Non-Asymptotic Existence Results

• Non-asymptotic existence results are explicit finite conditions guaranteeing solutions, estimator bounds, or system properties without relying on limiting arguments.
• They use precise methodologies such as Gaussian density sandwiches, metric entropy, and Lyapunov techniques to establish sharp thresholds within finite regimes.
• These results provide actionable insights for algorithmic reliability, statistical estimation, and robust numerical simulations under practical, finite conditions.

A non-asymptotic existence result refers to a theorem or characterization that guarantees the existence (or non-existence) of mathematical objects—such as solutions to equations, stationary distributions, or estimator risk bounds—under finite or explicit constraints, typically pertaining to sample size, system parameters, or computational resources. Unlike classical existence theorems that hold only in a limiting regime (as some parameter—sample size, time, system order—tends to infinity), non-asymptotic results specify explicit, quantifiable conditions and thresholds for existence that apply without taking any limit. Such results are foundational across probability, statistical inference, numerical analysis, dynamical systems, and applied mathematics, especially in contexts where practical performance and guarantees are required at finite scale.

1. Definition and Principles of Non-Asymptotic Existence

Non-asymptotic existence addresses the problem of confirming the existence of structures, solutions, or bounds for mathematical objects under finite conditions. What distinguishes it from classical existence proofs is the explicit characterization of the parameter regime and avoidance of asymptotic limiting arguments. The principle is to state (or prove) existence in terms of quantifiable metrics (sample size $n$, block length $N$, weights, parameters), typically paired with non-asymptotic inequalities, covering numbers, or explicit confidence intervals.

A prototypical example is the non-asymptotic posterior concentration result for Bayes estimators, where for any fixed $n$ one can bound the probability mass in small metric balls and risk for the Bayes estimator in terms of entropy and prior mass (Birgé, 2014). Similarly, a non-asymptotic existence result in numerical schemes establishes that for any fixed number $M$ of Monte Carlo samples, sharp Gaussian concentration bounds hold for errors of the Euler scheme (Lemaire et al., 2010).

The general structure is:

• Explicit quantitative bounds (not involving limits as $n\to\infty$),
• Applicability to finite parameter regimes,
• Often yields sharp thresholds partitioning parameter space into existence/non-existence domains.

2. Key Methodologies for Non-Asymptotic Existence Analysis

Methodologies employed for these results depend on the domain:

• Gaussian Density Sandwich and Modified Herbst Argument: Used by (Lemaire et al., 2010) for SDE discretization—this proceeds by first deriving two-sided Gaussian bounds for the transition density of the numerical scheme, then applying a modified Herbst argument adapted to measures dominated by these bounds to yield non-asymptotic concentration inequalities.
• Metric Entropy and Prior Mass Arguments: As in (Birgé, 2014), covering numbers (metric entropy functions $D(\cdot)$) and local prior mass assumptions enable precise control over the posterior contraction and risk bounds in Bayesian inference, for any fixed sample size.
• Variational Methods and Sub-Supersolution Construction: Used for elliptic systems with parameters, as in (Alves, 2020), where functional minimization over convex sets and careful sub-supersolution ordering gives sharp existence and non-existence domains.
• Integral Lyapunov Stability and Non-Coercive Functions: In nonlinear system stability, as in (Mironchenko et al., 2018), non-asymptotic existence is established using Lyapunov functions lacking coercivity, together with uniform integral inequalities that do not assume infinite time horizon or infinite-dimensional compactness.
• Reverse Chernoff-Cramér Bounds and Paley-Zygmund Inequality: For lower tail bounds in probability, (Zhang et al., 2018) inverts classical exponential upper bounds, using explicit mgf bounds and clever probabilistic inequalities to give matching lower bounds for finite sums of random variables.

3. Representative Domains and Example Results

Non-asymptotic existence results appear in a range of domains:

Paper arXiv id	Domain/Problem	Non-Asymptotic Existence Result
(Lemaire et al., 2010)	SDE Euler–Monte Carlo	Explicit Gaussian deviation bounds for error with finite $M$ samples
(Birgé, 2014)	Bayesian Nonparametrics	Rates for posterior contraction and risk of Bayes estimator for finite $n$
(Hayashi et al., 2015)	Markov Chain Randomness Extraction	Finite-blocklength bounds for error probability, variance, equivocation
(Ghaderpour, 2016)	Weighing Matrices & Orthogonal Designs	Algorithmic existence/non-existence per explicit parameter thresholds
(Honoré et al., 2016)	Recursive SDE Invariant Measure Approximation	Non-asymptotic confidence intervals for empirical measure and sharp CLT bounds
(Mironchenko et al., 2018)	Dynamical System Stability	Equivalence between integral stability and existence of non-coercive Lyapunov function

These results consistently illustrate that existence can be certified on the basis of measurable, finite system properties—entropy, variance, parameter inequalities, or explicit computational steps.

4. Sharpness, Thresholds, and Lower Bounds

A central theme is the identification of sharp thresholds in parameter space that demarcate existence and nonexistence. For instance:

• Weighing matrices (Ghaderpour, 2016): Non-existence of skew-symmetric designs when the weight cannot be written as a sum of three integer squares; existence holds for all sufficiently large matrix order when $k$ is a square.
• Multicomponent coagulation systems (Ferreira et al., 2021): Existence (or nonexistence) of stationary solutions tied to a precise inequality on kernel parameters $y+2p<1$.
• Bayesian estimators (Birgé, 2014): Non-asymptotic risk bounds with terms quantifying variance and model bias via metric entropy and KL divergence, with finite $n$.

Furthermore, results such as matching non-asymptotic lower and upper tail bounds for sums of random variables (Zhang et al., 2018) demonstrate sharpness—showing not only that certain bounds cannot be improved (up to constants) for finite values, but also giving matching rates.

5. Applications in Computation and Statistical Guarantees

Non-asymptotic existence theorems are essential in areas where algorithmic confidence and guarantees must be given at practical scale.

• Monte Carlo Simulation: In pricing, filtering, numerical SDEs, explicit confidence intervals and error bounds can be derived for any prescribed number of simulation paths—without relying on the central limit theorem (Lemaire et al., 2010, Honoré et al., 2016).
• Random Number Generation: Security levels and uniformity for privacy amplification protocols using Markov sources can be guaranteed for finite block-length—key in cryptographic protocols (Hayashi et al., 2015).
• Parameter Space Partitioning: Working in finite parameter regimes for elliptic PDE systems allows the practitioner to determine precisely whether positive solutions exist for their domain and parameter pair (Alves, 2020).
• Statistical Estimation: Bayes risk bounds and posterior concentration inform sample complexity guarantees in practical experiment design (Birgé, 2014).

6. Implication and Importance in Theory and Practice

The rise of non-asymptotic existence theorems reflects increased emphasis on deployable, robust mathematical theory for computation, statistics, information, and control. The ability to certify existence (and often uniqueness or optimality) in these settings provides a more nuanced and actionable foundation than classical asymptotic guarantees alone.

Such results tend to unify classical theory with modern computational demands:

• They clarify finite-sample, finite-resource boundaries between feasible and infeasible regimes.
• Parameter sharpness aids in optimal algorithmic design and resource allocation.
• Confidence intervals and deviation inequalities offer quantifiable guarantees directly usable in engineering, finance, statistics, and control.
• The adaptation of tools—entropy, tail bounds, Lyapunov methods—broadens applicability to high-dimensional, infinite-dimensional, or nonstandard contexts.

As mathematical, statistical, and computational sciences evolve toward probabilistic numerics and data-driven disciplines, non-asymptotic existence results provide essential infrastructure linking rigor with practical feasibility.