Convergence in probability and distribution proof $Z_n=(-1)^n Z$ where $Z \sim N(0,1)$.
How to compute $P(|Z_n -Z|)>e)$ for $e>0$.
Prove $Z_n$ converge in probability to Z and converge in distribution to Z.
Here is my solution.
Since $Z_n=(-1)^n Z$, if $n=2N$(N is interger) or n=0 then $Z_n-Z=0$.
Also if n=2N+1 then $Z_n-Z=-2Z$
Therefore $P(|Z_n -Z|)>e)=0$.
But I think something is missing in my logic...
 A: You correctly identified what $Z_n$ is according to the cases where $n$ is odd or even.
Actually, $(Z_n)_{n\geqslant 1}$ does not converge in probability to anything. Otherwise, we would have $Z_{n+1}-Z_n\to 0$ in probability but $\lvert Z_{n+1}-Z_n\rvert=2\lvert Z\rvert$ and the sequence $(2\lvert Z\rvert)$ does not converge to $0$ in probability.
However, convergence in distribution holds. To see this, I suggest to show that for each $n$, the distribution of $Z_n$ is equal to that of $Z$.
A: According to your question, we are supposed to show both $Z_n \stackrel{P}{\rightarrow} Z$, and  $Z_n \stackrel{D}{\rightarrow} Z$. However, the former is simply not true. If $n$ is odd, $P(|Z_n - Z| > \epsilon) = P(2|Z| > \epsilon)  \not \rightarrow 0$.
For the latter claim that $Z_n \stackrel{D}{\rightarrow} Z$, intuitively, this is true since $Z$ is symmetric about the y-axis. To formally prove this claim, you can show that the characteristic functions converge, i.e. $E(e^{itZ_n}) = E(e^{itZ})$.
Alternatively, we can show this from the definition of convergence in distribution. Let $F_n$ and $F$ be the cdf of $Z_n$ and $Z$, respectively. Then, $Z_n \stackrel{D}{\rightarrow} Z$ means that for any $x$ where $F$ is
continuous, $|F_n(x) - F(x)| \rightarrow 0$.
It is straightforward to verify that $F_n = F$, by considering the cases when $n$ is even or odd. When $n$ is even, clearly $F_n(x) = P((-1)^n Z < x) = P(Z<x) = F(x)$. When $n$ is odd, $F_n(x) = P((-1)^n Z < x) = P(-Z<x) =  P(Z > -x) = P(Z<x) =F(x)$.
Therefore, when $n\rightarrow \infty$, regardless of whether $n$ is even or odd, $\sup_x |F_n(x) - F(x)| = 0$, which is the definition of convergence in distribution.
