Proof of lemma 2.1. That q(· |X∗)is an element of L1(µ)for each realization of X∗. That it is also a random variable (i.e., a measurable map) follows easily from separability of L1(µ).22 To show thatEq(· |X∗) = f∗, we must
Proof of theorem 2.1. Regarding consistency, let P be ergodic, let(Xt)t≥0 be P-Markov and let X∗ ∼ ψ∗. Define Q(x) := q(· |x) − f∗, which is a mea-surable function from X to L1(µ). Note thatEQ(X∗) = 0 by lemma 2.1.
We need to show that
nlim→∞kfn∗− f∗k = lim
Our proof is an extention of that for the IID Banach space LLN, as given in Bosq (2000, thm. 2.4). To begin, fix e > 0 and choose a partition{Bj} of L1(µ)such that each Bjhas diameter less than e. For any L1(µ)-valued random variable Y, we let LJY :=∑Jj=1bj1{Y ∈ Bj}. We use the following fact, a proof of which can be found in Bosq (2000, pp. 27-28):
∃J ∈ N with EkQ(X∗) −LJQ(X∗)k < 2e (30) Our first claim is that
nlim→∞
22See, in particular, Bosq (2000, lemma 1.2).
To establish (31), we can use the real ergodic law (8) to obtain
almost surely, where the last equality follows immediately from the defi-nition ofE. Thus (31) is established.
Returning to (29), we have Using real-valued ergodicity again, as well as (31), we get
lim sup
Since e is arbitrary, the proof of (29) is now done.
Regarding the asymptotic normality result, this follows immediately from the Hilbert space CLT of Stachurski (2009, theorem 3.1), where x7→ q(· |x) corresponds to T0in that theorem. The only point that needs checking vis-a-vis that CLT is thatEq(· |X∗) = f∗, and this has already been verified in lemma 2.1.
Proof of theorem 4.1. We make use of the following fact: By the Markov property, for our state process(Xt)t≥0and integrable h : X→R we have
Eth(Xt+k) = Pkh(Xt)
where Et is expectation conditional on σ(X0, . . . , Xt), and Pk is the k-th iterate of the mapping h 7→ Ph. For convenience, let ¯h := h−R
hdψ∗ for any h : X→R. By the Markov chain CLT for V-uniformly ergodic kernels (Meyn and Tweedie, 1993, p 411), if g : X→R with g2≤V, then
n−1/2
∑
n t=1¯g(Xt) → N(0, v)
in distribution, where
v =:E ¯g2(X1∗) +2
∑
t≥2
E ¯g(X∗1)¯g(Xt∗) (32)
Here(X∗t)t≥0is a stationary version of the process (1). Regarding the stan-dard Monte Carlo estimator Enτ, it now follows that
n1/2(Enτ− Z
τdψ∗) = n−1/2
∑
n t=1¯τ(Xt) → N(0, v1) where v1is given by
v1 =:E ¯τ2(X∗1) +2
∑
t≥2
E ¯τ(X∗1)¯τ(X∗t) (33)
Regarding the look-ahead estimator, we claim that n1/2(EnPτ−R
τdψ∗) → N(0, v2), where
v2 =: E ¯Pτ2(X1∗) +2
∑
t≥2
E ¯Pτ(X∗1)Pτ¯ (Xt∗) (34)
To show this, observe that n1/2(EnPτ−
Z
τdψ∗) =n1/2(EnPτ− Z
Pτdψ∗) = n−1/2
∑
n t=1Pτ¯ (Xt)
where the second equality follows from (20).
Now note that P is also ˆV-UE, where ˆV := λV+L. To see this, observe that the drift inequality (17) holds with ˆV in place of V, because P is V-UE, and hence
P ˆV =λPV+L≤λ(λV+L) +L=λ ˆV+L
Second, the sublevel sets{Vˆ ≤α}are P-small, because the sublevel sets V are P-small, and
{Vˆ ≤α} = {λV+L≤α} = {V ≤ (α−L)/λ} Thus the CLT claim for n1/2(EnPτ−R
τdψ∗)will hold if we can show that (Pτ)2≤V. Using Jensen’s inequality, τˆ 2 ≤V and the drift condition (17),
(Pτ)2≤ Pτ2 ≤PV ≤λV+L =: ˆV
It remains only to show that v2 ≤ v1. To see this, consider first the term E ¯Pτ2(X1∗). Writing ψ∗τ forR
τdψ∗ and using Jensen’s inequality for con-ditional expectations, we obtain
E(Pτ(X1∗) −ψ∗τ)2 =E(E1(τ(X∗2) −ψ∗τ))2
≤EE1(τ(X2∗) −ψ∗τ)2
=E(τ(X2∗) −ψ∗τ)2 =E(τ(X1∗) −ψ∗τ)2
where the last step is by stationarity. In other words,E ¯Pτ2(X1∗) ≤E ¯τ2(X∗1). To complete the proof that v2 ≤ v1, then, it is sufficient to show that the autocovariance terms in v1and v2are equal. That is, for any t ≥2,
E(Pτ(X1∗) −ψ∗τ)(Pτ(X∗t) −ψ∗τ) =E(τ(X∗1) −ψ∗τ)(τ(Xt∗) −ψ∗τ) To see this, observe that
E(Pτ(X∗1) −ψ∗τ)(Pτ(Xt∗) −ψ∗τ) = E(E1τ(X2∗) −ψ∗τ)(Etτ(Xt∗+1) −ψ∗τ)
=EE1Et(τ(X2∗) −ψ∗τ)(τ(Xt∗+1) −ψ∗τ)
=E(τ(X2∗) −ψ∗τ)(τ(Xt∗+1) −ψ∗τ)
=E(τ(X1∗) −ψ∗τ)(τ(Xt∗) −ψ∗τ) The proof is done.
