1. Aufgabenblatt zur Vorlesung Probability Theory

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1 Aufgabenblatt zur Vorlesung By (Ω, A, P ) we always enote the unerlying probability space, unless state otherwise. 1. Let r > 0, an efine f(x) = 1 [0, [ (x) exp( r x), x R. a) Show that p f is integrable w.r.t. the Lebesgue measure λ 1 for every polynomial p. b) Compute the integrals x k f(x) λ 1 (x) for k = 0, 1, Let f(x, y) = 1/(2π) exp( (x 2 + y 2 )/2) enote the ensity of the two-imensional stanar normal istribution w.r.t. λ 2. a) Consier a ranom vector (U, V ) taking values in ]0, 1[ 2 with P (U,V ) = 1 [0,1] 2 λ 2. Define T : ]0, 1[ 2 R 2 by T (u, v) = 2 ln u ((cos(2π v), sin(2π v)), an put (X, Y ) = T (U, V ). Show that P (X,Y ) = f λ 2. b) Consier a two-imensional ranom vector (X, Y ) such that P X = P Y = N(0, 1). Prove or isprove P (X,Y ) = f λ Let F an F n with n N enote non-ecreasing functions on R with limits 0 an 1 for x an x, respectively. Assume that F is continuous. Show that pointwise convergence of (F n ) n towars F implies uniform convergence. 4. Consier the space L p = L p ([0, 1], B([0, 1]), λ), where 1 p < an where λ enotes the Lebesgue measure. a) Show that C([0, 1]) is ense in L p. Hint: Aufgabe 3.3 from Measure an Integration Theory (SS 2016) an algebraic inuction. b) For f L p an n N we efine I n,k = [(k 1)/n, k/n[ as well as Show that (f n ) converges to f in L p. f n = n n k=1 I n,k f λ 1 In,k. Abgabe: , 10:00 Uhr

2 Aufgabenblatt zur Vorlesung P -a.s. 1. Suppose that X n X, an let ε ]0, 1[. Show that there exists a set A A with P (A) 1 ε an lim X n (ω) X(ω) = 0. sup n ω A P P -a.s. 2. Suppose that X n X an X n X n+1 P -a.s. for every n N. Show that X n X. 3. Consier a probability space (Ω, A, P ) with a countable set Ω. Prove or isprove that almost sure convergence is equivalent to convergence in probability in this case. 4. Let Ω = ]0, 1[, A = B(Ω), an consier the uniform istribution P on Ω. Define for any istribution function F. a) Show that F is the istribution function of P X. X(ω) = inf{z R : ω F (z)}, ω ]0, 1[, b) Let U be uniformly istribute on ]0, 1[. Determine a measurable mapping T : ]0, 1[ [0, [ such that T U is exponentially istribute with parameter λ > 0. Abgabe: , 10:00

3 Aufgabenblatt zur Vorlesung 1. Let Q n enote the binomial istribution with parameters n N an p n ]0, 1[ such that λ = lim n n p n ]0, [. Show that (Q n ) n N converges weakly to the Poisson istribution with parameter λ. 2. Let f n an f enote probability ensities w.r.t. the Lebesgue measure λ 1. a) Suppose that f n converges to f λ 1 -almost everywhere. Show that f n λ 1 converges weakly to f λ 1. b) Provie an example, where f n λ 1 converges weakly to f λ 1 but f n converges to f only on a set of Lebesgue measure zero. 3. Consier the set { n P = λ k ε xk : n N, λ k > 0, k=1 n k=1 } λ k = 1, x k R of probability measures on (R, B). Prove that P is ense 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 P converges weakly to µ. 4. Give a proof of the implication (v) (i) in the Portemanteau Theorem. Hint: Employ a suitable partition of the range of f C b (M). Abgabe: , 10:00

