Prove that for probability measuresQ, Qn onP(Ω), we have Qn →w Q ⇔ ∀k∈NQn({k})→Q({k

TU Darmstadt Fachbereich Mathematik

Jakob Creutzig

WS 2006/07 15.12.06

9. Aufgabenblatt zur Vorlesung

”‘Probability Theory”’

1. – warming up

(a) LetX_n∼Γ(α_n, β_n), i.e.,P_Xn has Lebesgue density

g(x;αn, βn) :=1x>0x^αn−1·β_n^αne^−βn Γ(α_n) ,

where Γ denotes the Gamma function, andα_n, β_n>0. Assume thatα_n→ α,β_n→β withα, β∈[0,∞]. When does X_n converge in distribution?

(b) ConsiderNwith metricd(x, y) :=|x−y|. Show that in this metric space, every set{x}and thus any subsetA⊆Nis open; conclude thatB(N) = P(Ω). Prove that for probability measuresQ, Qn onP(Ω), we have

Qn

→w Q ⇔ ∀_k∈NQn({k})→Q({k}).

(c) Letpn∈(0,1) such that n·pn→λ >0. Show thatB(n, pn)→^w P oi(λ).

(Hint: Use (b) and convergence rules from Analysis. In particular, recall that, ifan→a >0, then (1−an/n)ⁿ →e^−a.)

2. LetF_n, F be distribution functions. Assume thatF is continuous and that Fn(x)→F(x) for allx∈R. Prove that

sup

x∈R

|Fn(x)−F(x)| →0.

(Hint: Show first that for fixed N, sup_x∈[−N,N]|Fn(x)−F(x)| →0; for this, use the uniform continuity ofF on [−N, N].)

3. Let Ω = ]0,1[, A=B(Ω), and consider the uniform distribution P on Ω. For a distribution functionF, define

X(ω) = inf{z∈R:ω≤F(z)}, ω∈]0,1[ . (a) Show thatX is a random variable withFX=F.

(b) Let U be uniformly distributed on ]0,1[. Determine a measurable map- pingT : ]0,1[→[0,∞[ such that T(U) is exponentially distributed with parameterλ >0. (This is a state-of-the-art way to simulate exponential distributions.)

4. Consider the set ofmolecular probability measures onR, i.e.,

P=nXⁿ

k=1

λk·δx_k:n∈N, λk>0,

n

X

k=1

λk = 1, xk ∈R o

.

Prove that P is dense in the set of all probability measures on (R,B) w.r.t.

weak convergence, i.e., for every probability measure µ on (R,B) a suitable sequence in Pconverges weakly toµ.

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