Counterexamples about Markov Chain

Question

Let $(X_k)_{k=1}^\infty$ be a sequence of iid random variables. Let $Y_n=X_{n}+X_{n-1}$, and $Z_n=\sum_{i=1}^n S_i$, where $S_n$ is the n-th partial sum of $(X_k)$. I think both $(Y_n)$ and $(Z_n)$ are not Markov Chain, but I can't find appropriate examples. Thanks!

Answer 1

Recall that a discrete prozess $(Z_n)_{n \in \mathbb{N}}$ is a Markov chain if

$$\mathbb{P}(Z_n = x_n \mid Z_{n-1} = x_{n-1},\ldots,Z_1 = x_1) = \mathbb{P}(Z_n = x_n \mid Z_{n-1} = x_{n-1}).$$

Hints for $(Y_n)_{n \in \mathbb{N}}$: Let $(X_k)_{k \in \mathbb{N}}$ be a sequence of independent identically distributed random variables such that $\mathbb{P}(X_k=1) =\mathbb{P}(X_k = 0)=1/2$.

1. Show that $$\mathbb{P}(Y_3 = 0 \mid Y_2 = 1) = \frac{1}{4}.$$
2. Show that $$\mathbb{P}(Y_3= 0 \mid Y_2 = 1, Y_1=0) = 0.$$
3. Conclude that $(Y_n)_{n \in \mathbb{N}}$ is not a Markov chain.

Hints for $(Z_n)_{n \in \mathbb{N}}$: Let $(X_k)_{k \in \mathbb{N}}$ be a sequence of independent identically distributed random variables such that $\mathbb{P}(X_k=1) = \mathbb{P}(X_k = 0)=1/4$ and $\mathbb{P}(X_k=1/2)=1/2$.

1. Show that $$\mathbb{P}(Z_3 = 3/2 \mid Z_2 = 1) = \frac{\mathbb{P}(X_3 = 0, X_2 = 0, X_1 = 1/2)}{\mathbb{P}(Z_2 =1)} = \frac{1}{6}.$$
2. Show that $$\mathbb{P}(Z_3 = 3/2 \mid Z_2=1, Z_1 = 0) = 0.$$
3. Conclude that $(Z_n)_{n \in \mathbb{N}}$ is not a Markov chain.