=[0,1)$ be the unit circle and let $f:{\mathbb{S}^{1}}\to [0,1]$ be a Borel function with Fourier coefficients
Define the convolution kernel
Then ${K_{f}}$ induces a stationary determinantal measure ${\mu _{f}}$ on ${\{0,1\}^{\mathbb{Z}}}$ via
The process is shift invariant under $\sigma {(x)_{n}}={x_{n+1}}$.
(5)
\[ {\mu _{f}}\big(\{x\in {\{0,1\}^{\mathbb{Z}}}\mid {x_{{n_{1}}}}=\cdots ={x_{{n_{k}}}}=1\}\big)=\det \hspace{-0.1667em}{\big(\widehat{f}({n_{i}}-{n_{j}})\big)_{1\le i,j\le k}}.\]Suppose that a function f is given in the form (4). The $n\times n$ Toeplitz matrix generated by f is defined by ${T_{n}}(f)={\big(\widehat{f}(i-j)\big)_{1\le i,j\le n}}$. Equivalently, ${T_{n}}(f)$ has the explicit form
\[ {T_{n}}(f)=\left(\begin{array}{c@{\hskip10.0pt}c@{\hskip10.0pt}c@{\hskip10.0pt}c@{\hskip10.0pt}c}\widehat{f}(0)& \widehat{f}(-1)& \widehat{f}(-2)& \cdots & \widehat{f}(-(n-1))\\ {} \widehat{f}(1)& \widehat{f}(0)& \widehat{f}(-1)& \cdots & \widehat{f}(-(n-2))\\ {} \widehat{f}(2)& \widehat{f}(1)& \widehat{f}(0)& \cdots & \widehat{f}(-(n-3))\\ {} \vdots & \vdots & \vdots & \ddots & \vdots \\ {} \widehat{f}(n-1)& \widehat{f}(n-2)& \widehat{f}(n-3)& \cdots & \widehat{f}(0)\end{array}\right).\]
We denote by $\mathbf{1}\in {\mathbb{R}^{n}}$ the vector whose all components are equal to 1.
For a $v=({v_{1}},\dots ,{v_{n}})\in {\mathbb{C}^{n}}$, we denote by ${D_{n}}(v)$ the diagonal matrix associated with v, defined by
\[ {D_{n}}(v):=\operatorname{diag}(v)=\left(\begin{array}{c@{\hskip10.0pt}c@{\hskip10.0pt}c@{\hskip10.0pt}c@{\hskip10.0pt}c}{v_{1}}& 0& 0& \cdots & 0\\ {} 0& {v_{2}}& 0& \cdots & 0\\ {} 0& 0& {v_{3}}& \cdots & 0\\ {} \vdots & \vdots & \vdots & \ddots & \vdots \\ {} 0& 0& 0& \cdots & {v_{n}}\end{array}\right).\]
Corollary 1 (see [5]).
Let $f:{\mathbb{S}^{1}}\to [0,1]$ be an integrable function. For every $n\ge 1$, define the Toeplitz matrix ${T_{n}}(f):={\big(\widehat{f}(i-j)\big)_{1\le i,j\le n}}$. Let $\varepsilon =({\varepsilon _{1}},\dots ,{\varepsilon _{n}})\in {\{0,1\}^{n}}$, and $C[\varepsilon \hspace{0.1667em}]:=\big\{x\in {\{0,1\}^{\mathbb{Z}}}:\hspace{2.83862pt}{x_{1}}={\varepsilon _{1}},\dots ,{x_{n}}={\varepsilon _{n}}\big\}$ be a cylinder.
