2. Error bounds for MCMC integration

This section contains our main result, Theorem 2.1, together with the required notation and assumptions. Let us start by specifying the general setting.

We assume that the state space $(\mathcal{X}, \mathcal{F}_\mathcal{X})$ is Polish with $\mathcal{F}_\mathcal{X}$ being countably generated. Let K be a Markov kernel and $\nu$ a probability measure, called the initial distribution, both defined on $(\mathcal{X}, \mathcal{F}_\mathcal{X})$ . Then, the Markov chain corresponding to K and $\nu$ , say $(X_n)_{n \in \mathbb{N}}$ , is defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P}_\nu)$ . In particular, such a probability space exists. We assume that $\pi$ is the unique invariant distribution of K and that $(X_n)_{n \in \mathbb{N}}$ is $\psi$ -irreducible. For definitions and further details we refer to [Reference Douc, Moulines, Priouret and Soulier4, Reference Meyn and Tweedie13].

Given $p \in [1, \infty)$ , we define $L^p (\pi)$ as the set of all functions $f \colon \mathcal{X} \to \mathbb{R}$ such that $\Vert f\Vert_{L^p(\pi)}^p = \pi(\vert f \vert^p) \lt \infty$ . Similarly, $L^\infty(\pi)$ denotes the set of functions with finite $\Vert f\Vert_{L^\infty(\pi)} = {\rm ess\,sup}_{x \in \mathcal{X}} \vert f(x) \vert$ . We follow the usual convention that two functions in $L^p(\pi)$ , with $p \in [1, \infty]$ , are considered as equal if they are equal $\pi$ -almost everywhere. Then, $(L^p(\pi),\Vert {\cdot}\Vert_{L^p(\pi)})$ is a normed space. Moreover, $L^2(\pi)$ , equipped with $\langle f, g\rangle_{L^2(\pi)} = \int_\mathcal{X} f(x) g(x)\,\pi(\mathrm{d} x)$ , is a Hilbert space with induced norm $\Vert {\cdot}\Vert_{L^2(\pi)}$ , and so is the closed subspace $L^2_0 (\pi) \,:\!=\, \{f \in L^2(\pi) \colon \pi(f) = 0\}$ .

Let $p \in [1, \infty]$ ; then the Markov kernel K induces a linear operator $K\colon L^p(\pi)\to L^p(\pi)$ via $f \mapsto Kf({\cdot}) = \int_\mathcal{X} f(x')\,K({\cdot}, \mathrm{d} x')$ . Indeed, the operator K is well defined and we have $\Vert K\Vert_{L^p(\pi) \to L^p(\pi)} =1$ ; we refer to [Reference Rudolf17, Section 3.1] for further details.

We denote by Id the identity on $L^2(\pi)$ . The following condition about the operator $\mathrm{Id}-K$ , restricted to $L^2_0(\pi)$ , is our main assumption.

We note that Assumption 2.1 is closely related to a spectral gap; there are, however, different definitions for a spectral gap:

• • Some authors (see, e.g., [Reference Andrieu, Lee, Power and Wang1, Reference Douc, Moulines, Priouret and Soulier4]) say K admits a(n) (absolute $L^2$ ) spectral gap if $\sup_{\lambda \in \mathcal{S}_0} \vert \lambda \vert \lt 1$ , where $\mathcal{S}_0$ is the spectrum of $K \colon L^2_0 (\pi) \to L^2_0 (\pi)$ . This is equivalent to the existence of some $m \in \mathbb{N}$ such that $\Vert K^m\Vert_{L^2_0(\pi) \to L^2_0(\pi)} < 1$ ; cf. [Reference Douc, Moulines, Priouret and Soulier4, Proposition 22.2.4].

• • On the other hand, a different definition, for instance used in [Reference Hairer, Stuart and Vollmer6, Reference Natarovskii, Rudolf and Sprungk14, Reference Rudolf17, Reference Rudolf and Schweizer18], is to say that K admits a(n) (absolute $L^2$ ) spectral gap if $\Vert K\Vert_{L^2_0(\pi)\to L^2_0(\pi)} \lt 1$ .

If K is reversible, which implies that the corresponding Markov operator is self-adjoint on $L^2(\pi)$ , then both definitions are equivalent.

