Family of large deviation principles

Question

The following question may be a bit imprecise in its formulation, I guess however the problem I have in mind is clear. Although to me it looks like a fairly standard question, I couldn't find any reference approaching it so far and hope someone here can help.

Assume that for every $\epsilon>0$, $\lbrace X^{\epsilon}_{n}\rbrace_{n}$ satisfies a LDP with rate function $I^{\epsilon}$. Also, suppose that for every $n\in\mathbb{N}$ we have tightness for $\lbrace X_{n}^{\epsilon}\rbrace_{\epsilon}$ and let $\lbrace X_{n}\rbrace_{n}$ be a family of limit points. Does convergence of the $I^{\epsilon}$ to some rate function $I$ in a reasonable sense (say $\Gamma$ or Mosco), already imply a LDP for $\lbrace X_{n}\rbrace_{n}$ with rate function $I$? What more is needed?
Remark: I'm here particularly interested in Schilder-type LDPs.

Answer 1

First, to see why this is not enough, suppose all variable involved take value in some compact interval. Suppose the sequence $X_n$ satisfies the LDP, with rate function $J(x)$, and let $X_n^\epsilon= X_n+1$ if $n>1/\epsilon$ and $X_n^\epsilon=X_n$ if $n<1/\epsilon$. Then $X_n^\epsilon\to_{\epsilon \to 0} X_n$. But $X_n^\epsilon$ satisfies the LDP with the $\epsilon$-independent rate function $J(x-1)$, while $X_n$ satisfies the LDP with rate function $J(x)$.

What is missing is some uniformity in the convergence of $X_n^\epsilon$ to $X_n$.