= Exponential equivalence
Coupled random elements $X_N,Y_N$ in a <metric space> are exponentially equivalent at <large-deviation speed> $a_N$ if for every $\eta>0$
$$
\limsup_N a_N^{-1}\log\mathbb P(d(X_N,Y_N)>\eta)=-\infty.
$$
If $X_N$ satisfies a <large deviation principle> with a <good rate function> $I$, then so does $Y_N$, with the same speed and <rate function>. For the upper bound, $\{Y_N\in F\}$ is contained in $\{X_N\in F^\eta\}\cup\{d(X_N,Y_N)>\eta\}$, where $F^\eta$ is the closed $\eta$-neighborhood. Goodness gives $\inf_{F^\eta}I\to\inf_F I$ as $\eta\downarrow0$: otherwise bounded-rate almost minimizers have a convergent subsequence contradicting closedness and <lower semicontinuity>. For an open-set lower bound, choose a ball about any $x\in G$ with a slightly larger ball still in $G$, subtract the superexponentially small error from the probability that $X_N$ belongs to the smaller ball, and shrink the ball. No <independence> assumption is required.
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