about large deviation

Exercise 1. (Counting dice rolls) /You may find (dH00, Theorem 1.3) a useful guide in this exercise. Consider rolling a die with six sides and, for k > 1, consider the random variable 1 - { if the k-th roll is a 6, otherwise. Suppose that the die is fair so that P[X* = 1] = and that each roll is independent of the others. As before, let Sn = {k=1 XX. (a) What is 11 := E[X1]? For r > 0, what is P[Sn 0, using Stirling's formula, give upper and lower bounds on log P[InSn Hl2c). (c) Show that, for any c> 0, the limit -I (c) := limnolog PISn-ul 2 exists and give an expression for Ic). Interpret this result as an asymptotic upper and lower bound on PILSn-ul>c] for large n. (d) Look at Cramr's theorem (Pha07, Theorem 2.1) or (dH00, Theorem 1.4]). Verify that your answer to (c) coincides with the direct answer provided by this theorem