Multifractal products of stochastic processes - Semantic Scholar

Multifractal products of stochastic processes: construction and some basic properties Petteri Mannersaloy VTT Information Technology P.O. Box 1202 FIN-02044 VTT Finland Tel. +358 9 456 5927 Fax. +358 9 456 7013 Email: [email protected]

Ilkka Norros VTT Information Technology P.O. Box 1202 FIN-02044 VTT Finland Tel. +358 9 456 5627 Fax. +358 9 456 7013 Email: [email protected]

Rudolf H. Riedi ECE Dept, DSPgroup Rice University MS 380 Houston TX 77251-1892 Tel: +1 713 348 3020 Fax: +1 713 348 6196 Email: [email protected]

Submitted to Applied Probability, August 2001

Manuscript correspondence y This work was supported by Academy of Finland, project 42535 1

2

P. Mannersalo, I. Norros, R. Riedi: Multifractal products of stochastic processes Abstract In various fields, such as teletraffic and economics, measured times series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a non-stationary process. To overcome this problem we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study

L2 -convergence, non-degeneracy and

continuity of the limit process. Establishing a power law for its moments we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.

Keywords: Stochastic processes, random measures, multifractals, teletraffic modeling. Classification (MSC2000): primary 60G57 (random measures), secondary 60G30 (continuity and singularity of induced measures).

3

submitted Applied Probability, August 2001

1 Introduction This study is strongly motivated by the search of new models for teletraffic. In various recent papers (see e.g. [RLV97, LVR97, FGW98]), it has been demonstrated that teletraffic often exhibits multifractal properties. There are many ways to construct random multifractal measures varying from the simple binomial measures to measures generated by branching processes (see e.g., [Man72, Man74, Fal94, AP96, Pat97, RCRB99]). In teletraffic modeling, we would like to have, in addition to a simple and causal construction, also stationarity of the increments. Unfortunately, most of the ‘classical’ multifractal models, in particular tree-based cascades, lack both of these properties. It should be noted that Jaffard has discovered the first multifractal with stationary increments, i.e., the Lévy processes [Jaf96]. However, the fact that the increments of these processes are actually independent severely limits their use as models for teletraffic. Moreover, Lévy processes have a linear multifractal spectrum while real data traffic exhibits strictly concave spectra [RLV97, LVR97, MN97, RCRB99]. There have been many studies on both deterministic and random cascade measures (see e.g. [Man74, HW92, Mol96, CM99]). In this paper, we generalize the cascade construction in a natural and stationary way. This kind of scheme was first studied by Kahane [Kah87] in a very general setup. In its simplest form our model is based on d bi which the multiplication of independent rescaled stochastic processes (i) d are piecewise constant (here denotes equivalence in distributions). It is instructive to compare it to a Fourier decomposition where one represents or constructs a process by superposition of oscillations i t .

( ) = ( )

=

sin( )

In multiplying rather than adding rescaled versions of a ‘mother’ process we obtain a process with novel properties which are best understood not in an additive analysis, but in a multiplicative one. Processes emerging from multiplicative construction schemes can easily be forced to have positive increments and exhibit typically a ‘spiky’ appearance. The so-called multifractal analysis describes the local structure of a process in terms of scaling exponents, accounting for (being adapted to) the multiplicative structure. It is tempting to view the multiplicative construction as an additive one — which opens the possibility to use linear theory — followed by an exponential. Such an approach, however, obscures what happens in the limit. As with the cascades, an infinite product of random processes will typically (almost surely at almost all times) be zero; equally, its logarithm tends to negative infinity. A non-degenerate limiting behavior can be observed for the product, though, by taking a distributional limit rather than pointwise limit. In simpler words, a multiplier (i) t should not be evaluated in points but should be seen as redistributing or re-partitioning mass. In the words of teletraffic modeling, (i) t can be thought of as a local change in the arrival rate where one is interested actually in the integrated ‘total load’ process. Consequently,

()

()

4

P. Mannersalo, I. Norros, R. Riedi: Multifractal products of stochastic processes

we will study

An (t) =

Z tY n

0 i=0

(i) (s) ds;

which converges to a well defined, non-degenerate and continuous process under suitable conditions. The paper is structured as follows: We start by studying the construction of multifractal measures based on iterative multiplication of stochastic processes as in An above, in particular convergence and non-degeneracy. Then, we consider a special case where the multipliers are independent rescaled versions of some mother process, looking at continuity, power laws of moments as well as LRD of the limiting process. Finally, we provide an application-friendly family based on piecewise constant, causal processes with exponentially distributed sojourn times as an example.

