mixing with respect to (G n, G + n ) n∈N and θ 0 p < ∞ for all p ≥ 1. Assumption 3.2. There exist constants L 1, L 2 > 0 such that for all θ, θ ′ ∈ R d and x, x ′ ∈ R m, Assumption 3.1 implies, in particular, that X 0 ∈ L r, for any r ≥ 1, thus, under Assumptions 3.1 and 3.2, Two important properties immediately follow from Assumptions 3.2 and 3.3.
where we have setã = aL 1 a + L 1.
Our aim initially is to estimate θ λ n − θ λ n 2, uniformly in n ∈ N. To begin with, an example is presented where explicit calculations are possible.
Example 3.4. Let d := 1, H(θ, x) := θ + x and let (X n ) n∈Z be a sequence of independent R-valued standard Gaussian random variables, independent of (ξ n ) n∈N. We observe that the function H satisfies Assumptions 3.2 and 3.3. Take θ 0 := 0. It is straightforward to check that, for any λ ∈ (0, 1), which clearly has variance It follows that This shows that the best estimate we may hope to obtain for sup n∈N θ λ n − θ λ n 2 is of the order √ λ. Our Theorem 3.5 below achieves this bound asymptotically as p → ∞.
Our main results may be stated as follows.
there exists C • (p) > 0 such that holds for a constant C • (p) that is explicitly given in the proof. It depends only on a, L 1, L 2, d, p and the process X through the quantities (7).
Proof. The proof of this theorem is postponed to Section 6.2.
The next result relates our findings in Theorems 3.5 to the problem of sampling from the probability law π.
where c 1 (κ), c 2 (κ) (given explicitly in the proof ) depend only on d, a, L 1, L 2 and on the process X.
Proof. The proof of this theorem is postponed to Section 6.3.

Independent data
When the data sequences (X n ) n∈Z are i.i.d., then the full rate is recovered under more relaxed conditions for the unbiased estimator of the gradient of U. More concretely, one may assume the following: Assumption 3.8. The process (X n ) n∈N is i.i.d. with X 0 2(ρ+1) and θ 0 2 being finite.
Assumption 3.9. There exists a mapping A : R m → R d×d such that and, for all θ, θ ′ ∈ R d and x ∈ R m, with the smallest eigenvalue of the matrix E[A(X 0 )] being a positive real number which is denoted by a.
It is clear then that properties (B1) and (B2) are still valid for the gradient h of U, with the only difference that the Lipschitz constant in (B1) is given by. This allows us to obtain the following result.
Proof. The proof of this Theorem is postponed to Section 7.

Related work and discussion
Rate of convergence. Theorem 3.6 significantly improves on some of the results in [24] in certain cases, compare also to [32]. In [24] the monotonicity assumption (9) is not imposed, only a dissipativity condition is required and a more general recursive scheme is investigated. However, the input sequence (X n ) n∈N is assumed i.i.d. In that setting, [24, Theorem 2.1] applies to (4) (with the choice δ = 0, β = 1 and d fixed, see also the last paragraph of Subsection 1.1 of [24]), which implies that holds whenever λ ≤ c 3 (ǫ/ ln(1/ǫ)) 4 and n ≥ c4 ǫ 4 ln 5 (1/ǫ) with some c 3, c 4 > 0. For the case of i.i.d. (X n ) n∈N see also the very recent [18]. Our results provide the sharper estimates (16) in a setting where (X n ) n∈N may have dependencies. One notes, further, that a noisy Langevin Monte Carlo algorithm (nLMC) with inaccurate drift is proposed in [6], where the drift is assumed to be a linear combination of the original gradient and of random noise represented by a dependent sequence of random vectors with non-zero means. Thus, a particular form of dependency is included in this approach. A convergence result, [6, Theorem 4], in Wasserstein-2 distance between nLMC and the target distribution π is provided, which is in agreement with our findings, i.e. rate of convergence equal to 1/2 is given when the biased term is eliminated. Thus, a somewhat improved rate is obtained in comparison to 1/2-ǫ in (15) but for a significantly smaller class of dependent data.
Choice of step size. It is pointed out in [27] that the ergodicity property of (2) is sensitive to the step size λ. Moreover, [19,Lemma 6.3 ] gives an example in which the Euler-Maruyama discretization is transient. As pointed out in [19], under discretization, the minorization condition is insensitive with appropriate sampling rate while the Lyapunov condition may be lost. An invariant measure exists if the two conditions hold simultaneously, see [19,Theorem 7.3] and also [27,Theorem 3.2] for similar discussions. In this work, an approach similar to [5] is chosen, in that strong convexity of U is assumed together with Lipschitzness of its gradient and, thus, the ergodicity of (2) is obtained.
More on exponential rates. The convergence rate of the Euler-Maruyama scheme (2) to its stationary distribution is comparable to that of the Langevin SDE (1), see [27] or the discussion after Proposition 3.2 of [24]. However, [24] proved the exponential convergence of (1) by using its version with the inverse temperature parameter, which is denoted by β, as follows under the condition that β > 2/m, where m appears in the (uniform in x) dissipativity condition (A.3) of [24], Let us consider the same setting with strongly convex h and β = 1. The constant m, which controls the quadratic term, plays a similar role to a in our Assumption 3.3. In [24], m has to be larger than 2, i.e., outside a compact set, the function U should be sufficiently convex. From the present paper it becomes clear that for the approximation (2) to work, strong convexity (a > 0) is enough. Our result is stronger and not included in [24], however, it requires global convexity. Note also that the dimension d has different effects on convergence rate: it is exp(−λn exp(Õ(d))) in [24] and exp(−λn)O(d 3/4 ) in our case, see Remark 6.7 immediately after the proof of Theorem 3.5.

The Langevin SDE and the Unadjusted Langevin Algorithm
By [22, Theorem 2.1.8], U has a unique minimum at some point θ * ∈ R d. Note that due to the Lipschitz condition (B1), the SDE (1) has a unique strong solution. It is a well-known result that the Langevin SDE (1) admits a unique invariant measure π. By [17,Theorem 4.20], one constructs the associated strongly Markovian semigroup (P t ) t≥0 given for all t ≥ 0, x ∈ R d and A ∈ B(R d ) by P t (x, A) = P (θ t ∈ A|θ 0 = x).
The following lemma from [9] with adapted statement provides the explicit bound of the second moment of the Langevin diffusion, which allows the analysis of the Wasserstein-2 distance between π and the aforementioned sampling algorithms. (i) For all t ≥ 0 and ϑ ∈ R d, For a fixed step size λ ∈ (0, 1], consider the Markov kernel R λ given for all A ∈ B(R d ) and θ ∈ R d by The discrete-time Langevin recursion (2) defines a time-homogeneous Markov chain, and for any n ≥ 1, and for any bounded (or non-negative) Borel function f : Lemma 5.2 below is also a result from [9] and along with Lemma 5.1 are presented here for completeness by using the notation of this article. In particular, Lemma 5.2 states that R λ admits a unique stationary distribution π λ, which may differ from π.
Lemma 5.2. Let Assumption 3.3 hold and thus (B2) is thereby implied. Then, for all λ <λ, whereλ is defined in (14), the following hold: (ii) The Markov kernel R λ has a unique stationary distribution π λ which satisfies whereã is defined in (13).
Then, one calculates where the last term is clearly zero. Thus, due to Lemma A.4, Moreover, one recalls that for λ < 2/(a + L 1 ) As a result, for θ which yields (20). Consequently, by Lemma A.5 below, Thus, one