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.



Family of large deviation principles
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 $X\_n$.


