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 $\eta >0$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 $\eta \downarrow 0$: 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.