The Ibragimov–Iosifescu conjecture for φ-mixing sequences

Astra constructs a strictly stationary real process $$X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots),$$ where the innovations $\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\varphi$-mixing, centered, square-integrable, and satisfies $$\operatorname{Var}(S_n)\to\infty.$$ Nevertheless there are times $n_j\to\infty$ such that $$\frac{S_{n_j}}{\sqrt{\operatorname{Var}(S_{n_j})}}\to0$$ in probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$. The same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.

Astra constructs a strictly stationary real process $$X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots),$$ where the innovations $\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\varphi$-mixing, centered, square-integrable, and satisfies $$\operatorname{Var}(S_n)\to\infty.$$ Nevertheless there are times $n_j\to\infty$ such that $$\frac{S_{n_j}}{\sqrt{\operatorname{Var}(S_{n_j})}}\to0$$ in probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$. The same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.