On Local Limit Theorems for the Sums of Independent Random Variables

Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables, ${\mathbf E}X_1=m$, ${\mathbf D}X_1=\sigma^2>0$, and ${\mathbf E}|X_1|^k<\infty$ for some integer $k\geqq 3$. The following theorem is proved:
Suppose that the variable $Z_n=\dfrac1{\sigma\sqrt n}\Bigl(\sum\limits_{j=1}^n{X_j-nm}\Bigr)$ has an absolutely continuous distribution with bounded density function $p_n(x)$ for some integer $n=n_0$. Then there exists a function $\varepsilon(n)$ such that lim $\varepsilon(n)=0$ and relation (1) is fulfilled.
A similar theorem is proved for the case when $X_1$ has a lattice distribution. Some consequences of these theorems concerning convergence to the normal law in the mean are discussed.