Let the system Ψ be a macroscopic statistical system whose microscopic state is described by a random field ϕ ( x , t ) , where x ∈ R 3 is the spatial coordinate and t ∈ R is the time coordinate. We consider some observable functional f ( ϕ ) of this random field, such as local density, temperature, or galaxy counts.

Two sampling schemes are defined:

1) Temporal sampling (single-point long time series):

Fix a spatial position x₀ and repeatedly observe the field value at that point at equally spaced instants t 1 , t 2 , ⋯ , t N , yielding the time series:

whose sample mean is:

2) Spatial sampling (single spatial snapshot):

Fix an instant t 0 and simultaneously observe the field values at spatial positions x 1 , x 2 , ⋯ , x N (separated sufficiently far apart so as to be approximately independent), yielding the spatial samples:

whose sample mean is:

The traditional ergodic theorem [1] answers the relation between “the same system at different times”: namely, whether the time average X ¯ N time converges to the ensemble average 〈 f 〉 as N → ∞ , under the core premises that the system possesses stationarity in the temporal dimension and that the trajectory is ergodic.

The principle proposed in this paper concerns the opposite direction: can the spatial average Y ¯ N space replace the time average X ¯ N time and yield equivalent statistical inference at the same order of convergence?

Intuitively, tossing a single coin many times and tossing many coins simultaneously should yield the same frequency of heads in the limit. However, this intuitive equivalence does not hold unconditionally—it depends on the system also possessing a certain “uniform” and “weakly correlated” structure in the spatial dimension, just as the convergence of the time average to the ensemble average depends on temporal stationarity and ergodicity.

The essence of the principle in this paper can be summarized as follows: the “time ensemble” equivalence relation implied by the ergodic theorem is extended, through spatial homogeneity and weak correlation conditions, to a “space → ensemble” equivalence relation, thereby establishing the statistical equivalence between “time” and “space”. It is worth emphasizing that this equivalence is statistical in nature—it concerns the equivalence of probability distributions (or low-order moments), rather than a symmetry transformation between spacetime coordinates, and therefore differs fundamentally from physical spacetime symmetries such as Lorentz symmetry and general covariance [3].

In short, this principle addresses the operational question of statistical inference—“whether a time average can be replaced by a spatial average for estimation”—rather than the ontological question of “whether time and space are symmetric”. It does not require the metric tensor to possess any Killing symmetry, nor does it involve isometric transformations of the spacetime manifold.

For this equivalence to hold rigorously, the system must simultaneously satisfy the following four conditions:

If the system Ψ satisfies the following conditions:

1) Stationarity. The first- and second-order statistics of the system do not vary with time, expressible as . E [ f ( ϕ ( x , t ) ) ] = μ for all t.

2) Spatialhomogeneity. The marginal distributions at different spatial positions are identical, expressible as: E [ f ( ϕ ( x , t ) ) ] = μ for all x.

3) Ergodicity. The time average converges almost surely to the ensemble average, expressible as: X ¯ N time → a .s . μ .

4) Weakcorrelation. The system is weakly correlated in the sense that its mixing coefficient decays exponentially. Let F − ∞ t be the σ-algebra generated by the observations up to time t, and let F t + k ∞ be the σ-algebra generated by the observations at time t + k and thereafter. The temporal mixing coefficient is defined as.

Similarly, for the spatial field, let F Λ and F Λ c be the σ-algebras generated by ϕ on a region Λ and its complement. The spatial mixing coefficient is defined as

We assume that both coefficients decay exponentially:

with C > 0 and θ > 0. This condition is used only to control the decay of the covariance function; it is not claimed to be necessary.

These four conditions together constitute the sufficient framework for the validity of the “Spacetime Statistical Equivalence Principle”.

Let ϕ ( x , t ) be a stationary random field on R 3 × R with a spatial translation action, and let f ( ϕ ) be an observable with finite second moment. Consider the following two sampling schemes:

1) Temporal sampling: fix x₀ and observe X i = f ( ϕ ( x 0 , t i ) ) , i = 1 , 2 , ⋯ , N , with sample mean X ¯ N time = 1 N ∑ i = 1 N X i .

