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Theory of Stochastic Processes

Lecture 13: Review (partially solved problems)

Tomonari SEI

July 13, 2017

Solutions to a part of recommended problems are given.

Lecture 8: Markov chain Monte Carlo

Problem 6.5.1 By induction, we obtain

i =0 i11 i

0.

Thenipi,i+1 =

0 i1 i

0 = i+1pi+1,i, i = 0, . . . , b 1.

For other pairs (i, j), i 6= j, ipij = 0 = jpji. Therefore the chain is reversible.

Problem 6.5.8 Suppose P is reversible. Let be the stationary distribution and D be thediagonal matrix with diagonal entries . Then S = DP is a symmetric matrix by definition

of reversibility. We obtain P = D1 S.

Conversely, suppose P = DS = (diSij). Define i = (1/di)/

k(1/dk). Then ipij =

idiSij = Sij/

k(1/dk) is symmetric. Therefore P is reversible.

Next suppose P = DS. Let T = D1/2PD1/2. Then T has the same eigenvalues as P .

Since T = D1/2SD1/2 is symmetric, the eigenvalues are real.

Finally, consider a transition matrix

P =

13 13 + 13 13

13

13

13

13

13 +

, ( 13 , 13 ).The stationary distribution is = ( 13 ,

13 ,

13 ). The chain is not reversible unless = 0. On the

other hand, P has the real eigenvalues 1, , 0.

Lecture 13: Review

Lecture 9: Stationary processes

Problem 2 The process is represented as

Xn = g(L)Zn, g(z) = (1 1z pzp)1.

Since all the roots of 1 1z pzp = 0 are outside of the unit circle, g(z) has theconvergence radius greater than 1. Then the spectral density is given by

f() =12

|g(ei)|2 = 12|1

pj=1 ie

ij|2.

by Theorem 2 of the lecture note of Lecture 9.

Problem 4 (i) Since A and B have the mean zero, the mean of Xn is

E[Xn] = E[A cos(n) + B sin(n)] = 0.

Since E[A2] = E[B2] = 1 and E[AB] = 0, the covariance of Xn and Xn+m is

E[XnXn+m] = E[cos(n) cos((n + m))] + E[sin(n) sin((n + m))]

=1

0

(cos(n) cos((n + m)) + sin(n) sin((n + m))) d

=1

0

cos(m)d =

{1 if m = 0,0 otherwise.

Therefore Xn is the white noise. In particular, it is stationary.

(ii) We have X0 = A, X1 + X1 = 2A cos(), and X1 X1 = 2B sin(). The first twoequations determine A and . If 6= 0 and 6= , then the third equation determines B andtherefore all Xn are determined. Even if = 0 or = , Xn = A cos(n) is determined.

Problem 5 For real numbers v1, . . . , vN ,N

j=1

Nk=1

vjvkc(j k) =N

j=1

Nk=1

vjvkE[XjXk] = E[(N

j=1 vjXj

)2] 0.

Hence the matrix C = {c(j k)} is non-negative definite*1. All the eigenvectors of C aregiven by vm = (N1/2e2imn/N )Nn=1 for m = 0, . . . , N 1. Indeed,

(Cvm)k = N1/2N

n=1

c(k n)e2imn/N

= N1/2e2imk/NN

j=1

c(j)e2imj/N , j = n k,

= Nf(m)(vm)k,

*1 The assumption c(n) = c(n N) is not necessary here.

2

Lecture 13: Review

where f(m) = N1N1

n=0 c(n)e2imn/N . The spectral decomposition of C is C =N1

m=0 Nf(m)vmvm and therefore

c(j k) =N1m=0

Nf(m)(vm)j(vm)k =N1m=0

f(m)e2im(jk)/N .

Problem 6 (i) It is easy to see that E[X] = N1

n E[Xn] = . Furthermore,

V [X] = E[(X )2] = N2N

j=1

Nk=1

E[(Xj )(Xk )]

= N2N

j=1

Nk=1

2(j k)

= N2N1

n=(N1)

(j,k):jk=n

2(n)

= 2N1N1

n=(N1)

(1 |n|/N)(n).

