Show that X∞ n=1 P(An

TU Darmstadt Fachbereich Mathematik

Wilhelm Stannat

WS 2007/08 14.11.07

Probability Theory 6. Aufgabenblatt Gruppen¨ubungen Aufgabe G18:

Determine the Dynkin-systemD(E) generated byEin the case whereEcontains two subsets Aand B of Ω. Show that D(E) and the σ-algebraσ(E) generated byE coincide if and only ifA∩B orA∩B^c orA^c∩B orA^c∩B^c are empty.

Aufgabe G19:

LetX be a real random variable and let f be a real Borel measurable function onR. Show thatX andf◦X are independent if and only iff◦X are constant a.s.

Aufgabe G20:

Let (An) be a sequence of independent events on (Ω,A, P) withP(An)<1 for alln∈NandP(S_∞

n=1An) = 1. Show that X∞

n=1

P(An) =∞.

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Haus¨ubungen Aufgabe H16:

LetPbe the uniform distribution on Ω ={0,1}^N. Forj= 1, . . . , NdefineA_j= {(ω1, . . . , ωN)∈Ω|ωj= 1}andAN+1={(ω1, . . . , ωN)∈Ω|ω1+. . .+ωN even}.

Show thatA1, . . . , AN+1 are dependent but anyN events out ofA1, . . . , AN+1

are independent.

Aufgabe H17:

Let (An) be a sequence of independent events and pn := P(An). Which as- sumptions for pn imply limn→∞1An= 0

(i) in probability, (ii) P-a.s.?

Hint: Use the lemma of Borel-Cantelli.

Aufgabe H18:

As in H5 consider the model for infinitely many fair coin tosses and

`n((xn)n∈N) := min{k≥1|xn=. . .=xn+k−1= 1, xk= 0}

Forr≥0 defineEn(r) :={`n ≥r}.

(i) With the lemma of Borel-Cantelli show for any increasing sequence of non-negative real numbersr1, r2, . . .withP_∞

n=12^−rn

rn =∞that P

· lim sup

n→∞ E_n(r_n)

¸

= 1.

(ii) Use (i) to conclude thatP[lim sup_n→∞En(log₂n)] = 1 and therefore with H5 that

P

· lim sup

n→∞

`n

log₂n = 1

¸

= 1.

(Hint for (i): Define the sequence (nk) inductively withn1= 1 andnk+1= nk+rnk. Then the eventsEnk(rnk), k≥1, are independent.)

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