Define $X_n$ to be position of a particle after $n$ seconds such after every second, particle moves up two $2$ with probability $1/2n$, it moves down $2$ units with probability $1/2n$ or it stays with probability $1-1/n$. What is $\lim_{n\to\infty} X_n$?
Using Characteristic functions, I was able to show that CF of the limit would be $\exp(-2\sin^2x) = \exp(\cos(2x)-1)$ but can anything be said about the distribution/density of the limit
Edit:
Define $Z_n$ st $P[Z_n=2]=P[Z_n=-2]=1/2n, P[Z_n=0]=1-1/n$
Then CF of $Z_n ={n-1\over n}+\cos2x/n=1-{1-\cos2x\over n}$
Since $X_n$ is sum of $n$ iid $Z_n$'s, $\phi_{X_n}=(1-{1-\cos2x\over n})^n$
$$\lim_{n\to\infty}\phi_{X_n}=\lim_{n\to\infty}\left(1-{1-\cos2x\over n}\right)^n=e^{\cos2x-1}=e^{-2\sin^2x}$$
Edit 2: $$P[X_n=0]=\sum_{p=0}^{\lfloor{n/2}\rfloor}{n\choose p,p,n-2p}{1\over 2n}^{2p}\left(1-\frac1n\right)^{n-2p}$$ Can we find this limit? If as $n\to\infty$ this is 1, then my calculations would be wrong.