(ii)

E[2] = N1N

n=1

E[(Xn X)2] = N1N

n=1

E[((Xn ) (X ))2]

= N1N

n=1

{E[(Xn )2] 2E[(Xn )(X )] + E[(X )2]

}= N1

{N2 2NE[(X )2] + NE[(X )2]

}= 2 V [X].

(iii) By (i) and Abels theorem (or monotone convergence theorem), we have

limN

NV [X] = limN

N1n=(N1)

(1 |n|/N)(n) =

n=(n) = f(0).

In particular, limN V [X] = 0. By (ii), we also have

limN

E[2] = limN

(2 V [X]) = 2.

3

Lecture 13: Review

Lecture 10: Martingales

Problem 12.1.3 E[Zn+1n1|Fn] = E[Zn+1|Zn]n1 = (Zn)n1 = Znn.E[Zn+1 |Fn] = E[Zn+1 |Zn] = G()Zn = Zn , where G is the generating function of Z1

Problem 12.1.5 E[(Yk Yj)Yj ] = E[E[Yk Yj |Fj ]Yj ] = E[(Yj Yj)Yj ] = 0.E[(Yk Yj)2|Fi] = E[Y 2k |Fi] 2E[E[Yk|Fj ]Yj |Fi] + E[Y 2j |Fi] = E[Y 2k |Fi] E[Y 2j |Fi].Suppose that E[Y 2n ] K for all n. Since E[(Yn Ym)2] = E[Y 2n ] E[Y 2m] for m < n, E[Y 2n ]is non-decreasing. Therefore there exists limn E[Y 2n ] K. In particular, for any > 0,there exists n0 such that |E[Y 2n ] E[Y 2m]| < for all n,m n0. Then E[(Yn Ym)2] =E[Y 2n ] E[Y 2m] < . Therefore Yn is a Cauchy sequence and converges in mean square*2.

Problem 12.1.6 By Jensens inequality, we have

E[u(Yn+1)|Fn] u(E[Yn+1|Fn]) = u(Yn).

This means u(Yn) is a submartingale. The processes |Yn|, Y 2n , Y +n are submartingales sincefunctions u(y) = |y|, y2, y+ = max(0, y) are convex, respectively.

Problem 12.2.1 We apply McDiarmids inequality. Let f((v1, w1), . . . , (vn, wn)) be the max-imum worth when the volume and worth of the i-th object are vi and wi. If an object (vi, wi)

is replaced with (vi, wi), then f changes at most M . Indeed, after the i-th object is removed if

necessary, the total cost is reduced at most M while the packed objects are feasible. Therefore

f((v1, w1), . . . , (vi, wi), . . . , (vn, wn)) M f((v1, w1), . . . , (vi, wi), . . . , (vn, wn))

and

f((v1, w1), . . . , (vi, wi), . . . , (vn, wn)) M f((v1, w1), . . . , (vi, wi), . . . , (vn, wn)).

Now, McDiarmids inequality implies that

P (|Z E[Z]| nM) 2 exp(n2/2), > 0.

Let x = nM to obtain

P (|Z E[Z]| x) 2 exp(x2/(2nM2)), x > 0.

*2 Here we used the completeness property of L2. This was beyond the scope of this lecture..

4

Lecture 13: Review

Lecture 11: Queues

Problem 8.4.1 (a) Let X1, X2, X3 be the service time of the two customers and you. Denotethe teller you chose by Z = 1, 2. Denote the exponential density function by f(x) = ex.

Then the probability we want is

p =12P (X1 + X3 > X2|Z = 1) +

12P (X1 < X2 + X3|Z = 2)

=12

x1+x3>x2

f(x1)f(x2)f(x3)dx1dx2dx3 +12

x1 X2) + P (X1 > X2, X1 < X2 + X3)

=

( + )2+

( + )2.

In particular, p(a) > p(b) > p(c).

Problem 11.2.3 The waiting time W of a customer arriving at time 0 is the sum of servicetime of the Q(0) people. Let X1, . . . , Xn be the service time of the Q(0) = n customers. Since

Q(0) has the stationary distribution, we have

P (W > x) =

n=1

(1 )nP (X1 + + Xn > x), x 0.