2 Multifractal products of stochastic processes In order to keep the presentation simple, we only consider 1-dimensional processes on the closed unit interval ; ]. Extensions to the real line R as well as to higher dimensions are not too difficult. For example, the 2-dimensional case is studied in [MN01]

[0 1

2.1 Construction

(t)gt2T with E(i) (t) = 1 8t 2 T; i = 0; 1; 2 : : : :

Let us consider a family of positive processes f (i)

Later, when studying particular properties of the process, we will usually assume that the (i) are stationary, but we do not require stationarity for the construction itself. Define the product processes

n(t) =:

n Y i=0

(i)(t)

and the corresponding cumulative processes

An (t) =:

Z

t

0

n(s) ds =

Z tY n

0 i=0

(i) (s) ds;

n = 0; 1; : : :

5


Instead of processes An , it is sometimes easier to study the equivalent positive measures defined on the Borel sets B of T :

n (B ) =:

Z

B

n(s) ds;

n = 0; 1; : : : ;

E( ( )) = 1 lim

B 2 B:

(t) = 0 for almost all t, almost

Notice that even though , n t n!1 n surely. This is seen by taking logarithm of the product,

log n(t) =

n X i=0

log (i) (t);

and interpreting it as a random walk. Because of the negative drift 1 at almost all t, this random walk will end up in minus infinity almost surely. This means that the possibly existing limit process of the cumulative processes is singular. However, it may be non-zero, and the convergence may happen in L1 .

2.2 Convergence Recall the following basic properties of discrete martingales (see e.g. [Wil91]).

(Mn : n 2 N ) be a supermartingale bounded in L1: supn E(jMn j) < 1. Then, almost surely, M = limn!1 Mn exists and is finite. Lemma 1. Let

Lemma 2. Let

(Mn : n 2 N ) be a martingale for which Mn 2 L2 for all n. Then it is

bounded in L2 if and only if X

and when this obtains Mn

E (Mk

Mk

1

)2 < 1;

! M almost surely and in L2.

( ) = lim

(1)

()

Lemma 1 implies immediately: For any fixed t, A t n!1 An t exists and is finite, almost surely. Considering the corresponding random measures, we see that An ! A weakly a.s. and that the almost sure convergence holds for any countable set of points in T . 1

By Jensen’s inequality we have

log(1) = 0.

E T [log (t)℄ < log E T [(t)℄ = log E

R

T

(t) dt =

6


Theorem 3. [Kah87] The random measures

n converge weakly a.s. to a random

measure . Moreover, given a finite or countable family of Borel sets

Bj on T , we

have with probability one:

8j (Bj ) = nlim (B ): !1 n j The a.s. convergence in countably many points at the same time can be extended to all points in T if we know that the limit process A is almost surely continuous. Conditions for continuity in a more specific setup are given later in Proposition 10. Corollary 4. If An

! A weakly a.s., and An and A are continuous almost surely, then

with probability one:

8t 2 T A(t) = limn!1 An(t):

~

Proof. For an arbitrary countable dense set T in T (e.g. rational numbers),

sup jAn(t) t2T

A(t)j

sup ~inf~ jAn(t~) A(t~)j + jAn(t) An(t~)j + jA(t) A(t~)j

t2T t2T

sup jAn(t~) A(t~)j + sup ~inf~ jAn(t) An(t~)j + jA(t) A(t~)j

t~2T~

t2T t2T

t~2T~

t2T

sup jAn(t~) A(t~)j + sup(jAn(t)j + jA(t)j);

where

A(t) denotes size of the jump at point t. Applying Theorem 3 and a.s. conti-

nuity of An and A completes the proof. Unfortunately, theorem 3 does not say anything about the L1 convergence; neither does lemma 1. In our setup, either of the following two cases is met: either An t ! A t in L1 for each given t or An converges to almost surely. The cases are called non-degenerate and degenerate, respectively. The question of non-degeneracy in a very general setup is studied by Kahane [Kah87]. He considers random measures R B n!1 B n for arbitrary Radon measures and positive martingales n . In our work, is the Lebesgue measure and our interests focus on the local scaling structure of the limiting process . In this paper, we study mostly the L2 -convergence which is the easiest to handle.

