Common Limit Theorem

• Common Limit Theorem is a framework describing asymptotic behavior of normalized sums, encompassing the weak law of large numbers and central limit theorem under various dependence structures.
• It generalizes classical results by incorporating extensions such as exchangeability, mixing conditions, and alternative normalizations that yield non-Gaussian or mixed-limit phenomena.
• The theorem applies broadly to fields like combinatorics, number theory, high-dimensional statistics, and interacting systems, highlighting both methodological advances and statistical consequences.

Across the cited literature, the expression “common limit theorem” denotes not a single isolated theorem but the canonical asymptotic laws governing normalized sums and related functionals, especially the weak law of large numbers and the central limit theorem, together with their analogues, refinements, and counterexamples in dependent, exchangeable, Markovian, high-dimensional, combinatorial, and number-theoretic settings. In the classical i.i.d. framework, if $X_1,X_2,\dots$ have mean $\mu$ and variance $\sigma^2<\infty$, then

$$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$

and

$$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$

Much of modern work asks how far these conclusions persist once independence, normalization, or the observable itself is changed (Pippenger, 2012).

1. Classical core and canonical formulations

In the narrowest and most classical sense, the common limit theorems of probability are the weak law of large numbers and the central limit theorem for i.i.d. sequences. One formulation takes centered and normalized variables $Z_1,\dots,Z_n$ with $E[Z]=0$. If $E|Z|<\infty$, then

$\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$

where $D=0$ almost surely; equivalently, $\mu$0 in probability. If $\mu$1, then

$\mu$2

In the usual uncentered form, these become

$\mu$3

and

$\mu$4

under existence of the mean for the first statement and of mean and variance for the second (Pippenger, 2012).

One notable feature of this classical presentation is that the assumptions are minimal in the i.i.d. setting: no stronger moment conditions are imposed than integrability for the weak law and finite variance for the CLT. A further point emphasized in the same work is methodological rather than merely formal. The proofs can be organized through smoothing by adding an independent random variable, Taylor expansion at the scale $\mu$5 or $\mu$6, and telescoping replacement of summands one by one, rather than through characteristic functions. That proof architecture makes explicit why the law of large numbers is a first-order cancellation phenomenon, whereas the CLT is a second-order fluctuation phenomenon (Pippenger, 2012).

2. Conditional-i.i.d. structure and mixing-based extensions

A large part of the modern theory consists of identifying settings in which the classical laws survive after replacing independence by a more structured dependence. Exchangeability is a paradigmatic example. For an infinite exchangeable sequence $\mu$7, de Finetti’s theorem reduces the problem to conditioning on the random directing measure $\mu$8, under which the variables become i.i.d. In the study of maxima of partial sums, this yields an exact analogue of the classical Erdős–Kac theorem when the Blum–Chernoff–Rosenblatt–Teicher conditions hold: $\mu$9 Under these conditions the limit remains

$\sigma^2<\infty$0

If only conditional centering is retained and the conditional variance is random, the limit becomes the corresponding scale mixture; if conditional drift is not zero, the asymptotic formula acquires explicit drift-mass terms. The conceptual lesson is that exchangeable limit theorems are conditionally classical and unconditionally mixed (Ruiz et al., 2015).

Finite-state hidden Markov models exhibit a different mechanism. The observed process is typically neither i.i.d. nor Markov, but under strict positivity of the channel matrix and irreducibility and aperiodicity of the hidden transition matrix, the relevant additive functionals inherit exponential $\sigma^2<\infty$1-mixing and exponential forgetting. For

$\sigma^2<\infty$2

one then obtains a law of large numbers,

$\sigma^2<\infty$3

a central limit theorem with Berry–Esseen-type error

$\sigma^2<\infty$4

an almost sure invariance principle, a law of the iterated logarithm, and a variant of Chernoff bound. In this setting the “common” limit theorems are recovered through blocking and mixing rather than exact independence (Han, 2011).

Related persistence of classical asymptotics appears for additive functionals of Markov operators and of path-dependent stochastic differential equations. For the former, exponential convergence to a unique invariant measure in Fortet–Mourier norm and a CLT for

$\sigma^2<\infty$5

are proved for bounded Lipschitz $\sigma^2<\infty$6 with $\sigma^2<\infty$7, using coupling and the Maxwell–Woodroofe theorem (Hille et al., 2015). For path-dependent SDEs, the segment process is handled via exponential ergodicity in a Wasserstein distance induced by a quasi-metric, and one obtains the strong law, central limit theorem, and law of the iterated logarithm for

$\sigma^2<\infty$8

with asymptotic variance

$\sigma^2<\infty$9

in the nondegenerate case (Bao et al., 2019).

