We will derive ergodicity results in the following sections based on Theorem 1 in the paper of Roberts and Rosenthal, 1 and we will use some proof techniques of Theorem 1 from Cui et al. 2 for part of our analysis. All random variables we will discuss below will be of the form X : Ω → X where X is a metric space with the associated Borel σ-algebra B(X ).Let X be the parameter space, assumed to have a metric space topology, and π : B(X ) → [0, 1] the target distribution to be sampled from by an adaptive MCMC algorithm. We will assume that π has a density f : X → [0, ∞). Let K γ denote a transition kernel that depends on an adaptation index γ ∈ Y, and assume that each K γ has π as an invariant distribution.We assume that for each fixed γ, an MCMC algorithm with K γ as the Markov transition kernel will eventually converge to π, that is lim n→∞ K n (x, .) − π T V = 0 where µ − ν T V = sup B∈B(X ) |µ(B) − ν(B)| is the total variation distance between two probability measures on X . Let X n be the random variable representing the state of the adaptive MCMC at iteration n, and let Γ n be the random variable representing the choice of kernel for updating from X n to X n+1 . The state of the algorithm is then modeled by the discrete-time stochastic process {(X n , Γ n )}, whose transition between steps is determined by the underlying rules of the algorithm. Finally, let G n = σ ({X 0 , . . . , X n , Γ 0 , . . . , Γ n }) denote the filtration generated by {(X n , Γ n )}. Thus, each Γ n+1 is a G n+1 -measurable random variable.Roberts and Rosenthal proved the following important result, which gives sufficient conditions for ergodicity of an adaptive MCMC.Theorem 1 (Theorem 1 in Roberts and Rosenthal 1 ). Consider an adaptive MCMC algo-