2) Spatial sampling: fix t₀ and observe Y j = f ( ϕ ( x j , t 0 ) ) , j = 1 , 2 , ⋯ , N , over a growing spatial region or a set of separated sites, with sample mean. Y ¯ N space = 1 N ∑ j = 1 N Y j

Assume that:

a) the field is stationary in time, E [ f ( ϕ ( x , t ) ) ] = μ for all t;

b) the field is spatially homogeneous, E [ f ( ϕ ( x , t ) ) ] = μ for all x;

c) the temporal process is ergodic, X ¯ N time → a .s . μ ;

d) the spatial field satisfies the exponential α-mixing condition of Section 2.2, and the relevant variables have finite (2 + δ)-moments.

Then both averages converge to the same ensemble mean μμ in probability, and their mean-square errors are both O(1/N), equivalently their standard deviations are both O(N^−1/2).

It should be emphasized that spatially identical marginal distributions alone do not ensure that the spatial average converges to the ensemble mean; the spatial mixing or ergodicity condition (d) is essential.

Let { Y j } j = 1 ∞ be a spatially correlated random field with ensemble mean E [ Y j ] = μ and summable spatial covariance,

or equivalently satisfying the spatial α-mixing condition of Section 2.2. The spatial sample mean is

Under the summable-covariance or mixing condition, its variance satisfies

The i.i.d. case corresponds to S sp = σ 2 and is used only as a reference, not as a general representation. Therefore, the mean-square error of the spatial average is O(1/N), and the corresponding standard deviation is O(N^−1/2). This is the optimal rate of convergence of the classical law of large numbers under the condition of finite second moments, and it serves as the benchmark for comparing the convergence order of the time average below.

Let { X t , t ∈ Z } be a real-valued stationary stochastic process representing the time series obtained by sampling at equally spaced instants along the time axis at a fixed spatial position x₀. According to the setup in Section 2.1, E [ X t ] = μ holds for all t (the first-order condition of stationarity). The time sample mean is:

Intuitive interpretation: this corresponds to tossing the same coin N times in succession (but allowing some correlation between successive outcomes, for example if the coin’s surface state changes slowly with time). Unlike spatial sampling, observations at different instants may be correlated with one another—this is precisely the fundamental difference between a time series and independent samples. To quantify this correlation, we introduce the autocovariance function:

where k is the time lag ( k ∈ Z ). Stationarity guarantees that γ ( k ) depends only on the lag k, and not on the absolute time t.

Intuitive interpretation: γ ( 0 ) = σ 2 is the variance of the process; γ ( 1 ) measures the degree of linear dependence between two adjacent observations; as k increases. γ ( k ) typically decays to zero—meaning that two observations separated by a long interval have “forgotten” each other and are approximately independent.

The variance of the time sample mean can be expanded as:

Intuitive interpretation: if all observations were completely independent, only the N terms with i = j would be nonzero, yielding σ 2 / N . However, correlation causes the off-diagonal terms ( i ≠ j ) to contribute to the variance as well—positive correlation increases the variance (because the samples tend to “cluster” away from the mean), whereas negative correlation decreases it.

Rearranging the double sum by the lag k = i − j , we obtain the exact expression:

Intuitive interpretation: the weight factor ( 1 − | k | / N ) reflects the proportion of times a given lag k is sampled in a finite sample. The smaller the lag (the closer k is to 0), the more sample pairs occur and the larger the weight; the larger the lag, the smaller the weight—which is natural: two observations separated by a long interval have fewer opportunities to meet in a finite sample.

In order for the sum in Equation (3.7) to converge to a finite value, the autocovariance function must be guaranteed to decay sufficiently rapidly with the lag. We adopt the standard strong mixing condition (α-mixing) in statistics, whose core idea is that events separated sufficiently far apart are almost independent.

Definition (α-mixing). Let F − ∞ t be the σ-algebra generated by the observations up to the time t, and let F t + k ∞ be the σ-algebra generated by the observations at time t + k and thereafter. For the spatial field, let F Λ and F Λ c be the σ-algebras generated by ϕ on a region Λ and its complement Λ^c, with d(Λ, Λ^c) ≥ k. The mixing coefficients are defined as in Section 2.2.

We assume that the mixing coefficients decay exponentially:

where C > 0 and θ > 0 are constants.