()

( ) = lim

(1)

d

0

()

7


Proposition 5. Suppose that the stationary processes (i) , i

= 0; 1; : : : , satisfy

E(i) (t) = 1; 8t 2 T; Var(i) (t) = Cov((i) (t1); (i) (t2)) =

2 < 1; 2 i (t1

8t 2 T; t2 ) ; 8t1 ; t2 2 T;

(2) (3) (4)

= 0; 1; : : : , are the normalized covariance functions with i (0) = 1. Then (An(1) : n 2 N ) is bounded in L2 if and only if where i , i

1 X n=0

R

an (1) < 1;

= 0t (t s)n(s) Qin=01(1 + 2i (s)) ds. holds true then An (t) ! A(t) in L2 for all t 2 [0; 1℄.

where

an (t)

(5)

Furthermore, if condition (5)

Note: Similarly as in Theorem 3, we could prove the simultaneous convergence for points in a countable set. In order to have L2 convergence for all points in T at the same time, we need extra conditions, like continuity of A.

Proof. By Fubini,

fAn(1)g is a martingale with respect to f(Fn; P)g, where Fn =

((0) ; (1) ; : : : ; (n) ) (an increasing sequence of -algebras). Since An (1) 2 L2 for any fixed n, lemma 2 applies and it is enough to study the criterium (1).

8


By the definition of An and assumptions (2)–(4)

E (An(1) = E = =

An

Z 1Z 1

0

Z 1Z 1

0

0

0

Z 1Z 1

0

1

(1))2

(n) (s1) 1 (n)(s2) 1 n 1(s1) n 1(s2) ds1 ds2

Cov (n) (s1); (n)(s2 E (n 1(s1)n 1(s2)) ds1 ds2

2

n

0

Z 1

= 22 (1 0

Thus, by Lemma 2

(js1

s2 j)

s)n (s)

n Y1

i=0 n Y1 i=0

(1 + 2i (js1

s2 j) ds1 ds2

(1 + 2 i(s)) ds:

(An(1) : n 0) is bounded in L2 if and only if

1 Z 1 X n=0

0

(1

x)n (x)

n Y1 i=0

(1 + 2 i(x)) dx < 1:

Since A is a positive, nondecreasing process, the above condition is sufficient for An being bounded in L2 for all t 2

(t)

[0; 1℄.

Explicit knowledge of the decay rates of the covariance functions simplifies the

L2-condition considerably:

Corollary 6. If there are positive constants , , b and C such that

exp( jbi sj) i (s) jCbi sj for all s 2

;

(6)

[0; 1℄, then (An(t) : n 0) is bounded in L2 if and only if b > 1 + 2:

The particular form of (6) is motivated by the processes we study in Section 3.

(7)

9


Proof. Sufficiency: Because i

(s) > 0 (i.e., (i) are positively correlated),

an (t)

Z

t

0

n (s)

Without losing generality, we can set

t

n Y1 i=0

(1 + 2 i(s)) ds:

= 1 and assume that C = 1, 2 (0; 1) and

1 + 2 < b < (1 + 2 )1=1 . First, we split the integration into parts and use the fact that i (s) 1 for all s. Let i (s) = minf1; jbi sj g and take an arbitrary m < n, then

an (1) =

Z

Z 1

0

n (s)

b n+m

0

n Y1 i=0

(1 + 2i (s)) ds

(1 + ) ds + 2

(

n

n 1 X i=m

(1 + )

2 n+m

i

Z

b n+i+1 b n+i

n (s)

n Y1

(1 + 2j (s)) ds

j =n+m i

1 + 2 n i(b n+m+i ) i n+m 2 n + (1 + ) b ds n (s) 2 1 + + + b i=0 ) ( nX m 1 Z b + + +1 2 b m i 1 + ds = (1 + 2)n b n+m + b n s 2 1 + + + b i=0 ( ) m 1 1 1)b(1 )m n X 2 b m i ( b 1 + = (1 + 2)nb n bm + b1 ; 2 1 1 + i=0 nX m 1Z

b n+m+i+1

)

n m i

n m i

n m i

(s) is decreasing with respect to both, s b 2 < 1g. Then and i. Next, let m = minfk 2 N : b1 1+1+ 2 where the second inequality holds because i k

an (1) where C

C (b; m)b n (1 + 2)n;

(b; m) < 1 does not depend on n. Thus P an(1) < 1.