Let us emphasise that either of the above definitions of a spectral gap implies that Assumption 2.1 is true, including in the non-reversible case. Spectral gap results were established for a number of MCMC methods, see for instance [Reference Andrieu, Lee, Power and Wang1, Reference Hairer, Stuart and Vollmer6, Reference Natarovskii, Rudolf and Sprungk14]; see also [Reference Rudolf17, Section 3.4] and [Reference Roberts and Rosenthal16, Theorem 2.1]. Moreover, we note that under Assumption 2.1 we cover the setting of [Reference Rudolf and Schweizer18, Theorems 1 and 2].

If for the initial distribution we have $\nu \ll \pi$ with Radon–Nikodým derivative ${\mathrm{d}\nu}/{\mathrm{d}\pi} \in L^q(\pi)$ for some $q \in[1, \infty]$ , then we set $M_q = \Vert{\mathrm{d}\nu}/{\mathrm{d}\pi}\Vert_{L^q(\pi)}$ . In particular, for $q = \infty$ we have $\sup_{A \in \mathcal{F}_\mathcal{X}}[{\nu(A)}/{\pi(A)}] \leq M_\infty$ , in which case $\nu$ is called $M_\infty$ -warm.

The following theorem is our main result, which shows that under Assumption 2.1 we have the optimal rate of convergence for the absolute mean error.

Theorem 2.1. Let Assumption 2.1 be true, $p \in (1,2]$ , and assume that $\nu$ is absolutely continuous with respect to $\pi$ with Radon–Nikodým derivative ${\mathrm{d}\nu}/{\mathrm{d}\pi} \in L^q(\pi)$ , where $q \gt 0$ satisfies $p^{-1} + q^{-1} =1$ . Then, for any $f \in L^p(\pi)$ and any $n \in \mathbb{N}$ ,

\[ \mathbb{E}_\nu[\vert S_n f - \pi(f)\vert] \leq \frac{C_p M_q\Vert f\Vert_{L^p(\pi)}}{n^{1-1/p}} , \]

where $C_p = 2^{2/p-1} \cdot (4s)^{1-1/p}$ and $M_q = \Vert{\mathrm{d}\nu}/{\mathrm{d}\pi}\Vert_{L^q(\pi)}$ .

Remark 2.3. Let $p\in (1,2]$ , $q \in [1, \infty)$ with $p^{-1} + q^{-1}=1$ , and $C_p, M_q$ as specified in Theorem 2.1. Given $c \in \mathbb{R}$ and $f \in L^p(\pi)$ , we set $f_c = f+c$ , and note that $S_n f_c - \pi(f_c) = S_n f - \pi(f)$ . Thus, for fixed $f \in L^p(\pi)$ and any $c \in \mathbb{R}$ we have

\[ \mathbb{E}_\nu[\vert S_n f_c - \pi(f_c)\vert] = \mathbb{E}_\nu[\vert S_n f - \pi(f)\vert] \leq \frac{C_p M_q\Vert f\Vert_{L^p(\pi)}}{n^{1-1/p}} , \]

even though $\Vert{\cdot}\Vert_{L^p(\pi)}$ is not invariant with respect to linear shifts.

3. Proofs

In this section we prove our main results. Recall that the chain $(X_n)_{n \in \mathbb{N}}$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P}_\nu)$ . For $p \in [1, \infty]$ we define $L^p(\mathbb{P}_\nu)$ as the set of random variables Y on $(\Omega, \mathcal{F}, \mathbb{P}_\nu)$ such that $\Vert Y\Vert_{L^p(\mathbb{P}_\nu)}^p = \mathbb{E}_\nu [\vert Y \vert^p] \lt \infty$ . As for the $L^p(\pi)$ spaces, we consider two random variables $Y_1, Y_2 \in L^p(\mathbb{P}_\nu)$ as equal if $Y_1=Y_2$ holds $\mathbb{P}_\nu$ -almost surely. Then, $(L^p(\mathbb{P}_\nu), \Vert \cdot\Vert_{L^p(\mathbb{P}_\nu)})$ is a normed space.

The first result of this section provides a bound for the mean squared error of $S_n h$ for the case where $(X_n)_{n \in \mathbb{N}}$ is stationary, i.e. where $X_1 \sim \pi$ , and h is a centred function.