Let $\varepsilon ={({\varepsilon _{i}})_{i=1}^{n}}\in {\{0,1\}^{N}}$ and let ${T_{n}}(f)={(\widehat{f}(i-j))_{1\le i,j\le n}}$ be the Toeplitz matrix generated by f. Then
\[ {\mu _{f}}(C[\varepsilon \hspace{0.1667em}])=\det \hspace{-0.1667em}\Big({D_{n}}(2\varepsilon -1)\hspace{0.1667em}{T_{n}}(f)+{D_{n}}(1-\varepsilon )\Big)=\det {A_{n}}(C[\varepsilon ]),\]
where ${A_{n}}(C[\varepsilon ])$ is the $n\times n$ matrix
\[ {A_{n}}(C[\varepsilon ])=\left(\begin{array}{c@{\hskip10.0pt}c@{\hskip10.0pt}c}(2{\varepsilon _{1}}-1)\widehat{f}(0)+(1-{\varepsilon _{1}})& \cdots & (2{\varepsilon _{1}}-1)\widehat{f}(1-n)\\ {} (2{\varepsilon _{2}}-1)\widehat{f}(1)& \cdots & (2{\varepsilon _{2}}-1)\widehat{f}(2-n)\\ {} \vdots & \ddots & \vdots \\ {} (2{\varepsilon _{n}}-1)\widehat{f}(n-1)& \cdots & (2{\varepsilon _{n}}-1)\widehat{f}(0)+(1-{\varepsilon _{n}})\end{array}\right).\]
2.4 The ψ-mixing coefficient
Let ${({\xi _{n}})_{n\in \mathbb{Z}}}$ be the coordinate process on $\Omega ={\{0,1\}^{\mathbb{Z}}}$, that is, ${\xi _{n}}(x)={x_{n}}$. For integers $n\le m$, define
the sigma-algebra generated by the coordinate maps ${\xi _{n}},\dots ,{\xi _{m}}$. Similarly, set
Then, for each $\ell \ge 1$, the ψ–mixing coefficient of ${\mu _{f}}$ is defined by
\[ {\psi _{{\mu _{f}}}}(\ell )=\underset{\substack{A\in {\mathcal{F}_{-\infty }^{0}},\hspace{0.1667em}B\in {\mathcal{F}_{\ell }^{+\infty }}\\ {} {\mu _{f}}(A){\mu _{f}}(B)\gt 0}}{\sup }\left|\frac{{\mu _{f}}(A\cap B)}{{\mu _{f}}(A){\mu _{f}}(B)}-1\right|.\]
We say that ${\mu _{f}}$ is ψ–mixing if
Theorem 2 (Fan–Liao–Qiu, 2022 see [5]).
Let $f:{S^{1}}\to [0,1]$ be an integrable function which is not identically 0 or 1. Then for every integer $\ell \ge 1$, the ψ–mixing coefficient of the stationary determinantal process ${\mu _{f}}$ satisfies
In particular, if ${\mu _{f}}$ is ψ–mixing, then $f\in {H^{1/2}}(\mathbb{T})$.
(7)
\[ {\psi _{{\mu _{f}}}}(\ell )\ge 1-\exp \hspace{-0.1667em}\left(-{\sum \limits_{n=\ell +1}^{\infty }}|n|\hspace{0.1667em}|\widehat{f}(n){|^{2}}\right).\]
Conversely, if there exists a constant $\tau \gt 0$ such that
then ${\mu _{f}}$ is ψ–mixing and its ψ–function satisfies the upper bound
(8)
\[ f\in {H^{1/2}}(\mathbb{T})\hspace{2em}\textit{and}\hspace{2em}\tau \le f(t)\le 1-\tau \hspace{1em}\textit{for all}\hspace{2.5pt}t\in \mathbb{T},\](9)
\[ {\psi _{{\mu _{f}}}}(\ell )\le \frac{2}{{\tau ^{2}}}\hspace{-0.1667em}\left({\sum \limits_{n=\ell +1}^{\infty }}|n|\hspace{0.1667em}|\widehat{f}(n){|^{2}}\right)\hspace{-0.1667em}\exp \hspace{-0.1667em}\left(1+\frac{2}{{\tau ^{2}}}{\sum \limits_{n=\ell +1}^{\infty }}|n|\hspace{0.1667em}|\widehat{f}(n){|^{2}}\right)\hspace{-0.1667em}.\]2.5 Main results
Lemma 3.
Let $A\in {M_{n}}$ and let ${\lambda _{1}},\dots ,{\lambda _{n}}$ denote the eigenvalues of A counted with their (algebraic) multiplicities (see [8, Definition 1.2.5]). Then
Theorem 3.
Let A be a Hermitian matrix of order n, and let B be a principal submatrix of A of order $n-1$. If
are the eigenvalues of A and B, respectively, then
Lemma 4 (see [3, page 75]).
Let A be an $n\times n$ Hermitian matrix with eigenvalues
and let B be a $k\times k$ principal submatrix of A with eigenvalues
Then, for each $s=1,\dots ,k$, one has
We formulate the main results of our work. The first result gives a uniform exponential estimate for cylinder probabilities. It shows that, under the non-degeneracy condition $\tau \le f\le 1-\tau $, the measure of every n-cylinder decays exponentially in n. This estimate will be used repeatedly to control the size of cylinder targets.
