7/27/2019 thi ht mn xc sut 2010

1/2

International Master in Mathematics.Probability Examination (3h).Documents are not allowed.

Exercise 1: Iet (p") b" r ""qrluo""tf probability measures on lR and p be a probability measure on lR.Assumethatforallo{bRthenu*Qa,fl--p(la,bl).Showthat,forallaelR'p"(l - oo'al) --p(l - *'"1)'Exercise 2: In this exercice, X, Xt, Xz .... a.re random variables defined on a probability spa.ce (Q,"4,F)taking values in the interval [-1, +1]. lB will denote the expectation with respect to lP. We assume that forall i > 1, lE(x) : E(x.) : 0.PART I : a concentration inequality for sums of random variablesLet eE r p-x e. - e-ncosh(z) : :j: and sinh(c) : zbe the hyperbolic sine and cosine functions.

1. Show that for all ) > 0, we have 12cosh()) < exp(;) .(You may use series expansions.)

2. Show that for all ) > 0 and t [-1,+1] thene)' < cosh()) + c sinh(.\) < *p(*) f rsinh(.\) ."2

3. Let X be a centered random variable with values in [-1, +1]. Show theit for all ,\ ) 0,\2^2.["'*] < e*p(]) a''a E[e-rx] < expl];

4. Show that for all a ) 0,F[x ) a] s "*p(-+) and IF[X 5 -o] 3 exp(-{).

5. Let Xi be independent centered random r,n"riables with ralues in I-1,+fl. Stton' that for all n andall a ) 0 we have rerrl $x,r t or . 2"*o. o''"^/n? ''-'- '(*t)'' .?=rPART II : an optimal bound ?

1. Let

2. Show that"-+ rrd@):"_ and o(z) = / 4(t)dt.\/2T J,*

1a()-9,4as ,r -+ co. (You may use some integration by paxts axgument.)

7/27/2019 thi ht mn xc sut 2010

2/2

Does there exist constanh n > | and .4 > 0 such that for all squences of centered independentrandom va.riables Xi witb values in [-1,+1], for all o ) 0 and all n'Ptl+txrl >al< Aexp(- na2)?' \ln -*

- Exercise 3:L. Let X be a random va.riable whose characteristic function / is equal to 1 on [-o,a] for some o > 0(we do not make any claim about the values of / outside [-4, c])' Show that P(X = 0) = l'Hint: note that IE(cos(tX)) :7lort [-a;o].2. Deduce that if x a.nd Y are independent random variables such that x + y and Y have the samelaw, then lP(X = 0) : 1.

Exercise 4: Let (e'1)",11 be a sequence of sqrrare integrable independent random variables, with the samedistribution, such that lE(e1) : o and E[(er)2] : 1' For all z 2 1' let us deffneXs=0, X*= QN^-t16^,where d lR. The questions of this exercise can be treated separately'

1. In this question, l0l < 1. Assume that the law of e1 is 3{(0, 1). what is the law of X" ? show thatthe law of X" weaklY converges.2. In this question' ldl < 1. Set M* : D;=t|i-Iei for n 2 1 and M6 = 0' Prove that for all n" X'"has the same distribution as M". Provd tliat M' a.s. converges. Deduce that the law of X" weakly

converges.3. Assume that IAI > 1. Prove that T-nXo a.s' conrges'4. Assume that ldl : 1. Find a real sequence (a6)4>1, such that the law of o"X" weakly coaverges tosome non degenerate law that you will specify.