Proof. Expanding $\mathbb{E}_\pi[\vert S_n h\vert^2]$ and using [Reference Rudolf17, Lemma 3.25], we get

\[ \mathbb{E}_\pi[\vert S_n h\vert^2] = \frac{1}{n^2}\sum_{j=1}^n\sum_{k=1}^n\mathbb{E}_\pi[h(X_j)h(X_k)] = \frac{1}{n^2}\sum_{j=1}^{n}\sum_{k=1}^n\big\langle h,K^{\vert j-k\vert}h\big\rangle_{L^2(\pi)}. \]

We have that $\sum_{j=1}^n\sum_{k=1}^nK^{\vert j-k\vert} = 2\sum_{\ell=1}^{n-1}(n-\ell)K^\ell + n\mathrm{Id} = 2\sum_{\ell=0}^{n-1}(n-\ell)K^\ell - n\mathrm{Id}$ . Moreover, by induction it follows that $\sum_{\ell=0}^{n-1}(n-\ell)K^\ell = (\mathrm{Id}-K)^{-1}\sum_{m=1}^{n}(\mathrm{Id} - K^m)$ , where all the operators are understood to act on $L^2_0(\pi)$ .

Hence, again considering all operators to act on $L^2_0(\pi)$ , the following identity holds:

(3.1) \begin{align} \sum_{j=1}^n\sum_{k=1}^nK^{\vert j-k\vert} & = 2(\mathrm{Id} - K)^{-1}\sum_{m=1}^{n}(\mathrm{Id} - K^m) - n\mathrm{Id} \nonumber \\ & = 2n(\mathrm{Id}-K)^{-1} - 2(\mathrm{Id} - K)^{-1}\sum_{m=1}^n K^m - n\mathrm{Id} \nonumber \\ & = n(\mathrm{Id} - K)^{-1}(\mathrm{Id} + K) - 2(\mathrm{Id} - K)^{-1}\sum_{m=1}^nK^m. \end{align}

Set $R_n =n (\mathrm{Id} - K)^{-1}(\mathrm{Id} + K) - 2(\mathrm{Id} - K)^{-1}\sum_{m=1}^nK^m$ . By the triangle inequality and the bounds $\Vert(\mathrm{Id}-K)^{-1}\Vert_{L^2_0(\pi) \to L^2_0(\pi)} \leq s$ and $\Vert K^m\Vert_{L^2_0(\pi) \to L^2_0(\pi)} \leq 1$ , which are true for any $m \in \mathbb{N}$ , we obtain $\Vert R_n\Vert_{L^2_0(\pi) \to L^2_0(\pi)} \leq 2sn + 2sn = 4sn$ . Since $\big\langle h,K^{\vert j-k\vert}h\big\rangle_{L^2(\pi)} = \big\langle h,K^{\vert j-k\vert}h\big\rangle_{L^2_0(\pi)}$ for any $j,k \in \{1,\ldots,n\}$ , it follows from (3.1) and the bound on $\Vert R_n\Vert_{L^2_0(\pi) \to L^2_0(\pi)}$ that

\[ \mathbb{E}_\pi[\vert S_n h\vert^2] = \frac{1}{n^2}\langle h,R_n h\rangle_{L^2_0(\pi)} \leq \frac{\Vert h\Vert_{L^2(\pi)}^2}{n^2}\Vert R_n\Vert_{L^2_0(\pi) \to L^2_0(\pi)} \leq \frac{4s\Vert h\Vert_{L^2(\pi)}^2}{n}, \]

which completes the proof.

Proposition 3.1. Let Assumption 2.1 be true, let $p \in [1, 2]$ , and assume that $(X_n)_{n \in \mathbb{N}}$ has initial distribution $\pi$ , i.e. $X_1 \sim \pi$ . Then, for any $f \in L^p(\pi)$ and any $n \in \mathbb{N}$ we have

\[ \mathbb{E}_\pi[\vert S_n f - \pi(f)\vert^p] \leq 2^{2-p}\bigg(\frac{4s}{n}\bigg)^{p-1}\Vert f\Vert_{L^p(\pi)}^p. \]

Proof. For any $g \in L^2(\pi)$ set $\bar{g} = g - \pi(g) \in L^2_0(\pi)$ . Moreover, we set $T_n f = S_nf-\pi(f)$ for $f \in L^p(\pi)$ with $p \in [1,2]$ .

