Theorem. A false-accept union bound without independence [ftip-007M]
✍️source

    For any joint law of a repeated verifier audit,

    \[ \Pr (\mathcal E_N^{\rm fa})\leq \sum _{i=1}^N \phi _i. \]

    If \(\iota _i=\Pr (S_i=0)>0\) for every \(i\), this becomes

    \[ \Pr (\mathcal E_N^{\rm fa})\leq \sum _{i=1}^N \iota _i\eta _{+,i}. \]

    Proof. 

      Apply the union bound to \(\mathcal E_N^{\rm fa}=\bigcup _i C_i^{\rm fa}\). The factorization \(\phi _i=\iota _i\eta _{+,i}\) follows from conditional probability whenever \(\iota _i>0\). If \(\iota _i=0\), then \(\phi _i=0\) while the corresponding conditional rate is undefined, so the first display remains the unconditional statement.

      Without a dependence assumption, matching marginal error rates do not give the equality in Theorem [ftip-007L]; the events \(C_i^{\rm fa}\) could coincide.

      This finite statement follows from the displayed hypotheses.