As a mixing condition including many interesting dynamic systems as special cases, $\mathcal{C}$-mixing condition has drawn significant attention in recent years. This paper aims to do some contributions on the following points. First, we show a Bernstein-type inequality under $\mathcal{C}$-mixing conditions. Compared with the pioneering work on this point, \citeA{hang2017bernstein}, our inequality is sharper under more general assumptions. Second, since the general definition of $\mathcal{C}$-mixing condition is based on a covariance inequality whose upper bound relies on some given $\mathcal{C}$-norm (see Definition \ref{def 1}), a natural difficulty arises when the $\mathcal{C}$-norm is infinitely large. Under this circumstances, we show some inequalities bounding the variance of partial sums without requiring finite $\mathcal{C}$-norms. Finally, up to our knowledge, there is few literature discussing central limit theorem under $C$-mixing conditions as general as that of \citeA{hang2017bernstein}. Thus, under \citeauthor{hang2017bernstein}'s $\mathcal{C}$-mixing conditions, we take one step forward on this point by deriving a central limit theorem with mild moment conditions. As for the applications, we apply the previously mentioned results to show Bahadur representation and asymptotic normality of weighted $M$-estimators.
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