Theorem 4.
Let $f:{\mathbb{S}^{1}}\to [0,1]$ be an integrable Borel function, and let ${\mu _{f}}$ be the associated stationary determinantal probability measure on ${\{0,1\}^{\mathbb{Z}}}$. Assume that there exists a constant $\tau \in (0,1/2)$ such that
\[ \tau \le f\le 1-\tau \hspace{2em}\textit{almost everywhere on}\hspace{2.5pt}{\mathbb{S}^{1}}\hspace{2.5pt}\textit{with respect to the Lebesgue measure}.\]
For every cylinder $C[\varepsilon ]:=C[{\varepsilon _{1}},{\varepsilon _{2}},\dots ,{\varepsilon _{n}}]$, $\hspace{2em}\varepsilon \in {\{0,1\}^{n}}$, the following bounds hold:
The next result is a conditional version of the previous cylinder estimate. It says that fixing additional coordinates decreases the measure at an exponential rate depending only on the number of newly fixed coordinates. This is the determinantal analogue of the finite-energy property for Gibbs measures.
Theorem 5.
Let $f:{\mathbb{S}^{1}}\to [0,1]$ be an integrable Borel function. Assume that there exists a constant $\tau \in (0,1/2)$ such that
Let $\mu ={\mu _{f}}$ be the stationary determinantal probability measure associated with f. Then for any pair of cylinders ${C_{1}}(\omega )\subset C(\omega )$ supported on finite intervals ${L_{1}}\supset L$, one has
We now state the main dynamical Borel–Cantelli result. The geometric assumption on the supports is the D-nested condition, while the probabilistic input is the quantitative mixing of the stationary determinantal measure. Together they imply the Sprindzuk condition and hence the strong Borel–Cantelli property.
Theorem 6.
Let $f:{S^{1}}\to [0,1]$ be a Borel function, and let $\langle {\{0,1\}^{\mathbb{Z}}},\hspace{0.1667em}\mathcal{B},\hspace{0.1667em}{\mu _{f}},\hspace{0.1667em}\sigma \rangle $ be the dynamical system generated by the stationary determinantal point process associated with f, where ${\mu _{f}}$ is the σ-invariant determinantal probability measure induced by the convolution kernel
and σ is the shift map.
Assume that there exist constants $C,a\gt 0$ and $\tau \in (0,1/2)$ such that
\[ |\widehat{f}(n)|\le C{e^{-a|n|}}\hspace{1em}(n\in \mathbb{Z}),\hspace{2em}\tau \le f(t)\le 1-\tau \hspace{1em}\textit{for a.e.}\hspace{2.5pt}t\in \mathbb{T}.\]
Suppose that there exists a fixed constant $D\in {\mathbb{N}_{0}}$ such that the cylinders ${C_{{L_{n}}}}(\omega )$, $n\ge 1$, are supported on finite intervals
and every pair ${L_{m}}$, ${L_{n}}$ is D-nested in the sense of Definition 3. Then,
3 Proof of Theorems 4 and 5
In this section we prove the cylinder estimates stated in the previous section. The first estimate gives a uniform exponential upper and lower bound for the measure of every finite cylinder. The second estimate is a finite-energy type bound, controlling the change of measure when a cylinder is refined by fixing additional coordinates. These estimates will be used in the proof of the Sprindzuk condition for D-nested cylinder targets.
3.1 Uniform bounds for cylinder probabilities
We first prove the uniform cylinder estimate. The key point is that the Shirai–Takahashi formula allows us to express each cylinder probability as a finite Toeplitz determinant. The non-degeneracy condition $\tau \le f\le 1-\tau $ then gives a uniform bound on the corresponding finite-dimensional operator.
Proof.
Fix $n\ge 1$ and a binary word
Consider the corresponding cylinder set
\[ C[\varepsilon ]:=\{x\in {\{0,1\}^{\mathbb{Z}}}:{x_{1}}={\varepsilon _{1}},\dots ,{x_{n}}={\varepsilon _{n}}\}.\]
By shift-invariance of the measure ${\mu _{f}}$, it suffices to consider cylinders supported on the index set $\{1,\dots ,n\}$.Let
be the Toeplitz matrix generated by f. For a vector $v\in {\mathbb{C}^{n}}$, denote by ${D_{n}}(v)$ the diagonal matrix with diagonal entries ${v_{1}},\dots ,{v_{n}}$.
