Assuming the following assumptions:
- $\underset{n \to \infty}{\limsup} \sum_{j=1}^n | 1- \varphi_{jn}(u)|< \infty$;
- $s_n = \sup_{1\leq j \leq n}|1- \varphi_{jn}(u)| \to 0$, as $n \to \infty$
I am trying to show a convergence involving characteristic functions $\varphi_{jn}$ : $$\sum_{j=1}^n | \varphi_{jn}(u) - \exp\{ \varphi_{jn}(u) -1 \} | \to 0 \quad(n \to \infty) $$
My attempt begins using the following: first note that the ch. f. of some random vector $X$ is such that $\varphi(u)= E[\cos(u'X)] + i E[\sin(u'X)]$, i.e., the $Re(\varphi(u))\leq 1$. So, $Re(\varphi(u)-1) \leq 0$. Thus, we are able to use this fact and conclude that: $$|P_1(z)- e^z|= |1+ z - e^z |\leq |z|^2, \quad z = \varphi_{jn}(u) -1 $$ i.e. \begin{equation}\label{taylor}\tag{T} |\varphi_{jn}(u) - \exp\{\varphi_{jn}(u) -1 \} |= |1+ (\varphi_{jn} -1 ) - \exp\{\varphi_{jn} -1 \} |\leq |\varphi_{jn}(u) -1|^2 \end{equation} Aggregating: $$\sum_{j=1}^n | \varphi_{jn}(u) - \exp\{ \varphi_{jn}(u) -1 \} |\leq \sum_{j=1}^n |\varphi_{jn}(u) -1||\varphi_{jn}(u) -1|\leq s_n \sum_{j=1}^n |\varphi_{jn}(u) -1|$$ Note that if $x_n\geq 0$ and $\limsup x_n =0$, then $\lim x_n =0$. So $$\limsup_{n \to \infty} \sum_{j=1}^n | \varphi_{jn}(u) - \exp\{ \varphi_{jn}(u) -1 \} |\leq \lim_{n \to \infty} s_n \limsup_{n \to \infty} \sum_{j=1}^n |\varphi_{jn}(u) -1|\overset{\hbox{ by 1 and 2}}{=}0$$
I am very unsure about my attempt and would like to know if there are any errors in my demo, mainly because, in fact, the fact mentioned above is true for $Re(z)<1$. Is this true for the case $Re(z)=0$ ?
I greatly appreciate your feedback.