In order to apply Davydov’s inequality [13], we assume that the relevant random variables possess finite moments of order 2 + δ for some δ > 0:

where δ > 0 , C ″ and θ ′ > 0 . In short, the exponential decay of the mixing coefficient, together with the finite (2 + δ)-moment assumption, implies the exponential decay of the covariance function. The same argument applies separately to the temporal process and to the spatial random field. See Davydov (1968) for the original bound [13], Doukhan (1994, Section 1.2) for a detailed treatment of α-mixing and its inequalities [14], and Cressie (1993, Chapter 2.3) for the spatial case [15].

Substituting the covariance upper bound | γ ( k ) | ≤ C ″ e − θ ′ | k | into the variance expression (3.7):

Using symmetry ( γ ( k ) = γ ( − k ) ), we split the sum into the k = 0 term and the two tails:

Key observation: the geometric series ∑ k = 1 ∞ e − θ ′ k and ∑ k = 1 ∞ k e − θ ′ k both converge to finite constants (since the common ratio e − θ ′ < 1 ). Therefore, the value in the square brackets is bounded and independent of N. Denoting this upper bound by the constant M, therefore,

and the corresponding standard deviation is O ( N − 1 / 2 ) . Although correlation exists in the time series, as long as the correlation decays sufficiently rapidly (exponentially), its effect on the variance of the sample mean is only a correction term; in the long run, the fluctuation amplitude of the time average still decreases as N^−1/2 in standard deviation, consistent with the behavior of independent samples.

The upper bound (3.12) only shows that the variance of the time average decays “no faster than” 1/N. To prove “same order” (i.e., that the two rates of convergence are exactly the same), we must also exclude a pathological case: the variance might decay faster than 1/N (for example, as 1/N²), which would imply that the time average is more efficient than the spatial average—such “super-convergence” is physically unnatural, but must be excluded theoretically.

We introduce the standard non-degeneracy condition. Under the exponential mixing condition considered in this paper, the autocovariance function is absolutely summable, so the following limit exists and is finite:

Intuitive interpretation: S is the sum of the autocovariance function, known as the “long-run variance” or the “zero-frequency spectral density” (up to a constant). Its positivity S > 0 means that the process does possess finite long-term fluctuations—if S = 0 , the process would be “super-ergodic” in some special sense (for example, perfectly negative correlation causing fluctuations to cancel completely), which is extremely rare in real physical systems and can be excluded by assuming continuity of the spectral density.

Under the conditions S > 0 and continuity of the spectral density at zero frequency, it can be shown (see Brockwell & Davis, 1991, Theorem 7.1.1; it can also be derived directly under mixing conditions [16]):

where the symbol “~” denotes asymptotic equivalence.

This result shows that the variance of the time average is asymptotically exactly equal to S / N , rather than some smaller order. The factor S can be understood as an “effective sample size reduction factor”—if the time series is positively correlated, then S > σ 2 , meaning that the effective number of independent samples is less than the nominal sample size N; if it is negatively correlated, then, and the effective number of samples is instead increased. In any case, as long as S is a finite positive number, the order of convergence remains. Therefore, under the exponential mixing condition, the time average cannot have a faster rate of convergence than the spatial average (independent samples)—the two are of the same order.

This means that there is no systematic difference between the rates of convergence of the time average and the spatial average—their statistical precision is asymptotically comparable at the same sample size.

Combining the upper bound (3.12) with the asymptotic expansion (3.14), we obtain

This means that there is no systematic difference between the convergence orders of the time average and the spatial average. However, this statement concerns the order of convergence only. The asymptotic variances are S time / N and S sp / N , respectively, S time and S sp need not be equal. Thus, equality of convergence order does not imply equality of asymptotic variance or of finite-sample precision.

From the variance upper bound (3.12) and the spatial average variance (3.2), combined with Chebyshev’s inequality, for any ε > 0 ,

Using the union bound,

Letting N → ∞ , the right-hand side tends to 0, i.e.:

that is, convergence to 0 in t probability. The standard deviation of this difference is of order O ( N − 1 / 2 ) .

In an expanding universe model, the physical distance r and the comoving distance x are related through the scale factor a ( t ) : r ( t ) = a ( t ) ⋅ x . Here a ( t ) increases monotonically with time. It is important to note that an FLRW universe can remain spatially homogeneous on each constant-time hypersurface. What fails in the expanding case is not spatial homogeneity, but the temporal stationarity of the chosen raw physical observables.

