Let $\{X_n\}_{n\in \mathbb{N}}$ be a strong mixing sequence (no stationarity) with exponentially decaying mixing rate. Further assume that $X_n$ has uniformly bounded forth moments. Does $$\frac{1}{n}\sum_{i=1}^n X_i -EX_i = O_P\left(\frac{1}{\sqrt{n}}\right) \ \ (*)$$ holds?
In order to give some background: If the sequence is i.i.d. then it is not hard to show the convergence with desired rate $\frac{1}{\sqrt{n}}$ by Tschebychef. In particular, my idea for the proof of $(*)$ uses Tschebychef as a first step and applies Rio's inequality, i.e. upper bound covariances with terms that contains the mixing coefficient. But from here I can not conclude the desired convergence rate..
Your assumptions are more than enough for the result you want. By Rio's covariance inequality, it suffices to show that $$ \sup_{n\geqslant 1}\frac 1n\sum_{1\leqslant i\leqslant j\leqslant n} \int_0^{\alpha(j-i)}Q_{X_i}(u)Q_{X_j}(u)du<\infty, $$ where $Q_X(u)=\inf\{t,\mathbb P(\left|X\right|>t)\leqslant u\}$. Using twice Cauchy-Schwarz inequality, , one has \begin{align} \int_0^af(u)g(u)du&\leqslant\left(\int_0^af(u)^2du\right)^{1/2}\left(\int_0^1g(u)^2\right)^{1/2}\\ &\leqslant a^{1/4}\left(\int_0^1f(u)^4du\right)^{1/2}\left(\int_0^1g(u)^2\right)^{1/2}. \end{align} Applying this for fixed $i,j$ to $a=\alpha(j-i)$, $f(u)=Q_{X_i}(u)$ and $g(u)=Q_{X_j}(u)$ gives, accounting that $\int_0^1Q_{X_i}(u)^4du=\mathbb E\left[X_i^4\right]$, that $$ \frac 1n\sum_{1\leqslant i\leqslant j\leqslant n} \int_0^{\alpha(j-i)}Q_{X_i}(u)Q_{X_j}(u)du \leqslant \frac 1n\sum_{1\leqslant i\leqslant j\leqslant n}\alpha(j-i)^{1/4}. $$ Then counting for a fixed $k$ the number of times where $j-i=k$ shows that the assumption $\sum_{k=1}^\infty \alpha(k)^{1/4}$ is sufficient.