Almost Sure Invariance Principle Rates

• Almost Sure Invariance Principle (ASIP) rates quantify the speed of coupling partial sums of dependent variables to Brownian motion with explicit pathwise error control.
• They utilize martingale approximations, spectral methods, and Young tower structures to achieve optimal convergence rates under various mixing and moment conditions.
• Applications span stationary processes, dynamical systems, and vector-valued models, offering rates from logarithmic loss to subpolynomial convergence.

The almost sure invariance principle (ASIP) provides a strong form of approximation of partial sums of dependent random variables or functional sequences by Brownian motion, including explicit control of the pathwise error rate. Rates in the ASIP quantify the speed and sharpness of this coupling—typically depending intricately on mixing, moment, and spectral properties of the underlying process or dynamical system. Recent research has produced a broad taxonomy of sharp rates for stationary and nonstationary processes, mixing sequences, deterministic and random dynamical systems, martingale and reverse martingale regimes, and vector-valued models.

1. Foundational Results and Classical Martingale Techniques

The foundational approach for rates in the ASIP in stationary and weakly dependent setups is martingale approximation, pioneered via the Maxwell–Woodroofe condition: $$A(X_0) := \sum_{k=1}^\infty k^{-3/2}\,\|E_0 S_k\|_2 < \infty$$ for a stationary, square-integrable, centered process $(X_k)_{k\in\mathbb{Z}}$. Here, $S_n = \sum_{i=1}^n X_i$, $E_0(\cdot) = E(\cdot\mid\mathcal{F}_0)$. Under this condition, there exists a unique martingale $(M_n)$ with stationary differences such that for all $n$

$$\| S_n - M_n \|_2^2 \ll n \sum_{k = n+1}^\infty k^{-3/2}\| E_0 S_k \|_2$$

and, crucially, for suitable slowly varying $b_n$ sequences,

$$\frac{S_n - M_n}{\sqrt{n} b_n^*} \to 0\quad \text{almost surely}$$

where $b_n^* = \sum_{k=1}^n \frac{1}{k b_k}$, assuming

$$\sum_{n = 1}^\infty \frac{1}{n b_n} \sum_{k = n+1}^\infty k^{-3/2}\|E_0 S_k\|_2 < \infty.$$

Optimally, this yields $o(\sqrt{n \log_2 n})$ rates for $(S_n - M_n)$, which transfer via Strassen embedding to the same almost-sure pathwise rate for the coupling $S_n - W(c n)$ by Brownian motion $W$, with $c^2 = E(D_0^2)$, and ensure law of iterated logarithm (LIL) behavior (Merlevède et al., 2011).

2. Rates in Dynamical Systems: Expanding and Intermittent Maps

For deterministic dynamical systems—specifically, uniformly expanding and intermittent (Pomeau–Manneville type) maps—a refined spectral approach yields optimal rates. For a uniformly expanding interval map $T$, with a spectral gap in bounded variation or Hölder spaces, and a mean-zero observable $f$ ($\int f\,dv = 0$), the ASIP holds: $$\sum_{i=0}^{n-1}[f \circ T^i - Z_i] = o(\sqrt{n \log\log n}) \quad \text{a.s.}$$ for a sequence of i.i.d. Gaussians $\{Z_i\}$, matching the classical i.i.d. logarithmic loss rate. For maps with neutral fixed points, optimal rates persist for observables satisfying an integrable tail condition. In both cases, these rates are best possible under the admissible mixing and moment hypotheses (Dedecker et al., 2011).

When the system admits a Young tower structure with return-time tail $m\{T > n\} = O(n^{-\beta})$, $\beta > 2$, for Hölder observables the almost-sure coupling to Brownian motion is accomplished at the rate $o(n^{1/\beta} L(n))$, with $L(n) = (\log n)^{1/\beta + \epsilon}$ for weak moments, or $L(n) \equiv 1$ when $\sum_n n^{\beta-1} m\{T>n\} < \infty$—again, achieving optimality relative to the associated renewal process (Cuny et al., 2018).

Table: Canonical ASIP Rates for Dynamical Systems

System Type	Return-Tail/Decay	ASIP Rate
Uniformly expanding	Exponential/spectral gap	$o(\sqrt{n\log\log n})$
Intermittent/Young Tower	Tail $n^{-\beta}$, $\beta>2$	$o(n^{1/\beta} L(n))$
Fast mixing (exponential tail)	$e^{-cn}$	$O((\log n)^2)$
Stretched-exponential	$e^{-c n^\gamma}$	$O((\log n)^{1+1/\gamma})$

3. Subpolynomial and Logarithmic Rate Regimes

For systems with very rapid mixing (exponential or sub-exponential light tails on return times), further refinements leveraging Markov shift codings and Bernoulli functionals achieve arbitrarily slow divergence in the ASIP remainder. For nonuniformly expanding or hyperbolic maps with an exponential tail, it is possible to construct a coupling such that for all $\epsilon > 0$,

$S_n = W_n + o(n^\epsilon) \textrm{ a.s.}$

which is "subpolynomial" and thus optimal up to arbitrary loss in the exponent (Korepanov, 2017). For dependent sequences with semi-exponential tails and subexponentially decaying coupling coefficients,

$$S_n - \sum_{i=1}^n N_i = O((\log n)^{\alpha}) \textrm{ a.s.}, \quad \alpha = 1 + \frac{1}{y_1} + \frac{1}{y_2}$$

where $y_1, y_2$ are parameters of the coupling decay and tail, respectively (Cuny et al., 2023).