Consider the matter density field ρ ( x , t ) . In an expanding universe, the background density satisfies:

For the same spatial point (fixed comoving coordinate x), the expectation of the time series is not a constant:

This directly violates condition (1) of Section 2.2—the first-order statistic varies with time. The failure is therefore a failure of temporal stationarity for the chosen observables, not a failure of spatial homogeneity.

Decomposing the time series into a deterministic trend term and a zero-mean fluctuation term:

its time sample mean is:

The first term (the sample mean of the trend term) does not converge to a constant, but rather to the average background value over the time interval [ t 1 , t N ] :

where T = t N − t 1 . This integral value depends on the starting and ending points of the observation window. Unless μ ( t ) happens to be constant (i.e., there is no expansion), otherwise:

Imagine measuring the temperature of a cooling object. If you average over different times, the average you obtain depends on when you start measuring and when you stop—not on some “intrinsic temperature” of the object. The time average loses its universality and becomes a function of the observation window. Taking de Sitter expansion ( μ ( t ) ) = μ 0 e H t ) as an example:

As N , T → ∞ , if H > 0 , this term diverges exponentially. Even if the time series is detrended, the variance of the residual is still affected by the scale change caused by expansion:

where the additional factor 1 / ( a ( t i ) a ( t j ) ) comes from the conversion between physical density and comoving density. Since a ( t ) increases with time, the covariance structure no longer depends only on the time difference | t i − t j | , but on absolute time; the spectral representation method fails, and the simple application of the mixing condition no longer holds.

Finally, the mean-square error of the time average is:

The trend deviation term does not decay with N—it remains O ( 1 ) . Therefore, the time average in an expanding universe does not converge to any fixed value, and the spacetime statistical equivalence fails completely.

Expansion introduces an irreversible arrow of time—the universe at different instants “is not the same system”, and its statistical properties drift with absolute time. This is like averaging a person’s height at age 20 with their height at age 60—the result has no statistical meaning. The Spacetime Statistical Equivalence Principle has a rigorous finite-sample theoretical foundation only in a non-expanding, stationary background.

In 1917, Einstein introduced the cosmological constant Λ into general relativity and obtained a static universe solution [10]:

where the scale factor α ( t ) = const . , and the matter density ρ and pressure p do not vary with time. This model is based on geometric symmetry and requires strict time-translation invariance and spatial homogeneity.

To present the similarities and differences between the two theoretical frameworks more clearly, their core features are systematically compared below as Table 1:

Table 1. Two theoretical frameworks compared.

The Spacetime Statistical Equivalence Principle can provide a statistical-mechanical interpretive foundation for Einstein’s static universe [10]: the static universe need not require strictly uniform matter at every point, but only that the statistical distributions of structures such as galaxies be stationary in space and time. This allows the static universe model to avoid conflicting with observations of local clustering phenomena.

It is important to distinguish between the mathematical result and its applicability to the actual universe. The results of Section 3 are theorems conditional on the assumptions of stationarity, spatial homogeneity, weak correlation, and ergodicity. Whether the actual universe satisfies these assumptions is an empirical question. In particular, the observed expansion, the cosmic microwave background, and galaxy-evolution data must be used to test whether the stationarity assumption is empirically viable. The present work does not claim that the actual universe satisfies the stationarity assumption; it explores the logical consequences that would follow if it did.

The theorem of Section 2.3 applies to a physical model consisting of a non-expanding, statistically steady-state background with local dynamical fluctuations. In such a model, the large-scale statistical state is stationary and homogeneous, while local regions are allowed to evolve through processes such as star formation and galaxy mergers. The ideal simultaneous spatial snapshot used in the spatial sampling scheme is a mathematical idealization: it corresponds to observing a large number of spatially separated regions at the same instant.

In actual galaxy surveys, observations lie on the light cone rather than on a constant-time hypersurface. A direct observational test of the corollaries therefore requires either a narrow redshift slice, so that evolutionary effects are negligible, or an explicit correction for cosmic evolution. This distinction is essential for making the proposed observational tests distinguishable from the theorem’s mathematical assumptions. Without such a correction or restriction, a light-cone survey cannot be regarded as a direct realization of the ideal spatial snapshot.