Find the limiting distribution of the following trinomial distribution

Question

Let $X_1,X_2,\dots$ be i.i.d. with the following probability density:

$$P(X_j=0)=1-\frac{\lambda}{n}$$ $$P(X_j=1)=P(X_j=2)=\frac{\lambda}{2n}$$

Define $Y_n=\sum_1^nX_j$. Find $\lim_{n\rightarrow\infty} \psi_{Y_n}(t)$.

I calculated the characteristic function of $Y_n$ as follows.

$$\psi_{X_j}(t)=E[e^{iX_jt}]= (1-\frac{\lambda}{n})e^0+\frac{\lambda}{2n}e^{it}+\frac{\lambda}{2n}e^{2it}$$

$$\psi_{Y_n}(t)=[\psi_{X_j}(t)]^n=[1+\frac{1}{n}(\frac{\lambda}{2}e^{it}+\frac{\lambda}{2}e^{2it}-\lambda)]^n$$

$$\lim_{n\rightarrow\infty} \psi_{Y_n}(t) = exp(\frac{\lambda}{2}e^{it}+\frac{\lambda}{2}e^{2it}-\lambda)$$

From here I am unable to simplify the characteristic function to that of any distribution I am familiar with, which leads me to believe I did something incorrectly.

Answer 1

You dropped a $\lambda$; the result should be $$\exp \left(\lambda( \frac{1}{2}e^{it}+\frac{1}{2}e^{2it}-1)\right).$$ Hopefully, that looks more familiar.

Re the below remark, this is the characteristic function of $Y+2Z$, where $Y$ and $Z$ are independent and both Poisson distributed with parameter $\lambda/2$.

To help see what's going on, it may be helpful to think of two independent Poisson processes $A$ and $B$ on the time interval $[0,1]$, both with rate $\lambda/2$, and to give the $A$ arrivals a weight of $1$ and the $B$ arrivals a weight of $2$. Then, as $n\to\infty$, in each time interval $[\frac {i-1}n, \frac{i}n]$, $i=1$, $\dots$, $n$,

• There will be no arrivals with probability $1-\frac{\lambda}{n}+O(n^{-2})$.
• There will be one $A$ arrival and no $B$ arrivals with probability $\frac{\lambda}{2n}+O(n^{-2})$.
• There will be one $B$ arrival and no $A$ arrivals with probability $\frac{\lambda}{2n}+O(n^{-2})$.
• All other combinations of arrivals, lumped together, have probability $O(n^{-2})$.

Up to events of probability $O(n^{-2})$, then, the total weight of arrivals in the time interval $[\frac {i-1}n, \frac{i}n]$ has the same distribution as $X_{i}$. So, you would expect the distribution of $X_1+\dots+X_n$ to converge to the distribution of the total weight of arrivals from $A$ and $B$ in the interval $[0,1]$. This is the distribution of $Y+2Z$.