10


> 0, be arbitrary. Then

Necessity: Let Æ , t > Æ

an (t)

Z

t Æ

(1 + 2 e b s)n ds 0 (1 + 2 e b (t Æ) )n+1 (1 + 2 )n (1 + 2 )Æ 1 = (1 + 2)n+1 (n + 1)bn 2 + 2 )n ; (1 +2 )Æ (1 (n + 1)bn

Æ

e

bn s

n

n

if n is large enough. Thus

P

a~n (t) diverges if b 1 + 2 .

Notice that the condition b >

1 + 2 is the same that appears in [KP76].

3 Self-similar Products 3.1 Invariance and Convergence

()

The analysis of the limiting process A t simplifies greatly provided that the multipliers (i) are connected through a rescaling property. More precisely, throughout this section we assume that the processes (i) are independent rescaled versions of some stationary mother process , i.e.,

(i) (t) =d (bi t); where b >

(8)

1 and E = 1. Two results follow straight from this rescaling property:

Proposition 7. Assume (8) with the mother process

(t) 2 L2 and that the scaled covariance function of

satisfies exp( jxj) (x) jCxj

for some positive numbers

1 + 2, where 2 = Var.

and . Then, An (t) converges in L2 if and only if b >

11


Proposition 8. Assume (8) and that

An (t) converges in L1 . Then, the limit process

A1 satisfies the recursion Z t 1 A(t) = (s) dA~(bs);

b

(9)

0

where

(i) the processes

and A~ are independent, and ~

(ii) the processes A and A are equally distributed. To relate to the classical cascades, let us note that ‘Mandelbrot’s martingale’ [KP76] can formally be written exactly as (8) with the only difference that has to be chosen non-stationary — it is constant over the intervals k=b; k =b (k integer) — and the values k=b form a sequence of i.i.d. random variables of mean . In this case, (9) may be reduced to

[

( )

( +1)

A(1=b) = 1b (0)A~(1);

(0)

~(1)

1

(10)

~(1) (( 1) ) = (1 )(( 1) )( ~( ) ~( 1))

where the random variables and A are independent, and A is equally distributed as A . =b =b k =b A k A k Generalizing (10) to A k=b A k and summing over k ; : : : ; b yields the invariance2 or recursion [KP76, equation 3], from which Kahane and Peyriere derive all their results. Their theorem 1 establishes four equivalent conditions for non-degeneracy of the limiting measure. The counterpart to the equivalence of conditions (), ( ) and ( ) of their theorem 1 is easy to establish using similar arguments as theirs (see Proposition 9 below). The main result of their theorem 1, however, the non-degenaracy condition ( Æ ) in terms of the moments of , remains an open problem in our case. At the moment, we can only state a sufficient and necessary condition for L2 convergence, given in Proposition 7, which is sufficient for non-degeneracy.

(1)

=1

( )

Proposition 9. Assuming (8), the following are equivalent:

() the a.s. martingale limit A(1) satisfies EA(1) = 1; 2

For cascades, the random variables A~(k )

A~(k 1) are independent.

12


( ) the a.s. martingale limit A(1) satisfies EA(1) > 0; ( )

the equation (9) has a solution A such that

EA(1) = 1.

~

Proof. Assume ( ). Then A can again be written as Z t 1 ~ (s) dA~ (bs); A~(t) =

b

where

0

~ is independent from , etc. Denote (0) = , (1) = ~ and so on. Then E(A(1)j

(0) ; : : :

;

(n)

)=

Z 1Y n

0 i=0

(i)(bi s) ds:

This martingale is uniformly integrable — but it is the same (in distribution at least) as

An (1)! Thus, holds. The remaining implications ) ) are obvious.

3.2 Continuity The following proposition gives sufficient conditions for the continuity of the limit process. Note that, in the random cascade case [KP76], the non-degeneracy is equivalent with the condition

E log < log b.

