= Simultaneous wavelet coefficient noise bound
{title2=$\tau_\lambda=\sigma q_\lambda\sqrt{2\log(2N/\delta)}$}
For coefficient weights $w_{\lambda i}$ and independent mean-zero <sub-Gaussian random variables> of parameter $\sigma$, put $q_\lambda^2=\sum_iw_{\lambda i}^2$. Multiplying moment-generating-function bounds shows that the weighted noise has parameter $\sigma q_\lambda$. The exponential Markov inequality, optimized in its parameter, gives tail bound $2\exp(-t^2/(2\sigma^2q_\lambda^2))$. At the displayed thresholds, a <union bound> over $N$ coefficients gives simultaneous control with <probability> at least $1-\delta$. No coefficient independence is required. Integrated wavelet weights satisfy $q_\lambda\leq n^{-1/2}$. Finite variance alone only supports the weaker Chebyshev choice $\tau_\lambda=\sigma q_\lambda\sqrt{N/\delta}$.
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