Proof of Theorem 5.1. Unbiasedness of the estimator fTn is equivalent to the claim thatEqT−1(· |XT−1) = fT. The proof is almost identical to that for f∗ given in the proof of lemma 2.1, and hence is omitted. Regarding consis-tency, the Banach-space law of large numbers (cf., e.g., Bosq, 2000, Theo-rem 2.4) implies that if(Ui)i≥1is anIIDsequence in L1(µ)with expectation
which is consistency result that we seek.
Finally, consider the issue of asymptotic normality. From Bosq (2000, the-orem 2.7), we can deduce that if(Ui)i≥1 is an IIDsequence in L2(µ) such
converges in distribution to a centered Gaussian on L2(µ). Once again we take Ui = qT−1(· |XiT−1), where (XiT−1)i≥1 IID∼ ψT−1. From the condition in Theorem 5.1 we have
Z
converges in distribution to a centered Gaussian on L2(µ).
References
[1] Aiyagari, S Rao, 1994. “Uninsured Idiosyncratic Risk and Aggregate Saving,” The Quarterly Journal of Economics, Vol. 109(3), pages 659-84.
[2] Bosq, Denis, 2000. Linear Processes in Function Space, Springer.
[3] Brock, W. A. and L. Mirman, 1972, “Optimal Economic Growth and Uncertainty: The Discounted Case,” Journal of Economic Theory, Vol.
4, pages 479-513.
[4] Devroye Luc and Gabor Lugosi, 2001 “Combinatorial Methods in Density Estimation” Springer-Verlag, New York.
[5] Hansen, Gary, D. and Edward C. Prescott, 1995, “Recursive Methods for Computing Equilibria in Business Cycle Models,” Chapter 2 in T.
F. Cooley, ed., Frontiers of Business Cycle Research, Princeton University Press.
[6] Henderson, G. Shane and Peter W. Glynn, 2001, “Computing den-sities for Markov chains via simulation,” Mathematics of Operations Research Vol. 26, pages 375-400.
[7] Khan, Aubik and Julia M. Thomas, 2008, “Idiosyncratic Shocks and the Role of Nonconvexities in Plant and Aggregate Investment Dy-namics.” Econometrica, Vol. 76(2), pages 395-436.
[8] Kristensen, Dennis, 2007. “Geometric Ergodicity of a Class of Markov Chains with Applications to Time Series Models,” mimeo, University of Wisconsin.
[9] Ljungqvist, Lars and Thomas J. Sargent, 2004. Recursive Macroeco-nomic Theory, MIT Press, second edition.
[10] McKeague, Ian W. and Wolfgang Wefelmeyer, 2000, “Markov chain Monte Carlo and Rao-Blackwellization,” Journal of Statistical Planning and Inference, 85, 171-182.
[11] Meyn, Sean P. and Richard L.Tweedie, 1993, Markov Chains and Stochastic Stability, Springer-Verlag: London.
[12] Nelson Charles and Charles Plosser, 1982, “Trends and random walks in macroeconomic time series: Some evidence and implications,”
Journal of Monetary Economics, vol. 10, pp. 139-162.
[13] Nishimura, Kazuo and John Stachurski, 2005, “Stability of Stochas-tic Optimal Growth Models: A New Approach,” Journal of Economic Theory, 122 (1), pp. 100–118.
[14] Saad, Yousef, (1992), Numerical Methods for Large Eigenvalue Problems, Halsted Press: New York, NY.
[15] Smith, Anthony M. and Fatih Guvenen 2006 “What do Labor and Consumption Data Jointly Tell About Labor Income Risk?” Unpub-lished manuscript.
[16] Stachurski, John, 2009, Economic Dynamics: Theory and Computation, MIT Press.
[17] Stachurski, John, 2008, “Continuous State Dynamic Programming via Nonexpansive Approximation.” Computational Economics Vol. 31(2), pages 141-160.
[18] Stachurski, John and Vance Martin, 2008, “Computing the Distribu-tions of Economic Models via Simulation,” Econometrica, Vol. 76(2), pages 443-450.
[19] Stachurski, John, 2009, “A Hilbert Space Central Limit Theorem for Geometrically Ergodic Markov Chains,” mimeo, Kyoto University.
[20] Tauchen, George, 1986, “Finite State Markov Chain Approximations to Univariate and Vector Autoregressions.” Economic Letters, Vol. 20, pages 507-532.