4 Aufgabenblatt zur Vorlesung 1. Let Q n = N(µ n, σ 2 n), where µ n R an σ n 0. By efinition, N(µ, 0) = ε µ. a) Show that (Q n ) n N converges weakly iff (µ n ) n N an (σ n ) n N converge. Determine the weak limit in case of convergence. b) Characterize the tightness of the set {Q n : n N}. 2. Consier ranom variables X n, X, Y n, Y on a common probability space. Prove or isprove a) X n X Y n Y X n + Y n X + Y. b) X, X n L 1 X n X (E(X n )) n N converges lim n E(X n ) = E(X). c) (X n ) n N u.i. X L 1 L X n X X 1 n X. 3. Let (Ω, A) enote a measurable space, an let M(Ω) enote the set of all probability measures on A. a) Show that efines a metric on M(Ω). (P, Q) = sup P (A) Q(A), P, Q M(Ω), A A b) Prove or isprove the continuity of the mapping ]0, [ M(R), λ π(λ), where π(λ) enotes the Poission istribution with parameter λ. c) Assume that Ω is a complete separable metric space an that A is the corresponing Borel σ-algebra. Characterize those spaces Ω, where convergence w.r.t. is equivalent to weak convergence. 4. Let Ω i = {0, 1}, A i = P(Ω i ), an P i = p ε 1 + (1 p) ε 0 for p ]0, 1[ an i N. Consier the corresponing prouct space (Ω, A, P ). Determine the istribution of the ranom variable X n : Ω {0,..., n} : ω {i {1,..., n} : ω i = 1}. Construct a ranom variable on (Ω, A, P ) that is geometrically istribute with parameter p. Construct ranom variables X an Y on (Ω, A, P ) that o not coincie almost surely but have the same istribution. Abgabe: , 10:00

5 Aufgabenblatt zur Vorlesung 1. Consier the probability space from Exercise 4.4 an efine a sequence of ranom variables, which take values in Z = {0,..., m 1}, given by X 0 (ω) = 0 an { X n 1 (ω) + 1 mo m if ω n = 1, X n (ω) = X n 1 (ω) 1 mo m if ω n = 0. Put p n,k = P ({X n = k}) for n N 0 an k Z. Derive a recursive formula for (p n ) n N0, an compute initial segments of this sequence numerically for ifferent values of m. Formulate conjectures concerning the convergence of (X n ) n N0 an (E(X n )) n N0. 2. Put I = ]0, 1[, an let B B(I k ) with λ k (B) > 0. The probability measure ν on B(I k ) that is efine by ν(a) = λ k (A B)/λ k (B) is calle the uniform istribution on B. Consier the simplex D = {(x, y) I 2 : x + y < 1}, an let K(x, ) enote the uniform istribution on the section D(x) for x I. a) Show that K is a Markov kernel from I to B(I). Use a ranom number generator to simulate the istribution µ K with ifferent choices of probability measures µ on B(I). In particular, take the uniform istribution µ on I. b) Determine a probability measure µ on B(I) such that µ K is the uniform istribution on D. Compute the expectation I 2 y (µ K)(x, y). 3. Consier a queue where, per time step, n new customers arrive with probability b n for n N 0, an, in case of a non-empty queue, the customer at the hea of the queue is serve an leaves. a) Choose an appropriate measurable space to moel the lengths of the queue at all times i N 0. Define the corresponing transition kernel. b) Suppose that initially the queue is empty. Derive a recursive formula for the probability of length k N 0 of the queue at time i N 0. Derive a formula for the probability of lengths (k 1,..., k i ) of the queue at times 1,..., i. 4. Let X 1 an X 2 be inepenent with P Xi = 1/2 (ε 1 + ε 1 ) for i = 1, 2. Verify that (X 1, X 2, X 1 X 2 ) is pairwise inepenent but not inepenent. Abgabe: , 10:00

