 Solution

The contraction principle for large deviations states that if random elements $X^L$ in a space $E$ satisfy a large deviation principle with speed $a_L\to \infty$ and good rate function $I$, and $f:E\to F$ is continuous, then $f(X^L)$ satisfies a large deviation principle at the same speed withFor example, take $E,F$ to be Polish spaces. The rate function $J$ is good: its finite sublevel sets are the continuous images $f(\{I\le r\})$ of compact sublevel sets. Compactness and lower semicontinuity ensure that a finite fibre infimum relevant to such a sublevel set is attained. The probability bounds follow by taking preimages of open and closed sets.