By Lemma 3.1 and the inequality $\Vert\bar{g}\Vert_{L^2(\pi)}^2 = \pi(g^2)-\pi(g)^2 \leq \pi(g^2) = \Vert g\Vert_{L^2(\pi)}^2$ , which is true for any $g \in L^2(\pi)$ , we obtain

\[ \mathbb{E}_\pi[\vert T_n g\vert^2] = \mathbb{E}_\pi[\vert S_n\bar{g}\vert^2] \leq \frac{4s}{n}\Vert\bar{g}\Vert_{L^2(\pi)}^2 \leq \frac{4s}{n}\Vert g\Vert_{L^2(\pi)}^2, \]

which implies that $\Vert T_n\Vert_{L^2(\pi) \to L^2(\mathbb{P}_\pi)} \leq \sqrt{{4s}/{n}}$ . Moreover, using the triangle inequality we see that $\Vert T_n\Vert_{L^1(\pi) \to L^1(\mathbb{P}_\pi)} \leq 2$ . Thus, by the Riesz–Thorin interpolation theorem, see [Reference Grafakos5, Theorem 1.3.4] in the setting $p_0 = q_0 = 1$ , $p_1 = q_1 = 2$ , and $\theta = 2- 2/p$ , we obtain that

\[ \Vert T_n\Vert_{L^p(\pi) \to L^p(\mathbb{P}_\pi)} \leq 2^{2/p-1}\bigg(\frac{4s}{n}\bigg)^{1-1/p}, \]

which implies the result.

Using a change of measure argument and Proposition 3.1 we are able to prove our main result. However, before we turn to the proof of Theorem 2.1 let us state another consequence of Proposition 3.1. We note that the upcoming result generalises [Reference Rudolf17, Theorem 3.41] by providing a bound for the mean squared error of $S_n f$ for $L^2(\pi)$ functions.

Corollary 3.1. Let Assumption 2.1 be true, let $p \in [1, 2]$ , and assume that $\nu \ll \pi$ with Radon–Nikodým derivative ${\mathrm{d}\nu}/{\mathrm{d}\pi} \in L^\infty(\pi)$ . Then, for any $f \in L^p(\pi)$ and any $n \in \mathbb{N}$ ,

\[ \mathbb{E}_\nu[\vert S_n f - \pi(f)\vert^p] \leq \frac{C_p M_\infty\Vert f\Vert_{L^p(\pi)}^p}{n^{p-1}} , \]

where $C_p = 2^{2-p}\cdot(4s)^{p-1}$ and $M_\infty = \Vert{\mathrm{d}\nu}/{\mathrm{d}\pi}\Vert_{L^\infty(\pi)}$ .

Proof. Since $\mathbb{P}_\nu$ and $\mathbb{P}_\pi$ only differ by the choice of the initial distribution of $(X_n)_{n \in \mathbb{N}}$ , we have

\[ \mathbb{E}_\nu[\vert S_n f - \pi(f)\vert^p] = \mathbb{E}_\pi\bigg[\frac{\mathrm{d}\nu}{\mathrm{d}\pi}(X_1)\vert S_n f - \pi(f)\vert^p\bigg] \leq \bigg\Vert\frac{\mathrm{d}\nu}{\mathrm{d}\pi}\bigg\Vert_{L^\infty(\pi)}\mathbb{E}_\pi[\vert S_n f - \pi(f)\vert^p]. \]

Hence, the result follows from Proposition 3.1.

Finally, we turn to the proof of Theorem 2.1.

Proof of Theorem 2.1. By the same change of measure argument as before and Hoelder’s inequality,

\[ \mathbb{E}_\nu[\vert S_n f - \pi(f)\vert] = \mathbb{E}_\pi\bigg[\frac{\mathrm{d}\nu}{\mathrm{d}\pi}(X_1)\vert S_n f - \pi(f)\vert\bigg] \leq \bigg\Vert\frac{\mathrm{d}\nu}{\mathrm{d}\pi}\bigg\Vert_{L^q(\pi)} \mathbb{E}_\pi[\vert S_n f - \pi(f)\vert^p]^{1/p}. \]

Now the desired result is a consequence of Proposition 3.1.