6 Aufgabenblatt zur Vorlesung 1. Consier ranom variables X n, X, Y n, Y on a common probability space, an assume inepenence of (X n, Y n ) for every n as well as inepenence of (X, Y ). Prove Cf. Exercise 4.2.a). X n X Y n Y X n + Y n X + Y. 2. a) Show that X an Y are inepenent iff E(f X g Y ) 2 E(f X E(f X)) 2 for all Borel-measurable mappings f, g : R R with f X, g Y L 2. Interpretation in terms of preiction problems? b) Construct a probability space together with square-integrable ranom variables X an Y on this space such that X an Y are uncorrelate but not inepenent, an E(X g Y ) 2 > E(X E(X)) 2 for every measurable mapping g that is ifferent from the constant mapping E(X). 3. Verify the facts that are state in Remark III Let (X n ) n N be i.i.. with X 1 L 2. Put S n = n i=1 X i. Show that 1 n 1 n (X i S n /n) 2 P -a.s. Var(X 1 ), i=1 i.e., the sample variance converges almost surely to the population variance. (Use the Strong Law of Large Numbers.) Abgabe: , 10:00

7 Aufgabenblatt zur Vorlesung 1. Consier a set D B with 0 < λ (D) <, a square-integrable function f : D R, an a sequence (U n ) n N that is i.i.. with U 1 being uniformly istribute on D. Put a = D f(x) x, M n = λ (D) n n f U k, n = a M n. k=1 a) Show that E( n ) 2 λ (D) n D f 2 (x) x. b) Suppose you only know a constant c > 0 such that f(x) c for every x D. Determine an integer n 0 such that P ({ n 10 2}) 10 4 for every n n 0. c) Let α ]0, 1/2[. Show that with probability one. lim n nα n = 0 2. Consier the situation of Remark III Prove the Glivenko-Cantelli Theorem. Hint: For ε > 0, choose N N with N 1/ε an consier the istribution function F at the knots x N k = inf{x R : F (x) k/n} for k = 1,..., N 1. Cf. Exercise Consier an inepenent sequence (X n ) n N of ranom variables with P ({X n = 0}) = 1 1 n log(n + 1), P ({X n = ±n}) = Show that (X X n )/n oes not converge almost surely. Hint: Consier the event lim sup n { X n = n}. 1 2n log(n + 1). 4. Use characteristic functions to verify the following facts. a) The sum of inepenent, normally istribute ranom variables is normally istribute, too. b) For ranom variables X n, X, Y n, Y such that (X n, Y n ) is inepenent for every n an (X, Y ) is inepenent we have Cf. Aufgabe 6.1. X n X Y n Y X n + Y n X + Y. Abgabe: , 10:00

8 Aufgabenblatt zur Vorlesung 1. a) Let µ n = B(n, p n ) enote a binomial istribution with parameters n N an p n ]0, 1[. Assume that lim n n p n = λ > 0. Use Fourier transforms to show that w µ n π(λ), where π(λ) enotes a Poisson istribution with parameter λ. b) Consier probability measures µ an µ n on B k as well as probability measures ν an ν n on w w w B l such that µ n µ as well as ν n ν. Conclue that µ n ν n µ ν. 2. Implement a program to visualize the Central Limit Theorem. Which of the limit theorems are in action? 3. Construct an inepenent sequence of ranom variables X n such that (i) X n L 2 an E(X n ) = 0 for every n N, (ii) the istributions P S n converge weakly to N(0, 1), where S n = n i=1 X i/ ( n i=1 Var(X i)) 1/2, an (iii) the ranom variables X nk = X k / ( n i=1 Var(X i)) 1/2 are not asymptotically negligible. 4. Let X an Y be inepenent an ientically istribute ranom variables with 0 < Var(X) <. Moreover, let Z = (X + Y )/a where a > 0. Show that P Z = P X implies (i) a = 2, (ii) ( ϕ X (x/2 n ) ) 2 2n = ϕ X (x) for all x R an n N, (iii) X N(0, Var(X)). Abgabe: , 10:00

