Limit theorems for a random walk of a special kind

Let $X_0\equiv0$, $X_1,\dots,X_n,\dots,$ be a Markov chain with the transition probabilities
\begin{gather*} \mathbf P\{X_{n+1}=m+1\mid X_n=m\}=p(n,m), \\ \mathbf P\{X_{n+1}=m\mid X_n=m\}=1-p(n,m). \end{gather*}

Recurrent relations are derived for the characteristic functions of the random variables $X_n$. On this basis for the cases $p(n,m)=\alpha+\varphi(n)$ and $p(n,m)=(n-m)/n$ Gärding's integral theorem (about the convergence of the appropriately normed and centered random variables $X_n$ to a normal random variable) is precised and a local limit theorem with an estimation of the speed of the convergence is proved