Proposition 10. If, in addition to (8), there exists a non-trivial integrable limit

limn!1 An, E log < log b, and EA(1) log A(1) < 1, then A is continuous. Proof. Denote by B the pure jump part of A:

B (t) =

X st

A(s):

A

=

13


By Proposition 9, (9) holds. Since no “new” jumps can be created in the integration, (9) holds for B as well. Denote g

E

X s1

g (B (s)) = E

= bE = bE

X s1

g

X s1

= E

X

= E

X

s1

s1

(x) = x log x. Now,

X

g

s1

1 (s)B~ (bs) b

1 (s=b)B~ (s) b

1 (s=b)B (s)(log(s=b) + log B (s) ) b

b

(s=b)B (s) log Bb(s) + E ((s=b) log(s=b))B (s) X

g (B (s))

s1

log(b)EB (1) + E( log)EB (1):

where the second equality uses the fact that

B has stationary increments. Since g is

superadditive,

E

X s1

g (B (s)) Eg (B (1)) Eg (A(1)) < 1:

log(b)EB (1) = E( log)EB (1). Using EB (1) EA(1) < 1 and E log < log b, we find EB (1) = 0 and B 0 a.s. It follows that

3.3 Scaling of moments

Next we consider general moments. Assuming that the possible jumps of behave q nicely enough, we can show that A t q tq logb . As straightforward corollaries, this gives almost the same necessary conditions for non-degeneracy and boundedness in Lp as [KP76].

E ()

Proposition 11. Assume (9) and that

A is non-degenerate. Let q >

0 be such that

14


A(1) 2 Lq , and assume that 1 X n=0

1 X

n=n

log(1 + (q; b n)) < 1

log(1

(q; b

n

(11)

)) > 1

(12)

(q; t) =: E sups2[0;t℄ j(0)q (s)q j and n = minfn : (q; b n) < 1g. Then

where

there exist constants C and C such that

Ctq

logb Eq

EA(t)q Ctq

logb Eq

8t 2 [0; 1℄:

(13)

To motivate (13) we recall the simple form (10) of the invariance of case of a cascade which implies for cascades

EA(1=bn)q =

b

q

A(t) in the

Eq n EA(1)q

(14)

This simple law is a direct consequence of the fact that the multipliers (i) of a cascade A are constant over intervals of length =bi. Proposition 11 claims that if the q-th power of the multipliers (i) do not oscillate too much over these intervals (see (11) and (12)) then roughly the same scaling law (13) holds.

1

Proof. Let us first establish (13) for the discrete points tn

=b

n,

n = 0; 1; 2; : : : . To

this end, set q E[A(b = E[A(b n )q ℄ E(0) bq

Rn (q )

) ℄:

n+1 q

For cascades, Rn is zero. Here, we show that it is not too large provided (11) and (12)

~

hold. Using equation (9) and equal distribution of A and A, we find

Rn (q )

=

b

q

E

" Z

0

b n

(s) dA~(bs)

!q

(0)q A~(b

)

n+1 q

#

:

15


Next, we use that for positive x we have Z

~(b

Since A

n+1

x(s) d(s)

I

sup jx(s)q (I )q C j: s2I

) = R0b dA~(bs), we find n

jRn(q)j b q E

Let n

C

q

sup j(s)q A~(b

s2[0;b n ℄

=

b

q

EA~(b

n+1 q

=

b

q

E[A(b

n+1 q

) (0)q A~(b

)j

n+1 q

!

n+1 q

) E sup j(s)q (0)q j s2[0;b n ℄

) ℄ (q; b n):

= minfn : (q; b n) < Eq g and assume that n > n . Set vn = EA(b n )q

for short. Applying the definition of Rn recursively we find

vn (q )

=

vn (q )

n Y1 i=n

Eq + Ri+1 (q) bq vi (q )

q n E = vn (q) bq

n n Y1

i=n

q +1 (q ) : 1 + bv R(qi)E q i

The product term is the ‘correction’ needed because the (i) are not constant over b-ary intervals. By conditions (11) and (12),

1 X

n 1 X bq Ri+1 (q ) ) 1 < log 1 Eq log 1 + v (q)Eq i i=n i=n 1 X

(q; b i ) log 1 + Eq < 1: i=n

Noticing that b

nq

(q; b

(Eq )n = (b n)q

i

logb Eq

= tnq log E b

q

shows that there exist posi-

tive constants C1 and C2 such that for all n

C1 tnq

logb Eq

EA(tn)q C2tnq logb Eq :

16


Finally, the bounds have to be extended for all

t

2 [0; 1℄.