Using the determinantal representation of cylinder measures, we have
where $\mathbf{1}=(1,\dots ,1)\in {\mathbb{R}^{n}}$.
(10)
\[ {\mu _{f}}(C[\varepsilon ])=\det \hspace{-0.1667em}\Big({D_{n}}(2\varepsilon -\mathbf{1})\hspace{0.1667em}{T_{n}}(f)+{D_{n}}(\mathbf{1}-\varepsilon )\Big),\]We first rewrite the matrix in (10) into a more convenient form. Set
Then the following identity holds:
where ${I_{n}}$ denotes the $n\times n$ identity matrix.
(11)
\[ {D_{n}}(2\varepsilon -\mathbf{1})\hspace{0.1667em}{T_{n}}(f)+{D_{n}}(\mathbf{1}-\varepsilon )=\frac{1}{2}\Big({I_{n}}+{D_{n}}(\eta )\hspace{0.1667em}{T_{n}}(g)\Big),\]Indeed, since
and
we obtain
\[ {D_{n}}(\mathbf{1}-\varepsilon )=\frac{1}{2}\big({I_{n}}-{D_{n}}(\eta )\big),\hspace{2em}{D_{n}}(2\varepsilon -\mathbf{1})={D_{n}}(\eta ).\]
Substituting these expressions into the left-hand side of (11), we compute
\[\begin{aligned}{}{D_{n}}(2\varepsilon -\mathbf{1})\hspace{0.1667em}{T_{n}}(f)+{D_{n}}(\mathbf{1}-\varepsilon )& ={D_{n}}(\eta )\cdot \frac{1}{2}\big({T_{n}}(g)+{I_{n}}\big)+\frac{1}{2}\big({I_{n}}-{D_{n}}(\eta )\big)\\ {} & =\frac{1}{2}\Big({D_{n}}(\eta )\hspace{0.1667em}{T_{n}}(g)+{I_{n}}\Big),\end{aligned}\]
which proves (11).Combining (10) and (11), we obtain
where
We now estimate $\det ({I_{n}}+A)$. Since ${D_{n}}(\eta )$ is diagonal with entries $\pm 1$, we have
Therefore,
The matrix ${T_{n}}(g)$ is a finite compression of the bi-infinite Toeplitz operator $T(g)$ acting on ${\ell ^{2}}(\mathbb{Z})$, hence
Moreover, $T(g)$ is unitarily equivalent to the multiplication operator by g on ${L^{2}}(\mathbb{T})$ (see, for example, [2], p. 179, Corollary 7.8), which yields
Using the assumption $\tau \le f\le 1-\tau $ almost everywhere, we obtain
Consequently,
Let ${\lambda _{1}}(A),\dots ,{\lambda _{n}}(A)$ denote the eigenvalues of A. Since the spectral radius satisfies $\rho (A)\le \| A\| $, it follows from (14) that
The eigenvalues of ${I_{n}}+A$ are $1+{\lambda _{i}}(A)$, and hence
By the triangle inequality,
\[ 1-|{\lambda _{i}}(A)|\le |1+{\lambda _{i}}(A)|\le 1+|{\lambda _{i}}(A)|,\hspace{2em}i=1,\dots ,n.\]
Combining these inequalities, we obtain
\[ {2^{n}}\hspace{0.1667em}{\tau ^{n}}\le |\det ({I_{n}}+A)|\le {2^{n}}\hspace{0.1667em}{(1-\tau )^{n}}.\]
By (12), we have
This completes the proof of Theorem 4. □3.2 Conditional cylinder estimates
We next prove a conditional version of the previous estimate. This result shows that the measure of a cylinder decreases exponentially when additional coordinates are fixed. Such a bound is needed later when two shifted cylinders overlap, since their intersection is again a cylinder supported on the union of the two intervals.
Proof.
We present the proof in the two pure cases, namely when the defining word consists only of ones or only of zeros. In these cases the matrices ${M_{\Lambda }}$ and ${M_{{\Lambda _{1}}}}$ are Hermitian positive definite, and the proof follows directly from the Cauchy interlacing theorem.
