Abstract
This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the \(\beta \)-Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.
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Notes
The subindices of \(X_{k,i}\) here start with 2, since, for example, in the case \(d_i=f(a_{i-1},a_i)\), \(i\ge 1\), the distribution of \(d_1(=f(0,a_1))\) is different from that of \(d_i=f(a_{i-1},a_i)\), \(i\ge 2\).
In [30, (3.5)], \(m_2^{k+l}\), \(m_2^{\mathbb {I}_h(\gamma _1)+\mathbb {I}_h(\gamma _2)}\) and the denominator \({\alpha }(k+l)\) shall be modified by \(m_2^{\frac{1}{2}(k+l)}\), \(m_2^{\frac{1}{2} (\mathbb {I}_i(\gamma _1)+\mathbb {I}_i(\gamma _2))}\) and \({\alpha }(k+l)+1-2{\alpha }\), respectively.
In [30, Corollary 2], the terms \(\frac{kl}{{\alpha }(k+l)}\) in the even and odd cases shall be modified by \(\frac{kl\sigma _Z^2}{{\alpha }(k+l)+1-2{\varepsilon }}\) and \(\frac{kl\sigma _d^2}{{\alpha }(k+l)+1-2{\alpha }}\), respectively.
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Acknowledgments
The author is grateful to Professor Dong Han for several valuable discussions.
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This work is supported by National Basic Research Program of China (973 Program, 2011CB808000, 2015CB856004), National Natural Science Foundations of China (11171216) and China Postdoctoral Science Foundation Funded Project (2015M581598).
Appendix
Appendix
Proof of (3.13)
By Assumption (H.1) and (H.2),
where \(\widetilde{Q}_{l,i}^{\overrightarrow{m}_l} = Q_{l,i}^{\overrightarrow{m}_l} - {\mathbb {E}}Q_{l,i}^{\overrightarrow{m}_l}\). Hence,
with \(\Vert \cdot \Vert _2\) denoting the standard \(L^2\) norm, which implies (3.13). \(\square \)
Proof of (3.21)
It is equivalent to prove that, as \(n\rightarrow {\infty }\),
First consider the nonsymmetric case. Let \({\alpha }_i=(a_{i-1}, d_i, b_i)\), \(i\ge 1\) and \(a_0=0\). By the independence and weak convergence of \({\alpha }_i\) in Assumption (H.1), it is not difficult to deduce that
where \(\widetilde{{\alpha }}_i\) are independent but with the same distribution as that of \((a+ \eta , d+\zeta , b+ \xi )\). Then, by (4.5) and the continuous mapping theorem ([19, Theorem 3.2.4]),
with F the continuous function defined as in (4.2).
Similarly,
On the other hand, by (3.3) and Hölder’s inequality
which implies the uniform integrabilities of \(n^{-2{\alpha }k} X_{k,n}X_{k,n+j}\), \(n^{-{\alpha }k} X_{k,n}\) and \(n^{-{\alpha }k} X_{k,n+j}\), \(n\ge 1\).
Therefore, by (6.3)–(6.5), we can apply the Skorohod representation theorem and the uniform integrability to take the limit and obtain (6.2) for the non-symmetric case. (See, e.g., [19, Theorem 3.2.4] for similar arguments for the bounded continuous mapping.)
The symmetric case can be proved analogously. In fact, with the uniform integrability (6.5), we only need to check the weak convergence of \(n^{-{\alpha }k} X_{k,n}\).
In the case that all \(d_i\) are independent of \(a_i(=b_i)\), let \(\beta _i = (d_i, a_i)\), \(i\ge 1\). In this case, \(X_{k,n}= F'(\beta _n,\ldots , \beta _{n+[\frac{k}{2}]})\) with \(F'\) defined as in (4.3). Then, it follows from similar arguments as above that
where \(\widetilde{\beta _i}\), \(1\le i\le [\frac{k}{2}]+1\), are independent but with the common distribution as that of \((d+\xi , a+\eta )\).
In the case that \(d_i=f(a_{i-1}, a_i)\), then \(\beta _i = (f(a_{i-1}, a_i), a_i)\), \(X_{k,n} \) is now a continuous function of \(a_{n-1}, \ldots , a_{n+[\frac{k}{2}]}\). As \((a_{n-1},\ldots , a_{n+[\frac{k}{2}]}) \overset{d}{\rightarrow } (\widetilde{{\alpha }_1}, \ldots , \widetilde{{\alpha }}_{[\frac{k}{2}]+2})\), by the continuous mapping theorem, we can obtain the weak convergence of \(n^{-{\alpha }k} X_{k,n}\). The proof is consequently complete. \(\square \)
Proof of (4.4)
Set \(N(k)= |\Psi _k|\), the number of sets in \(\Psi _k\) which is defined as in (2.3). N(k) is finite and depends only on k. Note that
Then, letting C denote the maximum of \(C_{l}^{\overrightarrow{m}_l,\overrightarrow{n}_l}\) over the finite sets in \(\Psi _k\) and setting \(\delta _{k} = \delta /([\frac{k}{2}] C N(k))\), we deduce that
Since by (2.1) and Assumption (H.3), \(\{ |Q _{l,i}^{\overrightarrow{m}_l,\overrightarrow{n}_l}| \}_{i\ge 1}\) is uniformly bounded, which implies that \({\mathbb {P}}(|Q _{l,i}^{\overrightarrow{m}_l,\overrightarrow{n}_l}| \ge n\delta _k) =0\) for n large enough. Hence, by Lemma 1.2.15 in [13],
yielding (4.4) as claimed. \(\square \)
Proof of (4.6)
Similarly to the proof of (6.7), we derive that
with \(\delta _k\) defined as in the proof of (6.7), which yields (4.6), due to the fact that \(\{Q_{l, i}^{\overrightarrow{m}_l,\overrightarrow{n}_l}\}_{i\ge 1}\) is uniformly bounded and \(n/{\lambda }_n \rightarrow {\infty }\).
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Zhang, D. Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations. J Theor Probab 30, 1076–1103 (2017). https://doi.org/10.1007/s10959-016-0683-7
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DOI: https://doi.org/10.1007/s10959-016-0683-7