Since

Aq () is a non-

decreasing process, it is an easy exercise to show that a correction factor large enough q is bq logb E . Thus

C

=

C2 b

q+logb Eq

C

=

C2 bq

logb Eq

are suitable constants for inequality (13).

Corollary 12. If A is non-degenerate and Proof. Denote function

xq , q

as in Proposition 11, then E logb 1:

= dbn e and '(q) = 1 q + logb Eq . By sub-additivity of the 2 [0; 1℄, A(1)q PNi=1 (Ii(n))q ; where Ii(n) = [ib n ; (i + 1)b n),

N

i = 0; : : : ; N . Applying stationarity of the increments and Proposition 11 gives

EA(1)q

CNbn(

q+log b Eq )

C~ bn(1

q+logb Eq )

= C~ bn'(q)

(q) 0 for all q 2 [0; 1℄. Since '(1) = 0, it follows that '0 (1 ) 0, i.e., E logb 1. for all n. This means that '

(1) 2 Lq , q > 1, and as in Proposition 11,

Corollary 13. If A is non-degenerate, A

Eq bq

1.

Proof. Denote

N

then

PN

i=1

= bbn .

(Ii(n) )q ; where Ii(n)

By super-additivity of the function

xq , q > 1, A(1)q

= [ib n ; (i + 1)b n), i = 0; : : : ; N . Applying stationarity

of the increments and Proposition 11 gives

EA(1)q

CNbn(

q+logb Eq )

C~ bn(1

q+logb Eq )

17


for all n. Thus

1

q + logb Eq 0.

E

(1)

q < bq 1 is sufficient to guarantee that A We conjecture that converges in Lq . This is indeed true for q . If the conjecture holds, then we may conclude that < , together with the existence of a finite q ’th moment for some q > , is b sufficient to guarantee that A be non-degenerate. Indeed, 0 < implies q < for some q > , thus the convergence of An in Lq and A t t. n An t In applications related to multifractals, we are usually interested in scaling properties. The deterministic partition function is defined as

E log 1 1

=2

1

(1+) 0 () 0 E ( ) = lim E ( ) =

1 log E PkN=01 (Ik(n))q n!1 n log jI (n) j (n) where Ik = [k 2 n ; (k + 1)2 n ), k = 0; : : : ; 2n 1. This may be the easiest scaling T (q ) =: lim inf

function to compute. The pathwise scaling properties as well as the multifractal spectra are left for future studies.

(1) 2 Lq , q > 1, and as in Proposition 11,

Corollary 14. If A is non-degenerate, A then T

(q) = q 1 logb Eq .

Proof. By Proposition 11,

lim n1 log2 EA(2 n) =

n!1

q + logb Eq :

Since A has stationary increments, 2X1 1 (n) q T (q ) = lim inf log E ( I ) 2 k n!1 n k=0 = lim n1 log2 2nEA(2 n)q n

=

q

1 logb Eq :

If the original mother process is positively correlated we can show that the multiplicative construction increases variance. This means that our construction preserves Long Range Dependence (LRD, see e.g. [RMV97]). We say that a square integrable process B with stationary increments is long range dependent if, for some > , B t t for all t.

Var ( )

1

18


(1) 2 L2 and

Proposition 15. Assume, in addition to (8), that A is non-degenerate, A

is positively correlated. Then Var At Var R0t (s) ds. Proof. Since

and A~ are independent, a simple manipulation and rearrangement gives

Z tZ t 1 ~ ~ (s1)(s2) dA(bs1 ) dA(bs2 ) t2 Var A(t) = E b2 0 0 Z tZ t 1 ((s1)(s2) 1) dA~(bs1 ) dA~(bs2 ) = E b2

Z tZ

0

0

t

E((s1)(s2) 1) ds1 ds2 t 1 +E b2 dA~(bs1 ) dA~(bs2 ) 0 0 Z Z t

+ =

Z tZ

t

0

0

E(( )( ) 1) ds1 ds2 ( )Cov dA~(bs1 ); dA~(bs2 ) + 12 Var A(bt)

s1 s2 0Z Z0 2 t t s ; s 1 2 b2 0 Z0 t

+Var

t2

0

(s) ds

b

:

The claim is proved if we can show that the both terms in the second last line are

Var A(bt) 0. Since both and A are positively correlated Cov(dA(s1 ); dA(s2)) 0 and (s1 ; s2) 0. (It is straightforward to replace the non-negative. Trivially,

somewhat heuristic infinitesimal covariance by a limit of finite increments). Rt

Corollary 16. If 0

(s) ds is long range dependent then A(t) is also.