Define
where ${T_{m}}(f)$ is the $m\times m$ Toeplitz matrix generated by f. By the Shirai–Takahashi formula for stationary determinantal processes,
Let ${\Lambda _{1}}$ be an extension of Λ:
Since ${M_{\Lambda }}$ is a principal submatrix of ${M_{{\Lambda _{1}}}}$ and both matrices are Hermitian, the Cauchy interlacing theorem (Lemma 4) yields
\[ {\Lambda _{1}}=({\varepsilon _{1}^{(1)}},\dots ,{\varepsilon _{m+k}^{(1)}})=\underset{{\Lambda _{l}}}{\underbrace{({\varepsilon _{1}^{l}},\dots ,{\varepsilon _{{k_{1}}}^{l}})}}\hspace{0.2778em}\underset{\Lambda }{\underbrace{({\varepsilon _{1}},\dots ,{\varepsilon _{m}})}}\hspace{0.2778em}\underset{{\Lambda _{r}}}{\underbrace{({\varepsilon _{1}^{r}},\dots ,{\varepsilon _{{k_{2}}}^{r}})}},\]
where ${k_{1}}+{k_{2}}=k$. Set
\[ A:={D_{{k_{1}}}}(2{\varepsilon ^{l}}-1),\hspace{1em}B:={D_{m}}(2\varepsilon -1),\hspace{1em}C:={D_{{k_{2}}}}(2{\varepsilon ^{r}}-1),\]
and
\[ {E_{l}}:={D_{{k_{1}}}}(1-{\varepsilon ^{l}}),\hspace{1em}E:={D_{m}}(1-\varepsilon ),\hspace{1em}{E_{r}}:={D_{{k_{2}}}}(1-{\varepsilon ^{r}}).\]
Then the matrix ${M_{{\Lambda _{1}}}}$ has the block form
\[ {M_{{\Lambda _{1}}}}=\left(\begin{array}{c@{\hskip10.0pt}c@{\hskip10.0pt}c}A{T_{ll}}+{E_{l}}& A{T_{l}}& A{T_{lr}}\\ {} B{T_{ml}}& BT+E& B{T_{mr}}\\ {} C{T_{rl}}& C{T_{r}}& C{T_{rr}}+{E_{r}}\end{array}\right),\]
with
Consequently,
and therefore
\[ \frac{{\mu _{f}}({C_{1}})}{{\mu _{f}}(C)}=\frac{\det ({M_{{\Lambda _{1}}}})}{\det ({M_{\Lambda }})}.\]
By Theorem 4,
Moreover,
(15)
\[ \tau \le {\lambda _{1}^{({\Lambda _{1}})}}\le \cdots \le {\lambda _{m+k}^{({\Lambda _{1}})}}\le 1-\tau .\]
\[ {\lambda _{i}^{({\Lambda _{1}})}}\hspace{0.2778em}\le \hspace{0.2778em}{\lambda _{i}^{(\Lambda )}}\hspace{0.2778em}\le \hspace{0.2778em}{\lambda _{i+k}^{({\Lambda _{1}})}},\hspace{2em}i=1,\dots ,m.\]
Using the product representation of determinants,
\[ \frac{{\mu _{f}}({C_{1}})}{{\mu _{f}}(C)}=\frac{{\textstyle\textstyle\prod _{i=1}^{m+k}}{\lambda _{i}^{({\Lambda _{1}})}}}{{\textstyle\textstyle\prod _{i=1}^{m}}{\lambda _{i}^{(\Lambda )}}}\ge {\prod \limits_{i=1}^{k}}{\lambda _{i}^{({\Lambda _{1}})}}\ge {\tau ^{\hspace{0.1667em}k}}.\]
Similarly,
\[ \frac{{\mu _{f}}({C_{1}})}{{\mu _{f}}(C)}\le {\prod \limits_{i=m+1}^{m+k}}{\lambda _{i}^{({\Lambda _{1}})}}\le {(1-\tau )^{\hspace{0.1667em}k}}.\]
Since $k=|{\Lambda _{1}}|-|\Lambda |$, we obtain
It remains to consider the mixed case, when both symbols 0 and 1 occur in the word ε. In this case the matrix
\[ {M_{\Lambda }}={D_{\Lambda }}(2\varepsilon -\mathbf{1}){T_{\Lambda }}(f)+{D_{\Lambda }}(\mathbf{1}-\varepsilon )\]
is not Hermitian in general. Put
Then
\[ {H_{\Lambda }}:={S_{\Lambda }}{M_{\Lambda }}={T_{\Lambda }}(f)-{D_{\Lambda }}(\mathbf{1}-\varepsilon )\]
