On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

Abstract

Abstract. We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite dimensional increments of the process. The distinct feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited for processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.

Conditioned limit theorems for products of positive random matrices

“…from Lemma 5.6 in Grama et al (2014). By choosing n 0 sufficient great, the first assertion of the lemma follows from (3.14), (3.15) and (3.16) taking into account that…”

“…First, we check that the weak invariance principle with rate stated in Grama et al (2014) (Theorem 2.1) may be applied to the sequence (ρ(g k , X k−1 )) k≥0 .The hypotheses C1, C2 and C3 of this theorem are given in terms of Fourier transform of the partial sums of S n ; combining the expressions (2.1), (2.2), (2.3) and the properties of the Fourier operators (P t ) t , we verify in the next section that these conditions are satisfied in our context. This leads to the following simpler but sufficient statement.…”

“…However, the studies on the subject of fluctuation was quite sparse a few years ago. Thanks to the approach of Denisov and Wachtel (2015) for random walks in Euclidean spaces and motivated by branching processes, I. Grama, E. Le Page and M. Peigné recently progressed for invertible matrices (Grama et al, 2014). Here we propose to develop the same strategy for matrices with positive entries by using Hennion and Hervé (2008).…”

“…We use the coupling argument to prove Theorem 1.2 and Theorem 1.3 in section 4. At last, in section 5 we check general conditions to apply an invariant principle stated in Theorem 2.1 in Grama et al (2014).…”

“…Inspired by a recent paper of I. Grama, E. Le Page and M. Peigné (Grama et al, 2014), we consider a sequence (g n ) n≥1 of i.i.d. random d×d-matrices with non-negative entries and study the fluctuations of the process (log |g n · · · g 1 x|) n≥1 for any non-zero vector x in R d with non-negative coordinates.…”