[Paper Review] Cram\'{e}r's moderate deviations for martingales with applications

Xiequan Fan, Qi-Man Shao|arXiv (Cornell University)|Apr 6, 2022

Advanced Harmonic Analysis Research25 references23 citations

TL;DR

This paper establishes Cramér's moderate deviation expansions for normalized martingales $X_n / \sqrt{\langle X \rangle_n}$ and standardized martingales $X_n / \sqrt{\mathbb{E}X_n^2}$ under a conditional Bernstein condition and a tail concentration condition on the predictable quadratic variation. The key result shows that for $x = o(n^{1/6})$, the tail probability $\mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x)$ is asymptotically equivalent to the standard normal tail $1 - \Phi(x)$, extending classical Cramér results to martingale settings with broader applicability to models like elephant random walks and autoregressive processes.

ABSTRACT

Let $(\xi_i,\mathcal{F}_i)_{i\geq1}$ be a sequence of martingale differences. Set $X_n=\sum_{i=1}^n \xi_i $ and $ \langle X angle_n=\sum_{i=1}^n \mathbf{E}(\xi_i^2|\mathcal{F}_{i-1}).$ We prove Cram\'er's moderate deviation expansions for $\displaystyle \mathbf{P}(X_n/\sqrt{\langle X angle_n} \geq x)$ and $\displaystyle \mathbf{P}(X_n/\sqrt{ \mathbf{E}X_n^2} \geq x)$ as $n o\infty.$ Our results extend the classical Cram\'{e}r result to the cases of normalized martingales $X_n/\sqrt{\langle X angle_n}$ and standardized martingales $X_n/\sqrt{ \mathbf{E}X_n^2}$, with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed.

Motivation & Objective

• To extend Cramér’s classical moderate deviation results for i.i.d. sums to the case of normalized martingales $X_n / \sqrt{\langle X \rangle_n}$, where $\langle X \rangle_n$ is the predictable quadratic variation.
• To establish precise asymptotic expansions for the relative error in the normal approximation of the tail probability $\mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x)$, improving upon prior results for standardized martingales.
• To provide a framework applicable to dependent and non-i.i.d. processes such as elephant random walks and autoregressive processes, where the classical Lindeberg or conditional moment conditions are not satisfied.
• To replace the restrictive uniform boundedness condition (1.3) with a more natural tail concentration condition (1.9) on the quadratic variation, enabling broader applicability.
• To derive moderate deviation principles and asymptotic confidence intervals for estimators in time series models, such as the autoregressive parameter in AR(1) processes.

Proposed method

• Introduce conditions (A1) and (A2): (A1) is a conditional Bernstein condition on higher-order conditional moments of martingale differences; (A2) is a sub-Gaussian tail bound on the deviation of the predictable quadratic variation $\langle X \rangle_n$ from its mean.
• Use martingale central limit theorem techniques and exponential moment bounds to control the cumulant generating function of the normalized martingale $X_n / \sqrt{\langle X \rangle_n}$.
• Apply a version of the Gärtner-Ellis theorem and exponential tilting to derive moderate deviation expansions for the logarithm of the tail probability.
• Establish a key inequality (Theorem 2.1) bounding $\left| \ln \mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x) - \ln(1 - \Phi(x)) \right|$ by terms involving $x^3(\epsilon_n + \delta_n)$ and $\delta_n |\ln \delta_n| + \epsilon_n |\ln \epsilon_n|$.
• Use the expansion to derive asymptotic equivalence $\mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x) \sim 1 - \Phi(x)$ for $x = o(n^{1/6})$, extending the classical Cramér result to martingales.
• Apply the results to two specific models: elephant random walks (with long-range dependence) and AR(1) processes, showing that the tail approximation holds under the new conditions.

Experimental results

Research questions

• RQ1 Can Cramér’s moderate deviation expansion, known for i.i.d. sums, be extended to normalized martingales $X_n / \sqrt{\langle X \rangle_n}$ under weaker moment and concentration conditions?
• RQ2 What conditions on the conditional moments and quadratic variation concentration are sufficient to ensure that the tail probability $\mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x)$ is asymptotically equivalent to the standard normal tail $1 - \Phi(x)$?
• RQ3 How does the range of validity of the moderate deviation expansion compare to the classical i.i.d. case, and can it be extended to $x = o(n^{1/6})$?
• RQ4 Can the results be applied to non-i.i.d. processes such as elephant random walks and autoregressive processes, where the classical Lindeberg condition fails?
• RQ5 What are the implications for statistical inference, such as constructing confidence intervals for parameter estimators in time series models?

Key findings

• Under conditions (A1) and (A2), the relative error in the normal approximation for $\mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x)$ is bounded by $c\left( x^3(\epsilon_n + \delta_n) + (1+x)(\delta_n |\ln \delta_n| + \epsilon_n |\ln \epsilon_n|) \right)$, where $\epsilon_n, \delta_n \to 0$ as $n \to \infty$.
• For $x = o(n^{1/6})$, the tail probability satisfies $\mathbb{P}(X_n / \sqrt{\langle X \rangle_n} > x) = 1 + o(1)$, which matches the classical Cramér result for i.i.d. sequences.
• The expansion (1.10) holds not only for normalized martingales but also for standardized martingales $S_n / \sqrt{n\sigma^2}$ under the same conditions, extending prior results.
• The results are applied to elephant random walks, where the conditional Bernstein condition and concentration of the quadratic variation are verified, and the moderate deviation expansion holds for $x = o(n^{1/6})$.
• For AR(1) processes, the method yields a moderate deviation principle for the least squares estimator $\hat{\theta}_n$, and confidence intervals $[A_n, B_n]$ are constructed with asymptotic coverage $1 - \kappa_n$.
• The asymptotic confidence interval $[A_n, B_n]$ for the autoregressive parameter $\theta$ is shown to have width $o(n^{1/6})$, consistent with the moderate deviation scaling.