Let $X$ have the negative binomial distribution with pmf
$f(x) = \binom{r+x-1}{x} p^r (1-p)^x, x=0,1,2,...,$
where $0<p<1$ and $r>0$ is an integer.
(a) Calculate the mgf of X.
(b) Define a new random variable by $Y=2pX$. Show that, as $p \downarrow0$, the mgf of Y converges to that of a chi squared random variable with 2r degrees of freedom by showing that
$$\lim_{p \to 0 } M_Y(t)=(\frac{1}{1-2t})^r, |t|< \frac{1}{2}$$
My attempt: I get the part (a) by refer to here: Deriving Moment Generating Function of the Negative Binomial?
My attempt of part b is:
$M_Y(t)=Ee^{tY}=Ee^{t2pX}=M_X(2pt)$, then I just put 2pt inside the answer of part (a). My question is about the value of t in part b. In part (a), we have $t<−log(1−p)$. Thus, I use $2pt < -\log(1-p)$, thus $t < -\frac{log(1-p)}{2p}$. When p goes to 0, I then get $t<1/2$. But in the textbook, it said $|t|<1/2$. Question 1: I don't know where the absolute value come from.
Question 2: the book said it converges to a chi squared random variable with 2r degrees of freedom. How can I see this? It's weird for me to understand mgf converges to a distribution directly, rather than some distribution's mgf converges to another distribution's mgf.
Source question: Casella statistical inference Exercise 2.38.