9 Aufgabenblatt zur Vorlesung 1. Consier a k-imensional ranom vector, whose istribution has a ensity with respect to the Lebesgue measure. State an prove a generalization of Theorem III.7.1 in this context. 2. Consier a normally istribute k-imensional ranom vector Y on (Ω, A, P ) with zero mean, which is not constant almost surely. Show that there exists an integer l k, a matrix L R k l of full rank, an an l-imensional stanar normally istribute ranom vector X on (Ω, A, P ) sucht that Y = LX. Note the ifference between this fact an the result that is establishe in the proof of Theorem III Let X = (X 1, X 2 ) be a two-imensional ranom vector, whose components X 1 an X 2 are stanar normally istribute. Prove or isprove that X is normally istribute. 4. Let X (j) = (X (j) t ) t [0, [ with j J enote stochastic processes on (Ω, A, P ) with state space (Ω, A ). Show that (X (j) ) j J is inepenent iff ( (j) (X t 1,..., X (j) t m ) ) j J 0 is inepenent for every finite set J 0 J an all m N an 0 t 1 < < t m. Abgabe: , 10:00

10 Aufgabenblatt zur Vorlesung 1. Let M = C([0, [) be equippe with the metric ρ that is introuce in Section IV.1, an let µ be the Wiener measure on B(M). Put Z(f) = 1 a) Show that Z : M R is B(M)-B-measurable. b) By 0 f(t) t, f M. ((t, f), (s, g)) = max( t s, ρ(f, g)) we efine a metric on [0, [ M. Show that (t, f) f(t) efines an B([0, [ M)-B measurable mapping [0, [ M R an that B([0, [ M) = B([0, [) B(M). c) Show that E(Z) = 0 an Var(Z) = 1/3. Hint: Fubini s Theorem. ) Determine the istribution of Z. Hint: Consier Riemann sums. 2. a) Let X = (X 1,..., X k ) enote a k-imensional normally istribute ranom vector such that Cov(X i, X j ) = 0 for 1 i n < j k. Show that (X 1,..., X n ) an (X n+1,..., X k ) are inepenent. b) Construct a two-imensional ranom vector X = (X 1, X 2 ) such that X 1 an X 2 are normally istribute an uncorrelate but not inepenent. In the sequel, X = (X t ) t [0, [ enotes a one-imensional Brownian motion on (Ω, A, P ). 3. Let T > 0 as well as Y t = X T +t X T. Show that Y = (Y t ) t [0, [ is a Brownian motion. Moreover, show that (X t ) t [0,T ] an Y are inepenent. 4. a) Let Z = sup 0 t T X t. Show that Z is A-B-measurable an P ({Z u}) T/u 2 for every u > 0. Hint: Kolmogorov s inequality. b) Show that lim t t α X t = 0 P -a.s. for every α > 1/2. Hint: Use a), Aufgabe 3, an the strong law of large numbers. c) Conclue that Y = (Y t ) t [0,T ] with Y t = t X 1/t for t > 0 an Y 0 = 0 is a Brownian motion. Abgabe: , 10:00

11 Aufgabenblatt zur Vorlesung 1. Let X = (X t ) t [0,T ] with T > 0 be a one-imensional Brownian motion. Moreover, efine Y t = t/t X T, Z t = X t Y t for t [0, T ]. a) Show that (Y t ) t [0,T ] an (Z t ) t [0,T ] are inepenent. b) Show that every finite-imensional marginal istribution of Z is a normal istribution as well as E(Z t ) = 0 an Cov(Z s, Z t ) = s (T t)/t for 0 s t T. c) Implement a program to simulate the istribution of (Y 0, Y 1/n,..., Y 1, Z 0, Z 1/n,..., Z 1 ) for any n N. For visualization use piecewise linear interpolation of U = (Y 0, Y 1/n,..., Y 1 ), V = (Z 0, Z 1/n,..., Z 1 ), as well as U + V. 2. Provie a proof of the Raon-Nikoym Theorem for two finite measures ν an µ such that ν µ by reucing this case to the statement that was establishe in the lecture. 3. Let (Ω, A, P ) = ([0, 1], B([0, 1]), λ) with λ enoting the Lebesgue measure. Put { X(ω) = 2ω 2 2ω, if ω < 1/2,, Y (ω) = 2ω 1, otherwise, for ω [0, 1]. Determine E(X Y ) an E(X Y = y) for y [0, 1]. 4. a) Let X an Y be i.i.. with E( X ) <. Determine E(X X + Y ). b) Consier an i.i.. sequence X 1, X 2,.... Determine where S k = k i=1 X i. E(X 1 σ({s n, S n+1,... })), Abgabe: , 10:00

