Ibragimov–Iosifescu conjecture for φ-mixing sequences

Let $(\Omega,\mathcal{F},\mu)$ be a probability space and let $X:=(X_k, k\in\mathbb{Z})$ be a strictly stationary sequence of real-valued random variables on it. For $m\in\mathbb{Z}$ put $\mathcal{F}^{m}:=\sigma(X_j, j\leqslant m)$ and, for $n\geqslant 1$, $\mathcal{F}_{m+n}:=\sigma(X_j, j\geqslant m+n)$, and define the $\phi$-mixing coefficients

$$\phi_X(n):=\sup\bigl\{\,|\mu(B\mid A)-\mu(B)| \;:\; m\in\mathbb{Z},\ A\in\mathcal{F}^{m}\ \text{with}\ \mu(A)>0,\ B\in\mathcal{F}_{m+n}\,\bigr\}.$$

Say that $X$ is $\phi$-mixing if $\phi_X(n)\to 0$ as $n\to\infty$.

Write $S_n:=\sum_{j=1}^{n}X_j$ and $\sigma_n^2:=E S_n^2$.

Assume that $X$ is $\phi$-mixing and that

$$E X_0=0,\qquad E X_0^2<\infty,\qquad \sigma_n^2=\operatorname{Var}(X_1+\cdots+X_n)\to\infty \quad (n\to\infty).$$

Then:

1. (Central limit theorem) $S_n/\sigma_n$ converges in distribution to the standard normal law $N(0,1)$ as $n\to\infty$.

2. (Weak invariance principle) The processes

$$W_n(t):=\frac{S_{[nt]}}{\sigma_n},\qquad t\in[0,1],$$

converge weakly, as $n\to\infty$, in $D[0,1]$ equipped with the Skorokhod topology, to a standard Brownian motion $W$ on $[0,1]$.