Limit Theorems for Stationary Random Processes

Stationary random processes form a foundational class of models in probability theory, characterised by probabilistic properties that remain invariant under time shifts. Limit theorems for such processes describe the long-term behaviour of aggregated quantities, most notably through versions of the law of large numbers, the central limit theorem and related invariance principles. These results rest on structural properties such as ergodicity, mixing conditions and martingale approximations, which ensure that dependence decays sufficiently rapidly. Over the past decades, advances have extended classical limit theorems from independent sequences to strongly dependent settings, encompassing random fields, Gaussian processes, Markov chains and processes with long memory. Applications span climatology, signal processing, financial time series and network traffic, where rigorous asymptotic results underpin inference, risk assessment and prediction. Recent work has focused on weakening dependence assumptions, exploring moment and projective conditions, and deriving functional limit theorems that yield convergence in path space to Gaussian or self-similar processes.

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New developments have introduced projective criteria under L¹ norms for central limit theorems of high-dimensional stationary random fields. By imposing minimal moment and projection bounds rather than classical mixing assumptions, these results provide verifiable conditions for the variance of partial sums to grow linearly and for Gaussian fluctuations to emerge. This approach has broadened the scope of limit theorems to include non-mixing structures and complex spatial dependence.

Recent studies on general functionals of Brownian local times have established limit theorems that extend beyond sums to more intricate path-dependent statistics. Under stationarity and moment conditions, normalisation of additive functionals of local times leads to Gaussian limits, with explicit expressions for asymptotic variances. These findings connect classical local time analysis with modern probabilistic tools, enabling applications to diffusion models and fractal occupation measures.

Investigations into p-domain functionals of stationary Gaussian fields have yielded both central and non-central limit theorems for weighted aggregates over growing index sets. By examining Hermite expansions and controlling long-range correlations, researchers have characterised regimes where Gaussian limits prevail or where non-Gaussian self-similar processes arise. These advances clarify the interplay of field smoothness, domain geometry and dependence strength, with implications for spatial statistics and physical modelling of random media.

Limit Theorems for Stationary Random Processes publication trend

The graph below shows the total number of articles in limit theorems for stationary random processes across all publications each year (not limited to Nature Index journals).

Technical terms

Stationary process: A stochastic process whose finite-dimensional distributions are invariant under time or index shifts.

Central limit theorem (CLT): A result stating that suitably normalised sums of dependent or independent random variables converge in distribution to a Gaussian law.

Invariance principle: A functional extension of the CLT, asserting convergence of rescaled partial-sum processes to a continuous Gaussian process in path space.

Strong mixing condition: A quantitative measure of dependence decay, ensuring that distant past and future events become nearly independent.

Martingale difference: A sequence of random variables whose conditional expectation given the past is zero, often used to approximate dependent processes.

References

1. Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions. Probability Surveys (2005).
2. Martingale approximations for random fields. Electronic Communications in Probability (2018).
3. Invariance principles for self-similar set-indexed random fields. Transactions of the American Mathematical Society (2014).
4. On the CLT for additive functionals of Markov chains. Electronic Communications in Probability (2020).
5. On the central limit theorem for stationary random fields under L1-projective condition. Electronic Communications in Probability (2022).
6. Limit theorems for general functionals of Brownian local times. Electronic Journal of Probability (2024).
7. Limit theorems for p-domain functionals of stationary Gaussian fields. Electronic Journal of Probability (2024).