Upper bounds of the Gärtner-Ellis theorem for the sequences of random variables

Let Y1,Y2,… be real valued random variables. The Gärtner-Ellis theorem gives sufficient conditions for a large deviations principle for the sequence {Yn/n}. Briefly, the theorem provides sufficient conditions for exponential upper bounds for the probabilities P(Yn/n∈F) for the closed sets F and lower bounds for the probabilities P(Yn/n∈G) for the open sets G. Our objective is to derive necessary and sufficient conditions for the upper bounds of the theorem.