Central limit theorem under variance uncertainty

We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $\xi_j$, perturbed by predictable multiplicative factors $\lambda_j$ with values in intervals $[\underline\lambda_j,\overline\lambda_j]$. It is assumed that the sequences $\underline\lambda_j$, $\overline\lambda_j$ are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a"worst"sequence $\lambda_j$, is described by the solution $v$ of one-dimensional $G$-heat equation. The main part of the proof follows Peng's approach to the CLT under sublinear expectations, and utilizes H\"{o}lder regularity properties of $v$. Under the lack of such properties, we use the technique of half-relaxed limits from the theory of viscosity solutions.


Introduction
Consider a sequence of independent one-dimensional random variables (ξ j ) ∞ j=1 with zero means and finite variances σ 2 j = Eξ 2 j > 0. Put s 2 n = n j=1 σ 2 j , ε > 0 and assume that the Lindeberg condition where ζ has the standard normal law.
In this paper we assume that the variance of ξ j is not known exactly and may belong to an interval. Our goal is to obtain the "least upper bound" L for the quantity (1.2) under such model uncertainty. The result, as well as its proof, are similar to those obtained by Peng [13,14] and the followers [11,20,8] under the nonlinear expectations theory paradigm. It appears that L can be described in terms of the solution v of a nonlinear parabolic equation, called G-heat equation. One of the objectives of the present paper is to show that this description also comes from a classical problem statement, and need not be linked to the nonlinear expectations theory.
The sequence λ j is bounded by a constant Λ.
Assumption 3. The sequences λ j , λ j satisfy the following stabilization condition: |} the Hausdorff distance between these intervals (see, e.g., [1,Chapter 2]). Condition (1.3) is equivalent to the following one: In the summability theory the transformation t n = p 1 a 1 + · · · + p n a n p 1 + · · · + p n , p n > 0  Denote by A n the set of adapted sequences λ n 0 = (λ j ) n j=0 with values in [λ j , λ j ]. Our goal is to describe the quantity In the context of the CLT under sublinear expectations, equation (1.7) appeared in [13]. It was called G-heat equation in [12]. As is mentioned in [5], such equation arises in various applications in control theory, mechanics, combustion, biology, and finance. It is known also as a Barenblatt equation: see, e.g., [9].
One can obtain (1.7) by considering λ j as a control sequence, writing down dynamic programming equations for discrete time finite horizon optimization problems λ j ξ j+1 , and passing to the limit as n → ∞. This approach was proposed in [18] in the case of identically distributed (multidimensional) random variables ξ j . However, in the present context, it seems that this method requires hypotheses, which are stronger than the Lindeberg condition. Thus, we follow Peng's approach, which takes equation (1.7) as a starting point, and utilizes a deep result on the existence of its solution in an appropriate Hölder class.
If λ = 0 then only the existence of a viscosity solution is guaranteed. Let us recall this result along with related definitions. Put Q • = [0, 1) × R and assume that f is a bounded continuous function: and for any (t, x) ∈ Q • and any test function holds true. To define a viscosity supersolution, one should consider a bounded lower semicontinuous (lsc) function v, a strict local minimum point of v − ϕ, and reverse the inequalities (1.9), (1.10). We will use the following comparison result. Consider a viscosity subsolution u and a viscosity supersolution w of (1.7), (1.8). Since we require (1.7) to be satisfied in the viscosity sense at the lower boundary of Q, by the accessibility theorem of [6], we have and by the comparison result of [7] (Theorem 1) it follows that u ≤ w on Q.
A bounded continuous function v : Q → R is called a viscosity solution of (1.7), (1.8), if it is viscosity sub-and supersolution. The existence of a continuous viscosity solution of (1.7), (1.8) for f ∈ C b (R) is well known from the theory of optimal control. The stochastic control representation of such solution can be found in [19] (Chap. 4, Theorem 5.2). It is interesting to compare Theorem 1 with related results obtained in the framework of sublinear expectations theory. Besides the original result of Peng [13,14], which is discussed in [18], we mention the papers [11,20,8], where the random variables were not assumed to be identically distributed. We will discuss only the result of [20], which extends [11]. The result of [8] concerns the multidimensional case.
Let us briefly describe the construction of a sublinear expectation space (Ω, H, E), which allows to rewrite the expression (1.6) in terms of a sublinear expectation. This construction is, in fact, the same as in [18,Section 4], where some more details are given. Consider the space of sequences Ω = {(y i ) ∞ i=1 : y i ∈ R}, and introduce the space of random variables H as follows: H = ∪ ∞ n=1 H n , where H n is some linear space (we do not go into details) of functions Y = ψ(y 1 , . . . , y n ) of n variables. Define the sublinear expectation by the formula Eψ(λ 0 ξ 1 /σ 1 , . . . , λ n−1 ξ n /σ n ). (1.11) Let Y i be the projection mappings: Y i (y) = y i . One can show that Y n is independent from (Y 1 , . . . , Y n−1 ) in the sense of sublinear expectations theory (see [15], Definition 3.10). By (1.11) we get the following representation for L : Let us apply Theorem 3.1 of [20] to the sequence Y i . We have Besides a condition, identical to Assumption 3, in [20] it is assumed that for some M > 0, δ > 0. Note that (1.12) was used in [20] in this form, although it was not clearly formulated (see condition (3) of Theorem 3.1 in [20]). The result of [20] tells us that L = Ef (Z), where Z is a G-normal random variable with By the characterization of the G-normal distribution (see, e.g., [15], Example 1.13) this is equivalent to the assertion of Theorem 1.
Note that if there is no model uncertainty: λ j = λ j = 1, then Theorem 1 reduces to the classical CLT, mentioned at the beginning of the present paper. This is not the case with the result of [20], since in this case the conditions (1.12), (1.13) are stronger then the Lindeberg condition. We also mention that [20] deals only with classical solutions of the G-heat equation, so the case λ = 0 is, in fact, not considered there. However, the sublinear expectations theory is able to handle the degenerate case via perturbation methods, see [14] (the proof of Theorem 5.1), [8] (the proof of Theorem 3.1).

