L¨ohr Winter term 2016/17
Exercises to the lecture Large Deviations
Exercise sheet 2
General LDPs & Exponential Tightness
Let (E, d) be a complete, separable metric space.
Exercise 2.1 (LDP: elementary (non-)examples). Let (µn)n∈N be a sequence in M1(E).
Decide in the following examples (with proof) if (µn)n∈N satisfies a large deviation principle. If so, determine the rate function and decide if it is good. Otherwise, decide if (µn)n∈N satisfies a weak large deviation principle.
(a) µn uniform distribution on [−n1,n1].
(b) µn uniform distribution on [−2n,2n].
(c) µn =µ∈ M1(E) for alln∈N(constant sequence).
(d) µn =δxn for a convergent sequence (xn)n∈N inE, i.e.xn→xfor somex∈E.
Exercise 2.2. LetX1, X2, . . .be i.i.d. Poisson distributed with parameter 1,Sn:=Pn
k=1Xk. We are interested in the existence of the following limit inR∪ {−∞}:
n→∞lim
1
nlogP {n1Sn ∈A}
. (1)
(a) Prove or disprove: the limit (1) exists for all open setsA.
(b) Prove or disprove: the limit (1) exists for all closed setsA.
(c) Calculate the limit (1) forA= [0,12[∪ {√
2} ∪[2,42].
Exercise 2.3 (LDP versus law of large numbers). Let (Xn)n∈N be anE-valued stochastic process. Assume that (Xn)n∈Nsatisfies a large deviation principle with good rate functionI (this is a short-hand for saying thatµn:=L(Xn) satisfies the LDP).
(a) Show that if I has a unique minimum at x0 ∈ E, then Xn converges almost surely to a random variableX. Calculate the distribution ofX.
(b) Show that if the minimum ofIis not unique, then it is possible that Xn does not converge in probability.
Exercise 2.4(LDP for convex combinations). Letm∈N. Fork= 1, . . . , m, letµkn∈ M1(E) and 0< pk <1 be such that (µkn)n∈N satisfies a large deviation principle with rate function Ik, andPm
k=1pk = 1. Defineµn:=Pm k=1pkµkn.
(a) Does (µn)n∈N satisfy a large deviation principle? If so, determine the rate functionI. If it does not in general, give a necessary and sufficient condition.
(b) If all Ik are good, isI necessarily good? If only I1 is good, is I necessarily good? If I is good, does this mean that one/all of the Ik have to be good?
(c) If we had allowedm=∞, would the answer to (a) be the same?
Please turn
Exercise 2.5 (exponential tightness: elementary (non-)examples). (a) Let µn = µ ∈ M1(E) for alln∈N. Is the sequence (µn)n∈N exponentially tight?
(b) Letµn∼ N0,1n be normally distributed with mean 0 and variance n1. Decide (with proof) if (µn)n∈N is exponentially tight.
(c) Prove or disprove: if the sequence (µn)n∈Nconverges weakly to a Dirac measureδxfor some x∈E, then it is exponentially tight.
Excercises will be discussed on 08.12.2016