12 Aufgabenblatt zur Vorlesung 1. In the sequel (X 0,..., X k ) enotes a (k + 1)-imensional normally istribute ranom vector. a) Assume that (X 1,..., X k ) is inepenent, an put I = {i {1,..., k} : σ 2 (X i ) > 0}. Show that E(X 0 (X 1,..., X k )) = Cov(X 0, X i )/σ 2 (X i ) (X i E(X i )) + E(X 0 ). i I Hint: Aufgabe b) Determine E(X 0 (X 1,..., X k )) without any further assumption on (X 1,..., X k ). c) Consier a one-imensional Brownian motion (Y t ) t [0, [. Determine E(Y t Z) for t 0, where Z = Y 1 or Z = 1 0 Y s s. 2. Consier the setting from Aufgabe At first, etermine the regular conitional istribution P X Y, an thereafter etermine the regular conitional istribution P i Y. 3. A point is picke uniformly at ranom from the surface of the unit sphere. With θ an φ enoting the longitue an latitue, respectively, etermine the regular conitional istribution of θ given φ an of φ given θ, respectively. 4. a) Consier a probability space (Ω, A, P ) with inepenent sub-σ-algebras G 1, G 2 A. Let Y : Ω R be G 1 -measurable an f : R Ω R be B G 2 -measurable an boune. Show that the mapping g : R R efine by g(y) = f(y, ) P for all y R is well-efine an that Ω E(f(Y ( ), ) G 1 ) = g Y. Hint: Algebraic inuction; at first consier suitable mappings of the form f(y, ω) = 1 A (y)1 B (ω). b) Let (W t ) t [0, [ be a Brownian motion, an let 0 s < t as well as Γ B. Use part a) to etermine P ({W t Γ} F W s ). Abgabe: , 10:00

13 Aufgabenblatt zur Vorlesung 1. Construct ranom variables X, Y L 2 on a common probability space an a measurable mapping ϕ : R R such that E( X ϕ Y ) < E( X E(X Y ) ). 2. Consier the situation of Example V.3.8. Moreover, we assume that (Y n ) n N is ientically istribute with P Y1 = 1/2 (ε 1 + ε 1 ). For α, β Z with α < 0 < β we set τ α = min{t N 0 : X t = α} an τ β = min{t N 0 : X t = β}. Show that P (τ α < ) = P (τ β < ) = 1, an P (τ α τ β ) = β β α an P (τ β τ α ) = α β α. 3. A martingale X is calle square-integrable if X t L 2 for all t I. Show that every square-integrable martingale has uncorrelate increments over isjoint intervals. 4. a) Let (Y s ) s N be an i.i.. sequence of ranom variables such that E( Y 1 ) <. Moreover, let F 0 = {, Ω} an F t = σ({y 1,..., Y t }) for t N. For any stopping time τ w.r.t. (F t ) t N we put τ Z = Y s. Show that Z L 1 an if E(τ) <. b) Consier the particular case s=1 E(Z) = E(Y 1 )E(τ), P Y1 = 1 2 (ε 1 + ε 1 ) an Determine E(τ). τ = inf{t N: t Y s = 1}. s=1 Abgabe: , 10:00