4. Reverse Martingales, Nonstationary Contexts, and Mixing Assumptions

For nonstationary sequences, skew products, and time-dependent or random environments, rates fundamentally depend on the control of mixing coefficients and nonuniformity in transfer-operator bounds. If, for instance, exponential $\phi$-mixing holds with uniformly bounded increments and variance $V_n = \Var(S_n)$, then, for every $\delta>0$,

$$S_n - \sum_{j=1}^n Z_j = o(V_n^{1/4+\delta}) \textrm{ a.s.}$$

with $Z_j$ independent Gaussians, and with explicit $L^2$ error bounds linked to the minimal variance growth (Hafouta, 2020, Dolgopyat et al., 2024). For random cocycles, the same $1/4$ rate appears, but for scalar observables martingale approximation on reverse martingale differences can yield improved rates $O(n^{1/4} (\log n)^{1/2+\epsilon})$ under tighter moment and spectral control (Dragičević et al., 2021, Hafouta, 2021).

For slowly mixing deterministic systems, e.g., Pomeau–Manneville maps with parameter $y \in (0, 1/2)$ and $m(T>n) = \Theta(n^{-1/y})$, the ASIP rate is $o(n^y (\log n)^{y + \epsilon})$, which strictly improves on the historical $O(n^{1/4})$ "barrier" (Cuny et al., 2018, Cuny et al., 2023).

5. Vector-Valued and Nonuniformly Hyperbolic Systems

In high-dimensional settings, or for time-dependent and nonuniformly hyperbolic dynamical systems, the same sophisticated spectral and complex cone technology applies. For vector-valued observables under uniform lower bounds on covariance growth, the ASIP can hold with arbitrarily small exponent loss: $$\sup_{1 \le k \le n}\left| S_k - \sum_{j=1}^k Z_j \right| = o(n^{1/2 - \epsilon})$$ for any $\epsilon > 0$, where each $Z_j$ is a Gaussian vector with the correct covariance (Hafouta, 2019). For Sinai billiards and dispersing billiards with slow mixing, two-sided Markov-tower constructions achieve ASIP at the rate $o(n^{1/B})$ (up to log corrections), with $B$ the power in the tail of the return time (Cuny et al., 2023, Stenlund, 2012).

6. Martingale and Cocycles in Random Walks and Lie Groups

For Markov-additive or cocycle structures such as random walks on linear groups, the martingale-coboundary decomposition and the associated projective criteria permit nearly optimal ASIP rates. If one can establish an $L^p$ structure ($p \in (2, 3]$) for the martingale differences, the ASIP holds with

$$S_n - \sum_{k=1}^n N_k = o(n^{1/p} (\log n)^{2/p + \varepsilon}) \textrm{ a.s.}$$

for any $\varepsilon>0$ (Cuny et al., 2019).

7. Information Rates in Symbolic and Mixing Systems

For $\beta$-mixing ergodic systems with information function $I_n(x)$ (logarithm of the measure of $n$-cylinder containing $x$), the ASIP holds with

$I_n = \sigma B(n) + O(n^{1/2-\delta})$

for all $$\delta < \min\left\{ \frac{p(w-2)-5w+1}{2p(w-2)+2w+2}, \frac{3}{3p+30} \right\}$$, where $w$ is the information moment index and $p$ is the polynomial decay rate of the $\beta$-mixing coefficient (Haydn, 2013). In the case of exponential mixing, the error exponent can be made arbitrarily close to $1/2$.

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Explicit ASIP rates depend delicately on the interaction between moment/coupling properties, decay of correlations and recurrence, and spectral or symbolic structure. In the optimal regimes (i.i.d., strong mixing, spectral gap systems), one recovers $O((n \log \log n)^{1/2})$ or $O(n^{1/4})$ rates. For systems with weaker mixing or heavy-tailed recurrence, the error exponent typically deteriorates: $n^{1/B}$ for return tails $n^{-B}$, logarithmic powers for semi-exponential tails, and subpolynomial rates for exponentially mixing towers. In vector-valued, random, and nonstationary frameworks, coupling constructions and block-partition arguments are essential for transferring these error rates. The Kolmogorov--Komlós–Major–Tusnády and Skorokhod approaches underpin the sharpest results achievable in current ASIP theory.