Let $k$ be an integer greater than or equal to 2. We say that random variables $(Y_n)_{n \geq 1}$ are $k$-independent if $$\mathbb{E}\left[\psi_1(Y_{n_1}) \cdots \psi_k(Y_{n_k})\right] = \mathbb{E}\left[\psi_1(Y_{n_1})\right] \cdots \mathbb{E}\left[\psi_k(Y_{n_k})\right]$$ for all indices $1 \leq n_1 < \cdots < n_k$ and for all bounded functions $\psi_1, \ldots, \psi_k : \mathbb{R} \rightarrow \mathbb{R}$. I.4.a) Prove that $k$-independence implies $j$-independence if $j \leq k$. I.4.b) What is $N$-independence for $N$ random variables $Y_1, \ldots, Y_N$? I.4.c) Let $Y_1$ and $Y_2$ be independent random variables with uniform distribution on $\{0,1\}$: for $n = 1,2$, $$\mathbb{P}(Y_n = 0) = \mathbb{P}(Y_n = 1) = \frac{1}{2}$$ Let $Y_3$ be the random variable on $\{0,1\}$ defined by $$Y_3 := Y_1 + Y_2 \quad \bmod 2$$ Prove that the random variables $(Y_1, Y_2, Y_3)$ are 2-independent but not 3-independent.
Let $k$ be an integer greater than or equal to 2. We say that random variables $(Y_n)_{n \geq 1}$ are $k$-independent if
$$\mathbb{E}\left[\psi_1(Y_{n_1}) \cdots \psi_k(Y_{n_k})\right] = \mathbb{E}\left[\psi_1(Y_{n_1})\right] \cdots \mathbb{E}\left[\psi_k(Y_{n_k})\right]$$
for all indices $1 \leq n_1 < \cdots < n_k$ and for all bounded functions $\psi_1, \ldots, \psi_k : \mathbb{R} \rightarrow \mathbb{R}$.
\textbf{I.4.a)} Prove that $k$-independence implies $j$-independence if $j \leq k$.
\textbf{I.4.b)} What is $N$-independence for $N$ random variables $Y_1, \ldots, Y_N$?
\textbf{I.4.c)} Let $Y_1$ and $Y_2$ be independent random variables with uniform distribution on $\{0,1\}$: for $n = 1,2$,
$$\mathbb{P}(Y_n = 0) = \mathbb{P}(Y_n = 1) = \frac{1}{2}$$
Let $Y_3$ be the random variable on $\{0,1\}$ defined by
$$Y_3 := Y_1 + Y_2 \quad \bmod 2$$
Prove that the random variables $(Y_1, Y_2, Y_3)$ are 2-independent but not 3-independent.