is Hermitian. Similarly,
\[ {H_{{\Lambda _{1}}}}:={S_{{\Lambda _{1}}}}{M_{{\Lambda _{1}}}}={T_{{\Lambda _{1}}}}(f)-{D_{{\Lambda _{1}}}}(\mathbf{1}-{\varepsilon _{1}})\]
is Hermitian. Since ${C_{{\Lambda _{1}}}}({\varepsilon _{1}})\subset {C_{\Lambda }}(\varepsilon )$, the matrix ${H_{\Lambda }}$ is a principal submatrix of ${H_{{\Lambda _{1}}}}$. Hence the Cauchy interlacing theorem applies. Moreover,
\[ \det {M_{\Lambda }}=\det {S_{\Lambda }}\hspace{0.1667em}\det {H_{\Lambda }},\hspace{2em}\det {M_{{\Lambda _{1}}}}=\det {S_{{\Lambda _{1}}}}\hspace{0.1667em}\det {H_{{\Lambda _{1}}}}.\]
Thus the signs coming from the zero symbols are completely accounted for by $\det {S_{\Lambda }}$ and $\det {S_{{\Lambda _{1}}}}$. Therefore, applying the same eigenvalue-ratio argument as above together with the Cauchy interlacing theorem, we obtain
This completes the proof of Theorem 5. □
4 Mixing estimates and proof of Main Theorem 6
In this section we prove the main dynamical Borel–Cantelli result. The proof has two steps. First, using the quantitative ψ-mixing estimate for stationary determinantal processes, we derive an exponential correlation bound for separated cylinder events. Second, this bound is combined with the geometric D-nested condition on the supports to verify the Schmidt–Sprindzuk condition.
Theorem 7 (Exponential cylinder mixing bound).
Let $f:{\mathbb{S}^{1}}\to [0,1]$ satisfy the following assumptions:
Then there exist constants ${c_{3}}\gt 0$ and $\theta \in (0,1)$ (depending only on a, C, τ) such that for any cylinder sets ${C_{1}}$ and ${C_{2}}$ whose supports are separated by a gap $d:={n_{2}^{-}}-{n_{1}^{+}}\in \mathbb{N}$, one has
Proof.
Let $\psi (\ell )$ denote the ψ–mixing coefficient of ${\mu _{f}}$ at separation ℓ (as in Fan–Liao–Qiu, Def. of ψ–mixing). By the definition of $\psi (\ell )$, for any $A\in {\mathcal{F}_{-\infty }^{0}}$ and $B\in {\mathcal{F}_{\ell }^{+\infty }}$ with ${\mu _{f}}(A){\mu _{f}}(B)\gt 0$,
If ${C_{1}}$ and ${C_{2}}$ are cylinder sets supported on disjoint coordinate blocks separated by $d:={n_{2}^{-}}-{n_{1}^{+}}\ge 1$, then ${C_{1}}\in {\mathcal{F}_{-\infty }^{0}}$ and ${C_{2}}\in {\mathcal{F}_{d}^{+\infty }}$. Thus (17) with $\ell =d$ gives
(17)
\[ \big|{\mu _{f}}(A\cap B)-{\mu _{f}}(A){\mu _{f}}(B)\big|\hspace{0.2778em}\le \hspace{0.2778em}\psi (\ell )\hspace{0.1667em}{\mu _{f}}(A){\mu _{f}}(B).\]Under (A2), Theorem 2 provides, for every $\ell \ge 1$,
Assumption (A1) gives $|\{\widehat{f}\}(n)|\le C{e^{-an}}$ for $n\ge 0$. Let $r:={e^{-2a}}\in (0,1)$. Then
for some ${K_{2}}={K_{2}}(a,C,\tau )\gt 0$ (since $S(\ell )\to 0$, the exponential factor in (19) stays bounded).