3. Weak independence is not enough

A recurrent misconception is that “weak independence” or approximate uncorrelatedness should suffice for the classical CLT. The sharpest counterexamples in the cited literature show that this is false even under identical marginals and finite variance. For pairwise independence, one explicit construction starts from an arbitrary target margin $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$0 satisfying a mild splitting condition: if $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$1, then $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$2 and there exists a Borel set $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$3 with $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$4 for some integer $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$5 such that

$$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$6

From auxiliary collision indicators $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$7 that are pairwise independent but not mutually independent, one builds pairwise independent identically distributed $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$8 with common margin $$\frac{X_1+\cdots+X_n}{n}\to \mu \quad \text{in probability}$$9, and the standardized sample mean satisfies

$$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$0

where $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$1, $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$2 is a standardized chi-square with $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$3 degrees of freedom, and

$$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$4

The limit is therefore explicit and generally non-Gaussian. In particular, the same paper notes that there exists a sequence of pairwise independent Gaussian random variables whose normalized sample mean has a distinctly non-Gaussian, asymmetric, heavy-tailed limit (Avanzi et al., 2020).

An analogous phenomenon persists under triplewise independence. A graph-based methodology constructs triplewise independent identically distributed sequences with arbitrary common margin $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$5 under the same type of mild condition, and for two concrete graph families the limit law is again explicit: $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$6 Here $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$7 is determined by the asymptotics of a graph-induced edge-count statistic. For complete bipartite graphs $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$8, $$\frac{X_1+\cdots+X_n-n\mu}{\sigma\sqrt n}\Rightarrow N(0,1).$$9 is standardized variance-gamma; for a graph with two hubs and $Z_1,\dots,Z_n$0 middle vertices, $Z_1,\dots,Z_n$1 is a Bernoulli-Gaussian mixture. These examples show that even triplewise independence can leave enough higher-order dependence on the $Z_1,\dots,Z_n$2-scale to destroy Gaussianity (Beaulieu et al., 2021).

The boundary is therefore precise. Mutual independence implies pairwise and triplewise independence, but neither pairwise nor triplewise independence can replace mutual independence in the classical CLT without further structural assumptions. The failure is not a normalization failure: the variance still scales linearly, and the limit can remain centered with unit variance while being non-normal (Avanzi et al., 2020).

4. Alternative normalizations and nonstandard fluctuation laws

The phrase “common limit theorem” also appears in contexts where the classical linear normalization is replaced by a qualitatively different transform. One example is the nonlinear shrinking map

$Z_1,\dots,Z_n$3

For i.i.d. nonnegative $Z_1,\dots,Z_n$4 and thresholds $Z_1,\dots,Z_n$5, the sums

$Z_1,\dots,Z_n$6

have a rigid weak-limit theory: any limit must be either degenerate,

$Z_1,\dots,Z_n$7

or compound Poisson with exponential jump distribution,

$Z_1,\dots,Z_n$8

No Gaussian, stable, or general infinitely divisible laws occur. The nonlinear normalization suppresses moderate contributions and leaves only rare exceedances, so the asymptotics become Poissonian rather than Gaussian (Jurek, 2019).

A second example is Laplace-type probability measures with density proportional to $Z_1,\dots,Z_n$9. When the unique maximum of the exponent is in the interior, the random vector $E[Z]=0$0 satisfies a weak law

$E[Z]=0$1

and a Gaussian fluctuation theorem

$E[Z]=0$2

When the maximum lies on a boundary $E[Z]=0$3, the limit law changes qualitatively: $E[Z]=0$4 has an exponential limit in the boundary-normal direction and a Gaussian limit in the tangential directions. The geometry of the maximizer, not just the large parameter $E[Z]=0$5, determines the fluctuation law (Łapiński, 2015).

Nonstandard observables produce further variants. For functions of marginal order statistics, the statistic

$E[Z]=0$6

admits a Bahadur-type linearization

$E[Z]=0$7

which yields both a CLT and a multivariate CLT (Babu et al., 2011). For the logarithm of the least common multiple of a uniformly random $E[Z]=0$8-element subset of $E[Z]=0$9, properly centered and normalized finite-dimensional distributions converge to Brownian motion when $E|Z|<\infty$0, $E|Z|<\infty$1, and $E|Z|<\infty$2 (Buraczewski et al., 2020).

