MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall Midterm Solutions

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/5.070J Fall 0 Midterm Solutios Problem Suppose a radom variable X is such that P(X > ) = 0 ad P(X > E) > 0 for every E > 0. Recall that the large deviatios rate fuctio is defied to be I(x) = sup (x log M()) for every real value x, where M() = E[exp(X)], for every real value. (a) show that I(x) = for every x >. Sice P(X > ) = 0, wehave M() = exp(x)dp X (x) exp()p (X ) = exp() We therefore obtai log M(), ad coclude that for x > I(x) =sup (x log M()) sup (x ) = (b) show that I(x) < for every E[X] x<. sice x E[X], we have that I(x) =sup (x log M()) = sup (x log M()) 0 Now, take ay E > 0 such that x < E, ad ote that M() = exp(x)dp X (x) exp(x)dp X (x) exp(( E))P(X > E) E Therefore log M() ( E) log P(X > E), ad we obtai I(x) =sup (x log M()) sup((x ( E)) log P(X > E)) log P(X > E) < 0 0 (c) show that lim E 0 P( E X ) = 0. Show that I() =. For ay E > 0, E M() = exp(x)dp X (x) = exp(x)dp X (x)+ exp(x)dp X (x) E exp(( E))P(X < E)+exp()P( E X ) exp(( E))( + P( E X )(exp(e) )) exp(( E))( + P( E X ) exp(e))

2 Let f(, E) deote the quatity above. For ay 0 ad E > 0, we have M() f(, E), ad we obtai that I() sup log f(, E) 0,E>0 Take = E log( P(X E)), so that + P( E X ) exp(e) =. We obtai that log f((e),e)= ( E) log, ad so log f((e),e) E log. Fially, ote that E = log( ) goes to as E goes to zero. P( E X ) Problem Recall the followig oe-dimesioal versio of the large Deviatios Priciples for fiite state Markov chais. Give a N-state Markov chai X, 0 with trasitio matrix P i,j, i, j N ad a fuctio f : {,..., N} R, the S i f(x i ) sequece = satisfies the Large Deviatios Priciple with the rate fuctio I(x) =sup (x log ρ(p )), where ρ(p ) is the Perro-Frobeius eigevalue of the matrix P =(e f(j) P i,j, i, j N). Suppose P i,j = π j for some probability vector π j 0, j N, j π j =. Namely, the observatios X for are i.i.d. with the probability mass fuctio give by π. I this case we kow that the large deviatios rate fuctio for the i.i.d. sequece f(x ), is described by the momet geeratig fuctio of f(x ),. Establish that the two large deviatios rate fuctios are idetical, ad thus the LDP for Markov chais i this case is cosistet with the LDP for i.i.d. processes. Proof. We have that P is exp(f(x ))π exp(f(x N ))π N P =.. exp(f(x ))π exp(f(x N ))π N Let v = [,..., ] T ad M() = E[exp(f(X ))]. The we have that P v = M()v Sice P has rak ad M() > 0, we have that M() is the Perro-Frobeius eigevalue of P. Thus, we have I(x) =sup (x log ρ(p )) = sup (x log M())

3 Problem (a) Suppose, X, 0 is a martigale such that the distributio of X is idetical for all ad the secod momet of X is fiite. Establish that X = X 0 almost surely for all. Proof. Let F be a filtratio to which the martigale X, 0 is adapted. The for ay, by tower property, we have E[(X X 0 ) ]=E[X ]+E[X 0 ] E[X 0 X ] = E[X ]+E[X 0 ] E[E[X 0 X F 0 ]] = E[X ]+E[X 0 ] E[X 0 E[X F 0 ]] = E[X ]+E[X 0 ] E[X 0 ] =0 by the fact that X, 0 has the same distributio ad that X has fiite secod momet. Thus, we have X = X 0 almost surely. (b) A ur cotais two white balls ad oe black ball at time zero. At each time t =,,... exactly oe ball is added to the ur. Specifically, if at time t 0 there are white balls ad B t black balls, the ball added at time t +is white B t with probability ad is black with the remaiig probability.i particular, sice there were three balls at the begiig, ad at every time t exactly oe ball is added, the + B t = t +, t 0. Let T be the first time whe the proportio of white balls is exactly 50% if such a time exists, ad T = if this is ever the case. Namely T =mi{t : = } if the set of such t is o-empty, ad T = otherwise. Establish a upper boud P(T < ). Proof. First, we will establish that is a martigale. Let F t be a filtratio W to which t is adapted. The we have that [ ] E + B t ad that + B t E[ F t ]= + + B t + B t + + B t + B t + + B t ( + + B t ) = = ( + B t )( + + B t ) + B t

4 Thus,, t 0 is a martigale. Sice X t T, the optioal stoppig theorem gives that X T is almost surely well defied radom variable ad E[X T ]=E[X 0 ]. Thus, we have E[X T ]= P(T < )+αp(t = ) =E[X 0 ]= P(T < ) +α( P(T < )) = () where 0 α is the fractio for t ad it exists by Martigale covergece theorem. By (), we have that P(T < ) <, thus P(T < ) P(T < ) α = 0 P(T < ) P(T < ) P(T < ) 4

5 MIT OpeCourseWare J / 6.65J Advaced Stochastic Processes Fall 0 For iformatio about citig these materials or our Terms of Use, visit: