ON THE TESTABILITY OF THE ERGODIC HYPOTHESIS ISAAC LOH Abstract. We show that stationary, Polish space-valued stochastic processes can be uniformly approximated by weakly mixing processes, non-ergodic processes, and non-mean-ergodic processes. Therefore, the ergodic hypothesis|that time averages will converge to their statistical counterparts| is not testable in the general case. We also show that the set of weakly mixing processes is generic and the set of strongly mixing processes is meagre in a rich space of stationary stochastic processes. 1. Introduction The ergodic hypothesis, that time averages asymptotically converge to statistical averages, is fundamental in many economic and statistical settings. In this paper, we consider the testability of the ergodic hypothesis under minimal assumptions apart from stationarity. Namely, we ask if there exists a statistical test with nontrivial power that distinguishes between ergodicity and nonergodicity. We provide an answer in Theorem 1 which states that any stationary stochastic process taking on values in a Polish space can be approximated in a very strong sense by stationary ergodic stochastic processes. This theorem is proved by applying a result on uniform approximation in ergodic theory. The approximation result also holds very generally with non-ergodic processes, as well as mean-ergodic and non-mean-ergodic processes. This implies that in our very general setting, the power of any test for ergodicity (or alternately non-ergodicity, mean-ergodicity, non- mean-ergodicity) cannot exceed its size, despite the number of time periods or the number of observations available. Our results contrast with the tests for ergodicity which are proposed in [8] and [9] for processes that are Markovian in nature, and demonstrate that the consistency results achieved therein depend on the Markovian assumptions. Our approximation theorem complements several well known results on the existence of station- ary processes having a certain specified marginal distribution. [12] and [19] show the existence of stationary processes with a finite state space arising from measure preserving, ergodic, and ape- riodic transformations that have a certain marginal distribution. Similarly, [18] and [2] study the problem of coding a stationary process onto another process with a given marginal distribution, when the coded process takes on values in a finite set and a countable set, respectively. Our findings particularly parallel those of [17], which shows that marginal distributions for stationary processes on a finite state space can be matched with periodic measures and closely approximated with ergodic measures. In contrast with [17], we allow for stochastic processes taking on values in an arbitrary Polish space, and prove that those processes can be approximated in a uniform sense with other processes satisfying ergodicity, non-ergodicity, and related conditions. The implication of our main result for the testability of the ergodic hypothesis supplements existing work on testability when ergodicity is a maintained assumption in place of the usual i.i.d. assumption for data. [20] and [24] investigate the discernability of families of stationary ergodic processes, whereas we study the discernability of stationary processes with mixing characteristics from those without mixing. Finally, we apply our result to studying the genericity of various kinds of mixing conditions in a space of stationary stochastic processes taking on values in a given Polish space. We show that, within this large space of processes, weak mixing is a generic property but strong mixing is meagre. [21] showed that the set of ergodic measures for stationary stochastic processes was a generic set 1 2 ISAAC LOH in the set of stationary measures endowed with the weak topology, whereas the set of strongly mixing measures was meagre. Our observation corroborates these findings in a space of stationary stochastic processes, as opposed to their push-forward measures. It also parallels, and makes use of, results on the approximation and genericity of measure preserving transformations possessing certain mixing properties (e.g. [13] and [11]). 