5. Specialized analogues in combinatorics, arithmetic, high dimension, and interacting systems

In several domains, common limit theorems take the form of model-specific Gaussian approximations with explicit scalings, variance conditions, or mixture structure. For the length $E|Z|<\infty$3 of the longest common subsequence of two independent random words over a finite alphabet, a CLT holds under the variance lower bound

$E|Z|<\infty$4

and the paper gives the quantitative Wasserstein estimate

$E|Z|<\infty$5

for any $E|Z|<\infty$6. The same work stresses that the analogous problem for two independent uniform random permutations has Tracy–Widom rather than Gaussian fluctuations, so there is no universal common limit law across all LCS-type models (Houdré et al., 2014).

In probabilistic number theory, the empirical density of pairs with prescribed greatest common divisor has a genuine CLT with nonclassical scaling. For i.i.d. $E|Z|<\infty$7 uniform on $E|Z|<\infty$8, and fixed $E|Z|<\infty$9,

$\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$0

Here the $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$1 scale reflects the local dependence created by shared indices among pair indicators, and the same paper also proves an LDP for the empirical coprimality density (Mehrdad et al., 2013).

In high-dimensional multivariate analysis, a new CLT is proved for the normalized trace of a product of four independent Wishart matrices: $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$2 This theorem is then used to construct a test for the common principal components hypothesis, with a studentized statistic asymptotically normal under the null and diverging to $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$3 in probability under fixed alternatives when $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$4 (Tsukuda et al., 2020).

Interacting particle systems with common noise generate a different type of modification. For weakly interacting jump-diffusions driven by both empirical measure interaction and a common factor, the fluctuation theorem is conditional Gaussianity rather than unconditional Gaussianity. For square-integrable path functionals $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$5,

$\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$6

so the limit is a Gaussian mixture indexed by the realized common factor (Budhiraja et al., 2014).

6. Conceptual boundaries and statistical consequences

Taken together, these works delineate a precise boundary for the idea of a common limit theorem. The classical Gaussian limit is robust under some structural replacements for independence—conditional i.i.d. representations, exponential mixing, coupling-based ergodicity, or Wasserstein ergodicity—but it is not robust to arbitrary weakening of independence, nor to arbitrary changes of normalization, nor to arbitrary geometric constraints.

One recurring theme is that conditional Gaussianity often survives where unconditional Gaussianity does not. Exchangeable sequences lead to de Finetti mixtures; common-factor systems lead to Gaussian mixtures; boundary-concentrated Laplace measures lead to an exponential law in one direction and Gaussian laws in the others (Ruiz et al., 2015, Budhiraja et al., 2014, Łapiński, 2015). Another theme is that identical marginals and finite variance are not enough: pairwise and triplewise independence can still leave hidden global dependence that persists at the $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$7-scale and produces non-Gaussian limits (Avanzi et al., 2020, Beaulieu et al., 2021).

A further misconception is that every “central” asymptotic regime is Gaussian. Rare-event transforms such as $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$8 produce only degenerate or compound Poisson limits; some combinatorial models fall into Tracy–Widom rather than Gaussian universality; arithmetic pair statistics require $\frac{Z_1+\cdots+Z_n}{n}\Rightarrow D,$9 rather than $D=0$0 normalization (Jurek, 2019, Houdré et al., 2014, Mehrdad et al., 2013). This suggests that the most stable content of the phrase “common limit theorem” lies not in a single universal limiting distribution but in a methodological template: identify the correct centering, the correct scale, the effective dependence structure, and the relevant asymptotic object—Gaussian, mixed Gaussian, compound Poisson, chi-square-based, variance-gamma, Brownian, Tracy–Widom, or degenerate.

The statistical implication is direct. Classical inference routinely invokes asymptotic normality for means, scores, test statistics, and empirical functionals. The cited counterexamples show that such reasoning can be badly misleading when mutual independence is replaced by pairwise or triplewise independence, or when latent environments induce mixture structure. Conversely, the positive results show that asymptotic normality can often be recovered once the right structural conditions are identified. In that technical sense, the modern theory of common limit theorems is a theory both of persistence and of failure: it specifies exactly when classical asymptotics extend, exactly when they fracture, and exactly what replaces them when they do.