Download Problem set 4

Aalto University
Department of Mathematics and Systems Analysis
Exercise set 4
K Kytölä & A Karrila
MS-E1602 Large Random Systems, 2017/IV
Exercise session: Tue 14.3. at 12-14
Solutions due: Mon 20.3. at 10
Exercise 1. Let n ∈ N, and let C ∈ Rn×n be a symmetric1 and positive definite2,
matrix and let m ∈ Rn be a vector. Define a function p : Rn → R by
1
1
>
−1
p(x) = exp − (x − m) C (x − m) ,
Z
2
where Z is a constant.
R
(a) Calculate Rn p(x) dn x, and show that p is a (correctly normalized) probability
p
density on Rn if Z = (2π)n/2 det(C).
Hint: Do first a change of variables (translation) to reduce to the case m = 0. Then do an orthogonal
change of variables to a basis in which C is diagonal.
(b) Choose Z as in part (a), and suppose that ξ = (ξ1 , ξ2 , . . . , ξn ) is a random
3
vector in Rn , which has probability
p : Rn → R as above.
i θ·ξdensity
P Calculate the
, for θ ∈ Rn where θ · ξ = nj=1 θj ξj denotes
characteristic function ϕ(θ) = E e
the inner product.
(c) Let ξ be the random
vector as in (b), and let a1 , . . . , an ∈ R. Show that the
P
linear combination nj=1 aj ξj is a random number with Gaussian distribution.
Exercise 2. Suppose that B = (Bt )t≥0 is a standard Brownian motion and define
(k)
three other stochastic processes W (k) = (Wt )t≥0 , k = 1, 2, 3, by setting
(1)
= Bs+t − Bs
(2)
= λ−1/2 Bλt
(
tB1/t , when t > 0
=
0
, when t = 0
Wt
Wt
(3)
Wt
where s ≥ 0 and λ > 0 are constants. Show that all these three stochastic processes
(k)
W (k) = (Wt )t≥0 , k = 1, 2, 3, are also standard Brownian motions.
Hint: You may use results about existence and uniqueness of a standard Brownian motion.
Exercise 3. Suppose that B = (Bt )t≥0 is a standard Brownian motion. Define a
stochastic process X = (Xt )t∈[0,1] by setting Xt = Bt − tB1 .
(a) Show that X is a Gaussian process. Calculate the mean function and the covariance function of X, i.e., t 7→ E[Xt ] and (s, t) 7→ Cov[Xs , Xt ] = E[Xs Xt ]−E[Xs ] E[Xt ].
(b) Show that X and B1 are independent.
Hint: The distribution of X is determined by the finite dimensional marginals P[Xt1 ∈ A1 , . . . , Xtn ∈ An ],
for 0 ≤ t1 < · · · < tn ≤ 1 and A1 , . . . , An ⊂ R Borel. Consider also using the Gaussianity of the processes
(recall: Gaussians are independent iff they are uncorrelated).
1Symmetric
means Cij = Cji for all i, j = 1, . . . , n.
Positive definite means that for
vector v ∈ Rn \ {0} we have v > Cv > 0.
R any non-zero
3This means that P[ξ ∈ E] =
n
p(x) d x for all Borel sets E ⊂ Rn .
E
2
Exercise 4. Suppose that X = (Xt )t∈[0,1] is a continuous and Gaussian process, for
which E[Xt ] = 0 for all t ∈ [0, 1] and Cov[Xs , Xt ] = s(1 − t) for all 0 ≤ s ≤ t ≤ 1. Let
Y ∼ N (0, 1) be a random variable independent of X. Define a stochastic process
W = (Wt )t∈[0,1] by setting Wt = Xt + tY .
(a) Show that W has the same finite dimensional distributions as a standard
Brownian motion on the time interval [0, 1].
The event {W1 = y} has zero probability, but it is natural to define conditioning on this event by the following limiting procedure. For any ε > 0, the event
{|W1 − y| < ε} has positive probability, so we can consider the conditional distribution of the process W given the event {|W1 − y| < ε}, and then take the (weak)
limit of this conditional distribution as ε & 0.
(b) Show that the conditional distribution of the process W given {W1 = y} is
Gaussian, and calculate their mean and covariance functions.
Note: Interpret the distribution of the process as the collection of its finite dimensional distributions, and conditioning as defined by the limiting procedure above.
Exercise 5. Let B = (Bt )t∈[0,∞) be a standard Brownian motion. For t ≥ e define
p
λ(t) = t log(log(t)).
√
|Bn |
= 2, where the lim sup is taken
(a) Show that almost surely lim supn→∞ λ(n)
along n ∈ N.
Hint: Observe that the restriction to Brownian motion to integer times, (Bn )n∈N , is a random
walk with Gaussian steps. Recall the law of iterated logarithm proven earlier.
(b) Find a function λ̃ : N → [0, ∞) with the following properties:
λ̃(n)
λ(n)
→ 0 as
n → ∞, and almost surely maxs∈[n,n+1) |Bs − Bn | ≤ λ̃(n) except for finitely
many values of n ∈ N.
Hint: You may use the fact that P maxs∈[0,1] Bs > r =
√2
2π
R∞
r
1
2
e− 2 v dv for r ≥ 0.
(c) Show that almost surely we have (with lim sup taken along t ∈ [0, ∞))
|Bt | √
lim sup
= 2.
t→∞ λ(t)
(d) Show that almost surely
√
|Bt |
lim sup q
=
2.
t→0
t log | log(1/t)|
Hint: Use (c) and the last part of Exercise 2 above.
Exercise 6. Let S be a finite or countably infinite set. Define % : S N × S N → [0, ∞)
by the formula4
X
%(ω, ω 0 ) =
2−i
for ω, ω 0 ∈ S N , ω = (ωi )i∈N , ω 0 = (ωi0 )i∈N .
i∈N
ωi 6=ωi0
(a)
(b)
(c)
(d)
4The
Show that % is a metric on the set S N .
Show that the projections πi : S N → S defined by πi (ω) = ωi , are continuous.
Show that (S N , %) is complete.
Show that (S N , %) is separable.
notation S N means the set of all functions N → S, i.e., of all sequences ω = (ωi )i∈N such
that ωi ∈ S ∀i ∈ N.