2. Main Result Throughout, we consider a Polish space X and say that a X -valued stochastic process X = (X (~!)) :(Ω~; F~; P)~ ! Q X is stationary if its finite dimensional distributions are shift t t2Z t2Z k invariant, i.e. for each k ≥ 1 and measurable H 2 X , P ((Xt+1;:::;Xt+k) 2 H) is invariant with respect to t. Let X Z denote Q X , and let T : X Z !X Z be the left shift map. We say that a t2Z stationary process is ergodic if for all k 2 N and cylinder sets A ⊂ X , one has t 1 X a:s: lim I(T jX 2 A) = P (X 2 A) t!1 t j=1 Alternately, we say that a stationary process is weakly mixing if, for all Borel measurable A; B ⊂ X Z, t (1) lim E 1A(T X)1B(X = P (A)P(B) ; t!1 t62I0 1 where I0 is some zero density subset of N . Note that if A is a shift invariant set, putting B = A in (1) implies that P (A) = P A2, and thus P (A) 2 f0; 1g. Thus, Theorem 9.5.6 of [16] implies that weak mixing implies ergodicity. For some equivalent characterizations of weak mixing, see [13] or [26]. Define a process as strongly mixing if it satisfies (1) with I0 = ;. In the particular case where X = R, we call a stationary stochastic process mean ergodic if t 1 X a:s: lim Xj = E [X0] ; t!1 t j=1 provided that the integral on the right exists. A stochastic process X, is said to be aperiodic if the probability that :::;X−1;X0;X1;::: is a periodic sequence is zero. Our result applies approximation results from ergodic theory (see the canonical result of [22] and also [3] for an overview). Some relevant mathematical terminology is introduced further along in Section A. Let us formalize a notion of statistical testing in our setting. Let X be a X -valued, stationary and measurable stochastic process with law P 2 P (where the set P is to be specified later) on the space X Z. As in [6], we are interested in hypothesis testing problems of the form H0 :P 2 P0 H1 :P 2 P1; where P0 ⊂ P is the set of probability measures on Ω for which the null hypothesis is deemed to hold and P1 = P n P0 is its complement. A possibly randomized statistical test will be denoted by t a map 't : X ! [0; 1], where t indicates the time dimension of the test. The corresponding size of the test is given by Z sup E['t(X1;:::;Xt)] = 't(x1; : : : ; xt) dP(x): P2P0 Ω 1 1 Pt Recall that I0 ⊂ N is said to have zero density if limt!1 t j=1 1j2I0 = 0. ON THE TESTABILITY OF THE ERGODIC HYPOTHESIS 3 We will show that when P0; P1 are chosen to test ergodicity (or non-ergodicity) of the stochastic process X, one has under mild conditions sup EP ('t) ≤ sup EP ('t)(2) P2P1 P2P0 for any sequence f'tgt2N of tests and any sample size t. In other words, the power of the test 't cannot exceed its size, for any t. For this reason, one can say that the null hypothesis H0 is non-testable. A simple argument (see [23], Remark 2) then shows that the nontestability result i i n holds even if 't is allowed to depend not only on X1;:::;Xt, but an iid sample (X1;:::;Xt )i=1 of size n, where n is arbitrarily large. The definition of P and its constituent sets P0 and P1 clearly play a significant role in establishing (2). We let P denote the set of standard Borel measures on (X Z; B1) corresponding to distributions of stationary X -valued stochastic processes X. Lemma 5 below implies that P contains all of the pushforward measures induced on X Z by Borel measurable stochastic processes. We then let PE ⊂ P denote the subset of distributions which correspond to X -valued stochastic processes which are additionally ergodic, and PWM those which are weakly mixing. In the special case where X = R and P contains only distributions induced by stochastic processes X which are integrable in the sense that E [jX0j] < 1, we let PME ⊂ P denote the subset of distributions which correspond to mean ergodic stochastic processes. By Corollary 6 below, PE ⊂ PME. In the analysis to follow, we consider several null hypotheses. The first sets P0 = PE. This is to say that the stochastic process in question is ergodic, under the null hypothesis. The other scenarios c c we consider are that P0 = PE, PME, and PME. We will show that (2) holds in all scenarios. In fact, this is implied by our main result Theorem 1, which is stronger. The theorem states that, given a stationary X -valued stochastic process X and law P, we can find a sequence (Xk) of ergodic (or non-ergodic) stationary stochastic processes that uniformly approximate X. The proof of our main theorem uses uniform approximation results from the theory of measure preserving transformations. The specific results results that we use depend on aperiodicity of the underlying transformation, which by analogy would suggest that our main theorem would be valid only for aperiodic processes. However, Theorem 1 shows not only that periodicity is not a hindrance to ergodic approximation, but that the approximation can always be done with aperiodic transformations. This has the added benefit of implying that all of our non-testability results still apply when P is further restricted to contain only the set of aperiodic stochastic processes. As aperiodicity is a light assumption, this negates what would otherwise be a straightforward workaround to the finding of non-testability.