We now assume that the random variables $X_1, \ldots, X_N$ are independent, so that they are $k$-independent for all $k \in \{2, \ldots, N\}$. We now want to establish the following bound: there exist numerical constants $\alpha, \beta > 0$ (independent of $K \geq 1$ and $N$) such that for all $t \geq 0$, $$\mathbb{P}\left(|S_N| \geq t\sqrt{N}\right) \leq \beta \exp\left(-\alpha t^2/K^2\right)$$ I.6.a) Justify that it suffices to consider the case $K = 1$, which we will do in the next three questions. I.6.b) Let $k$ be the largest even integer in $\{1, \ldots, N\}$ less than or equal to $\frac{2t^2}{e^2}$. Justify that (27) is satisfied if $$e \leq t \leq \frac{e}{\sqrt{2}}\sqrt{N}$$ I.6.c) Under hypothesis (32), prove that we have (31) with $$\beta = 2e, \quad \alpha = e^{-2}$$ I.6.d) Conclude that there exist numerical constants $\alpha, \beta > 0$ such that (31) is verified for all $t \geq 0$.
We now assume that the random variables $X_1, \ldots, X_N$ are independent, so that they are $k$-independent for all $k \in \{2, \ldots, N\}$. We now want to establish the following bound: there exist numerical constants $\alpha, \beta > 0$ (independent of $K \geq 1$ and $N$) such that for all $t \geq 0$,
$$\mathbb{P}\left(|S_N| \geq t\sqrt{N}\right) \leq \beta \exp\left(-\alpha t^2/K^2\right)$$
\textbf{I.6.a)} Justify that it suffices to consider the case $K = 1$, which we will do in the next three questions.
\textbf{I.6.b)} Let $k$ be the largest even integer in $\{1, \ldots, N\}$ less than or equal to $\frac{2t^2}{e^2}$. Justify that (27) is satisfied if
$$e \leq t \leq \frac{e}{\sqrt{2}}\sqrt{N}$$
\textbf{I.6.c)} Under hypothesis (32), prove that we have (31) with
$$\beta = 2e, \quad \alpha = e^{-2}$$
\textbf{I.6.d)} Conclude that there exist numerical constants $\alpha, \beta > 0$ such that (31) is verified for all $t \geq 0$.