4 Examples In order to introduce an application friendly process family, i.e., parsimonious and causal, easy to generate and analyze, we consider Markov jump processes which satisfy the following assumptions.

19


1.

(i) () =d (bi );

=01

1

i ; ; : : : ; where b > and is a positive, stationary, positive correlated, piecewise constant Markov process.

2. The transition rates are bounded both above and below:

P ((t) constant on [t; t + ) j (t) = x) = exp( (x)); (15) where min (x) max for all x in the state space of . 3. A = lim An is non-degenerate. If we assume further that the variance of is finite and that its covariance decays exponentially fast, then b > 1 + 2 is a sufficient condition for non-degeneracy (by Corollary 6). On the other hand, assuming boundedness of together with the independent exponential distributed constant periods, guarantees that the assumptions of proposition 11 are satisfied and the moments behave according to EA(t)q tq log E . q

(t) with transition rates 1 and 2 on the state space S = fS1 ; S2 g. In order to have E((t)) = 1, the transition Example 17. Consider a stationary two-state Markov process

rates must satisfy the equation

2 S1 1 S2 + 1 + 2 1 + 2

= 1:

The covariance is given by

Cov((t); (s)) = 2 e

(1 +2 )js tj ;

where

2

2 S12 + 1 S22 = + 1 2

1:

Constructing a family f (i) g from the mother process

(i) () =d (bi); means that the processes

i = 0; 1; : : : :

(i) , i = 0; 1; : : : , are independent two-state Markov pro-

cesses with transition rates figure 1.

by changing time

bi 1 and bi 2 . A realization of this construction is seen in

20

P. Mannersalo, I. Norros, R. Riedi: Multifractal products of stochastic processes 0.35

0.005

0.3 0.004 0.25 0.003

0.2 0.15

0.002

0.1 0.001 0.05 0.2

0.4

0.6

0.8

0.2

1

0.4

Figure 1: Example 17. On the left, a realization of process

1=2, S1 = 1=3, S2 = 7=6 and b = 4.

0.6

0.8

A7 (t) with 1

1

= 2, 2 =

On the right, the corresponding incremental

process at resolution 0.001.

Example 18. Let the mother process

be a piecewise constant process with Exp( )

distributed i.i.d. lengths of constant periods. For each interval we draw independently a random value M from a common distribution satisfying

E(M ) = 1. Thus, the process

(n) (t) is a piecewise constant process whose covariance is given by Cov((n)(0)(n) (x)) = Var(M )e

bn jxj

= 2 n(x):

A realization is shown in figure 2. Notice that this process is a generalization of Mandelbrot’s martingale analyzed in [KP76]. Instead of having a deterministic division of the interval, we split according to a Poisson process. Even though these two examples seem to be quite similar, there are some differences which can also be observed visually. The first process has some sort of periodic structure due to the fact that after drawing the initial state of a multiplier the only randomness is in the lengths of the constant periods. The second process is clearly burstier. The obvious reason is the unboundedness of the multipliers.

21

submitted Applied Probability, August 2001 1.2 0.02 1 0.015

0.8 0.6

0.01 0.4 0.005 0.2

0.2

0.4

0.6

0.8

1

0.2

0.4

Figure 2: Example 18. On the left, a realization of process

4, and M Gamma(3; 1).

0.6

0.8

A7 (t) with

1

= 1, b =

On the right, the corresponding incremental process at

resolution 0.001.

5 Concluding remarks The mathematical analysis of multifractal products of stochastic processes is far from complete. The aim of this paper was to give some basic definitions and properties in the general case, and show how this construction can be applied in the case of rescaled mother processes. The results given in this paper are about global behavior. The pathwise analysis is going to be more difficult. However, we are hoping to complete soon a study on pathwise multifractal properties of the family introduced in this paper.

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