Concentration bounds for overlapping binomial sums

Suppose I have some random variables $X_1, \dots, X_n$ (say they are iid Bernoulli-$p$ for simplicity). Consider the overlapping sums $Y_i = X_i + \dots + X_{i+w}$ for some fixed $w$. I want to bound the probability that there exists some $i$ for which $Y_i \geq q$.

By the Union Bound this is clearly at most $(n-w+1) \times P(Y_1 \geq q)$. Are there any better bounds available, that take into account the fact that adjacent windows are closely correlated?

Some keywords or references would be great

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