Proof of Theorem 1
(i) We first consider the case f ∈ C 2+α (R), α > 0 and λ > 0. Put Since the solution v of (1.7), (1.8) belongs to v ∈ C 1+α ′ /2,2+α ′ (Q), we can apply Taylor's formula: where t j = t j +β(t j+1 −t j ), X j = X j +γ(X j+1 −X j ), β, γ ∈ [0, 1]. By the independence of X j and ξ j+1 we conclude that E(v x (t j , X j )(X j+1 − X j )) = 0. Thus, We can rewrite J n as J 1 n + J 2 n , where From the definition of v we see that J 1 n = 0. Furthermore, from the stabilization condition (1.3) it follows that n → ∞, since the second derivative of v is uniformly bounded. On the other hand, choosing a sequence with an arbitrary λ 0 , we get an opposite inequality Combining all these results, we conclude that (2.14) Now consider I n = I 1 n + I 2 n + I 3 n : By the Hölder continuity of v t we have Using the inequality E|ξ j+1 | α ′ ≤ (Eξ 2 j+1 ) α ′ /2 = σ α ′ j+1 , and the independence of λ j and ξ j+1 , we obtain the estimate From (1.5) it follows that I 1 n → 0. Furthermore, since the sequence λ j is bounded and the second derivative of v is uniformly bounded, by the Lindeberg condition we get The last term I 3 n is estimated with the use of the Hölder continuity property of v xx : Therefore, lim So, we have proved the theorem in the case f ∈ C 2+α , λ > 0.
(ii) Now assume that λ = 0. Put By part (i) of the proof, we infer that in the classical sense. Let v be the continuous viscosity solution of the limiting problem v t + 1 2 The desired result is a consequence of the relations which we are going to prove.
Take ϕ ∈ C 2 (R 2 ) and assume that z = (t, x) ∈ Q is a strict local maximum point of v − ϕ on Q. Then there exist sequences ε k → 0, z k = (t k , x k ) ∈ Q such that z k → z, v ε k (z k ) → v(z), and z k is a local maximum point of v ε k − ϕ on Q: see [2] (Chap. 5, Lemma 1.6).
Thus, for sufficiently large k, We have proved that v is a viscosity subsolution of (2.17). Similarly, one can prove that v is a viscosity supersolution of (2.17). By the comparison result of [7], mentioned in Section 1, we have v ≤ v on Q. The converse inequality v ≥ v is immediate from the definition. We infer that v = v = v is a continuous viscosity solution of (2.17), and the second equality in (2.18) holds true: This completes the proof of Theorem 1 in the case λ = 0.