\[ S(\ell )\hspace{3.33333pt}\le \hspace{3.33333pt}{C^{2}}{\sum \limits_{n=\ell +1}^{\infty }}n{r^{\hspace{0.1667em}n}}\hspace{3.33333pt}=\hspace{3.33333pt}{C^{2}}\hspace{0.1667em}\frac{{r^{\ell +1}}\big((\ell +1)-\ell r\big)}{{(1-r)^{2}}}\hspace{3.33333pt}\le \hspace{3.33333pt}{K_{1}}\hspace{0.1667em}(\ell +1)\hspace{0.1667em}{e^{-2a(\ell +1)}},\]
with ${K_{1}}:={C^{2}}{(1-{e^{-2a}})^{-2}}$. Substituting this in (19) we obtain
(20)
\[ \psi (\ell )\hspace{3.33333pt}\le \hspace{3.33333pt}{K_{2}}\hspace{0.1667em}(\ell +1)\hspace{0.1667em}{e^{-2a(\ell +1)}},\]Fix any $c\in (0,2a)$ and set $\theta :={e^{-c}}\in (0,1)$. Using the standard bound ${\sup _{x\gt 0}}x{e^{-\gamma x}}=1/(e\gamma )$ for $\gamma \gt 0$, we have
\[\begin{aligned}{}(\ell +1){e^{-2a(\ell +1)}}& ={e^{-c(\ell +1)}}\cdot (\ell +1){e^{-(2a-c)(\ell +1)}}\le \\ {} & \le \frac{1}{e(2a-c)}\hspace{0.1667em}{e^{-c(\ell +1)}}=\frac{1}{e(2a-c)}\hspace{0.1667em}{\theta ^{\hspace{0.1667em}\ell +1}}.\end{aligned}\]
Hence, from (20),
\[ \psi (\ell )\hspace{3.33333pt}\le \hspace{3.33333pt}\frac{{K_{2}}}{e(2a-c)}\hspace{0.1667em}{\theta ^{\hspace{0.1667em}\ell +1}}\hspace{3.33333pt}\le \hspace{3.33333pt}{K_{3}}\hspace{0.1667em}{\theta ^{\hspace{0.1667em}\ell }},\]
for a constant ${K_{3}}={K_{3}}(a,C,\tau ,c)\gt 0$ (absorbing the extra factor θ into ${K_{3}}$). Putting $\ell =d$ in (18) yields
\[ \big|{\mu _{f}}({C_{1}}\cap {C_{2}})-{\mu _{f}}({C_{1}}){\mu _{f}}({C_{2}})\big|\hspace{3.33333pt}\le \hspace{3.33333pt}{K_{3}}\hspace{0.1667em}{\theta ^{\hspace{0.1667em}d}}\hspace{0.1667em}{\mu _{f}}({C_{1}}){\mu _{f}}({C_{2}}).\]
Finally, enlarging the constant if necessary to cover finitely many small d, we obtain (16) with ${c_{3}}:={K_{3}}$. □Lemma 5.
Assume that f satisfies assumptions (A1)–(A2) of Theorem 7. Let ${C_{1}}$ and ${C_{2}}$ be cylinders supported on intervals ${L_{1}}$ and ${L_{2}}$, respectively. Then there exist constants ${c_{4}}\gt 0$ and ${\theta _{4}}\in (0,1)$ such that
Proof.
There are two cases depending on the position of ${L_{1}}$ and ${L_{2}}$: disjoint and intersecting.
1) Let ${L_{1}}$ and ${L_{2}}$ be disjoint. First assume that ${k_{1}}\lt {m_{2}}$. Applying Theorem 7, we have
\[ \left|\mu \left({C_{1}}\cap {C_{2}}\right)-\mu \left({C_{1}}\right)\mu \left({C_{2}}\right)\right|\le {c_{3}}{\theta ^{{m_{2}}-{k_{1}}}}\mu \left({C_{1}}\right)\mu \left({C_{2}}\right).\]
By Theorem 4,
Therefore,
\[\begin{aligned}{}\left|\mu \left({C_{1}}\cap {C_{2}}\right)-\mu \left({C_{1}}\right)\mu \left({C_{2}}\right)\right|& \le {c_{3}}{\theta ^{{m_{2}}-{k_{1}}}}{(1-\tau )^{{k_{2}}-{m_{2}}+1}}\mu \left({C_{1}}\right)\\ {} & \le {c_{4}}{\theta _{4}^{{k_{2}}-{k_{1}}}}\mu \left({C_{1}}\right)\\ {} & ={c_{4}}{\theta _{4}^{\delta \left({L_{1}},{L_{2}}\right)}}\mu \left({C_{1}}\right).\end{aligned}\]
Here we used the fact that, for ${k_{1}}\lt {m_{2}}$,
The case ${k_{2}}\lt {m_{1}}$ is treated similarly. In this case,
and the same argument gives
2) Let ${L_{1}}$ and ${L_{2}}$ be intersecting. If ${C_{1}}\cap {C_{2}}=\varnothing $, then
\[ \left|\mu \left({C_{1}}\cap {C_{2}}\right)-\mu \left({C_{1}}\right)\mu \left({C_{2}}\right)\right|=\mu ({C_{1}})\mu ({C_{2}}).\]
Using Theorem 4, we obtain
Since ${L_{1}}\cap {L_{2}}\ne \varnothing $, we have
Hence
Now suppose that ${C_{1}}\cap {C_{2}}\ne \varnothing $. Then ${C_{1}}\cap {C_{2}}$ is a cylinder supported on ${L_{1}}\cup {L_{2}}$, and
Applying Theorem 5, we have
Since
we get
On the other hand, by Theorem 4,
Therefore,
The proof is complete. □
Proof of Theorem 6..