Since Xi has the exponential distribution with parameter , the distribution of X1 + +Xnis the gamma distribution. Therefore

P (X1 + + Xn > x) =

x

ntn1

(n 1)!etdt.

5

Lecture 13: Review

and

P (W > x) =

x

n=1

(1 )n ntn1et

(n 1)!dt

=

x

(1 )

( n=1

(t)n1

(n 1)!

)etdt, = ,

= (1 )

x

e()tdt

= (1 )e()x

= e()x.

We have P (W x) = 1 e()x. In particular, P (W = 0) = P (W 0) = 1 .

Problem 11.3.1 The problem is to determine the mean of Q(D). By Theorem 13.3.5 ofPRP, the generating function of Q(D) is given by

G(s) = (1 )(s 1) MS((s 1))s MS((s 1))

,

where MS() is the moment generating function of a typical service time. In this problem,

the service time is the constant d. Therefore MS() = ed, and

G(s) = (1 )(s 1) e(s1)d

s e(s1)d= (1 )(s 1) e

(s1)

s e(s1),

where = d is used. The mean length of Q(D) is G(1). By Taylors expansion around

s = 1, we obtain

G(1 + r) = (1 )r 1 + r + O(r2)

1 + r 1 r 122r2 + O(r3)

= (1 ) 1 + r + O(r2)

1 122r + O(r2)

= 1 +(

+12

2

1

)r + O(r2)

= 1 +12

(2 )1

r + O(r2).

Hence G(1) = 12(2 )/(1 ).

6

Lecture 13: Review

Lecture 12

Problem 2 Typo: the definition of Z should be Z =N

i=1 bi1(Wti Wti1).Denote Z = ZN and Ft = {Ws}st. By induction,

E[ZN ] = E[ZN1 + bN1(WtN WtN1)]= E[Zn1] + E[bN1E[WtN WtN1 |FtN1 ]]= E[Zn1]

= E[Z0] = 0.

Similarly,

V [ZN ] = E[(ZN1 + bN1(WtN WtN1))2]= E[E[(ZN1 + bN1(WtN WtN1))2|FtN1 ]]= E[Z2N1 + b

2N1(tN tN1)]

=N

i=1

E[b2i1](ti ti1).

Problem 4 The OU process is represented as

Xt = et(

X0 + t

0

esdWs

).

Thus, given X0 = x, the mean is E[Xt] = etx 0 as t , and the variance is

V [Xt] = E[(Xt etx)2]

= 2e2tE

[( t0

esdWs

)2]

= 2e2t t

0

e2sds (Ito isometry)

= 21 e2t

2

2

2(t ).

Thus N(0, 2/(2)) must be the stationary distribution. Indeed, if X0 N(0, 2/(2)),then Xt = et(X0 +

t0

esdWs) is the sum of independent random variables having

N(0, e2t2/(2)) and N(0, 2(1 e2t)/(2)). Therefore Xt N(0, 2/(2)).

Problem 5 (a) d(W 3t ) = 3W 2t dWt + 3Wtdt. Thus W 3t = 3 t0

W 2s dWs + 3 t0

Wsds.

(b) dF (Wt) = f(Wt)dWt + (1/2)f (Wt)dt. Thus F (Wt) = t0

f(Ws)dWs + (1/2) t0

f (Ws)ds.

7

Lecture 13: Review

Problem 7 For any f : Rd R, we have

E[df(Xt)] = E

i

(if)dXi(t) + (1/2)i,j

(ijf)(dXi(t))(dXj(t))

= E

i

(if)i + (1/2)i,j

(ijf)

a

iaja

dtBy using E[f(Xt)] =

f(y)p(t, y)dy and the integral-by-parts formula, we have

f(y)tpdy =

f(y)

i

(ip) + (1/2)i,j

ij{

a

iajap}

dy.Since f is arbitrary, the result follows.

Problem 9 The drift term is

=12g

(log

g

)+

12

g

(1

g

)=

12g

(

g

2g

) g

4g2=

2g g

2g2

The diffusion term is = 1/

g. Then the right hand side of the forward equation is

() + 12(2) =

(

2g g

2g2

)+

12

(

g

)=

(

2g g

2g2

)+

12

(

g g

g2

)= 0.

8