# How to obtain tail bounds for a linear combination of dependent and bounded random variables?

I am looking for tail bounds (preferably exponential) for a linear combination of dependent and bounded random variables.

consider $$K_{ij}=\sum_{r=1}^N\sum_{c=1}^N W_{ir}C_{rc}W_{jc}$$ where $i \neq j$, $W\in \{+1, -1\}$ and $W$ follows $\operatorname{Bernoulli}(0.5)$, and $C=\operatorname{Toeplitz}(1, \rho, \rho^2, \ldots, \rho^{N-1})$, $0 \leq \rho < 1$.

I will be to happy if you give me any pointer to how I can evaluate the moment generating function of $K_{ij}$ to have bound for $Pr\{K_{ij} \geq \epsilon\}\leq \min_s\exp(-s\epsilon)E[\exp(K_{ij}s)]$ based on chernoff bound ans $s \geq 0$.

For a hint you can look at my another question entitled as How to obtain tail bounds for a sum of dependent and bounded random variables?'' which is a special case of this problem where $C_{rc}=1, 1 \leq c \leq N, 1 \leq r \leq N$. Thanks a lot in advance.

This is Hanson Wright. Let $$\sigma=W_i$$ and $$\sigma' = W_j$$ then by Jelani's Lecture Notes:
$$\Pr[|\sigma^T C \sigma'|>\lambda] = O(\exp(-c\lambda^2/\|C\|_F^2) + \exp(-c'\lambda/\|C\|^2)).$$
where $$c$$ and $$c'$$ are some constants.
Here $$\|C\|_F^2=N\sum_{i=0}^{N=1} \rho^{2i}=N\frac{p^{2N}-1}{\rho^2-1}$$ is the Frobenius norm, and $$\|C\|=\sum_{i=0}^{N-1}\rho_i=\frac{\rho^N-1}{\rho-1}$$ is the spectral norm.