Suppose that the intervals ${L_{n}}=[{m_{n}},{k_{n}}]$, $n\in \mathbb{N}$, satisfy the D-nested condition. It is important to note that no assumptions are made regarding the relationship between n and m, nor between the measures of ${C_{n}}(\omega )$ and ${C_{m}}(\omega )$.
We denote by ${L_{n}}-(m-n)$ the segment formed by shifting ${L_{n}}$ to $(m-n)$ to the left, i.e.
Without loss of generality, assume that ${L_{m}}$ lies in the D-neighborhood of ${L_{n}}$, that is, ${L_{m}}\subset {N_{D}}({L_{n}}).$ The opposite case is obtained by interchanging m and n. We estimate the δ-asymmetric distance between ${L_{m}}$ and ${L_{n}}-(m-n)$. By the D-nested property, we have
Consider the quantity
\[\begin{aligned}{}{R_{mn}}& =\mu ({\sigma ^{-m}}{C_{m}}(\omega )\cap {\sigma ^{-n}}{C_{n}}(\omega ))-\mu ({\sigma ^{-n}}{C_{n}}(\omega ))\mu ({\sigma ^{-m}}{C_{m}}(\omega ))\\ {} & =\mu ({C_{m}}(\omega )\cap {\sigma ^{m-n}}{C_{n}}(\omega ))-\mu ({C_{n}}(\omega ))\mu ({C_{m}}(\omega )).\end{aligned}\]
By applying Lemma 5 to the cylinders ${C_{m}}(\omega )$ and ${\sigma ^{m-n}}{C_{n}}(\omega )$, we obtain
Hence
Since D is fixed, replacing ${c_{4}}$ by a larger constant if necessary, we get
We first sum over the pairs satisfying ${L_{m}}\subset {N_{D}}({L_{n}})$. Since ${R_{mn}}={R_{nm}}$, the remaining pairs are treated by interchanging m and n. Absorbing the resulting factor 2 into ${c_{4}}$, for $N\gt M$ we have
\[\begin{aligned}{}{\sum \limits_{m=M}^{N}}{\sum \limits_{n=M}^{N}}|{R_{mn}}|& \le {c_{4}}{\sum \limits_{m=M}^{N}}\mu ({C_{m}}(\omega )){\sum \limits_{n=M}^{N}}{\theta _{4}^{|m-n|}}\\ {} & \le {c_{4}}{\sum \limits_{m=M}^{N}}\mu ({C_{m}}(\omega )){\sum \limits_{j=-\infty }^{\infty }}{\theta _{4}^{|j|}}.\end{aligned}\]
Since
\[ {\sum \limits_{j=-\infty }^{\infty }}{\theta _{4}^{|j|}}=1+2{\sum \limits_{j=1}^{\infty }}{\theta _{4}^{j}}=\frac{1+{\theta _{4}}}{1-{\theta _{4}}}\lt \infty ,\]
we obtain, again increasing ${c_{4}}$ if necessary,
\[ {\sum \limits_{m=M}^{N}}{\sum \limits_{n=M}^{N}}|{R_{mn}}|\le {c_{4}}{\sum \limits_{m=M}^{N}}\mu ({C_{m}}(\omega )).\]
Thus the condition $(SP)$ is satisfied. Therefore, if
then the sequence ${\{{C_{n}}(\omega )\}_{n\ge 1}}$ is strongly Borel–Cantelli. Theorem 6 is proved. □Example 1.
Let $r\in (0,1)$ and choose $\varepsilon \in \mathbb{R}$ such that
Define
Then f satisfies the assumptions of the Theorem 7.
Indeed, since
we have
Hence, for any
one gets
Also,
If we set
then
Thus assumptions (A1) and (A2) are satisfied.
Therefore, by the theorem, there exist constants ${c_{3}}\gt 0$ and $\theta \in (0,1)$ such that for any cylinder sets ${C_{1}},{C_{2}}$ whose supports are separated by a gap $d\in \mathbb{N}$,
In particular, the corresponding ψ–mixing upper bound decays exponentially.
