LU -factorization and probabilities Vincent Vigon 6 septembre 2007 - - PowerPoint PPT Presentation

LU-factorization and probabilities

Vincent Vigon 6 septembre 2007

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 1 / 22

Introduction

LU-factorization of A : A = LU where L is Lower triangular U is Upper triangular. For unicity we need to precise that L have diagonal entries equal to 1. Our subject : A = I − P.

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 2 / 22

Questions

The ”developped” LU-factorization allways exist. When the ”true” LU-factorization exist ? When the LU-factorization is unic ? When the LU-facrorization is associative : (LU)f = L(Uf ) ? When we LU-facrorization is commutative : LU = UL ? Probabilistic interpretation of all these...

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 3 / 22

Markov staff

P sub-markovian (P1 ≤ 1) on E denumerable. P considerated as a transition kernel : Ex[1{X0=xo , X1=x1 , ... , Xt=xt}] = I(x, xo)P(xo, x1)...P(xt−1, xt) When P1 = 1, the markov process can die. the potential kernel relative to P by : U(x, y) =

∞

• t=0

Pt(x, y) = Ex

• t

1{Xt=y} (no links with U).

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 4 / 22

An altitude

a : E → R, x y ⇔ a(x) ≤ a(y) Complicate examples : Simple examples : E ⊂ Z and ” ” =≤.

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 5 / 22

Descending processes

גo goes from a state to the following state at the same altitude until X cross under X0. ג′ goes from a state to the following state at an inferior altitude. ג goes from a state to the following state at a strictly inferior altitude.

ρ ' ρ ζ

First mininum Last mininum Death time

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 6 / 22

K o, K ′, K their transition kernel. V o, V ′, V their potential kernel. Example

1 2 3

1/2 3/4 1/2 1/3 1/4 2/3

K o =   1/4 1/2 1/2 2/3 1/3   K ′ =   1/4 3/4 1/2 1/2 2/3 1/3   K =   3/4  

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 7 / 22

K as series

Let A ⊂ E. We denote by PA(x, y) = P(x, y)1{x∈A}1{y∈A} and UA =

• t

Pt

A

with A = { x} = {y : y x} : Px(x, y) = P(x, y)1{yx} and Ux =

• t

Pt

x

We have K(x, y) = UxP(x, y)1{x≻y} K ′(x, y) = PU≻xP(x, y)1{xy} + P(x, y)1{xy} K o(x, y) = PU≻xP(x, y)1{x∼y} + P(x, y)1{x∼y} V (x, y) = U≻yP(x, y) + 1{x=y} V ′(x, y) = Uy(x, y) V o(x, y) = Uy(x, y)1{x∼y}

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 8 / 22

Sweveling

Kernels K, V ... are functions of P K[P](x, y) = UxP(x, y)1{x≻y} =

• a
• n

Pn

x(x, a)P(a, y)1{x≻y}

V[P](x, y) = U≻yP(x, y) + 1{x=y} We define K (x, y) = K[PT ](y, x) V (x, y) = V[PT ](y, x) Those double transpositions give : K (x, y) = PUy(x, y)1{x≺y} V (x, y) = PU≻x(x, y) + 1{x=y}

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 9 / 22

LU-factorizations

The developped one : K ′ + K + P = K K ′ always true The true one : (I − P) = (I − K )(I − K ′) when K < ∞ The three factors one : (I − P) = (I − K )(I − K o)(I − K) when K < ∞ The inverse one : U = V ′V = VV oV always true

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 10 / 22

Proof of the ”inverse” factorization

V (x, y) = Ex

S

• t=0

1{Xt=y}

X0=x y S

V ′V (x, y) = Ex

• t∈•
• S
• t=0

1{Xt=y}

• θt

x y

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 11 / 22

Interpretation of K

t ∈ {oplit on y} ⇔ Xt = y ≻ X0 and X y on [1, t].

1 y x

3 oplits on y

K(x, y) = PUy(x, y)1{x≺y} = Ex[♯{oplit on y}]

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 12 / 22

Existence

Theorem : LU-factorization is possible if (and only if) there is no state which is : 1/ recurrent 2/ undescendable 3/ Reacheable from below. E finite : these conditions just depend on the graph of P.

3/4 1/2 1/3 2/3 1/4 1/2

P =   1/2 1/2 1/3 2/3 3/4 1/4   K ′ =   1/2 1/2 1 1/4   K =   2/3 3/2 ∞   K K ′ + P = K ′ + K Yes. I − P = (I − R)(I − R′) impossible....

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 13 / 22

Known theorem on M-matrices

M-matrix is A = λI − Q with ρ(Q) ≤ λ If h is an egein vector associate to ρ(Q), then 1 λ h(y) h(x)Q(x, y) is sub-Markovian and so 1 λ h(y) h(x)A(x, y) is a generator Theorems : For a M-matrix, LU-factorization is possible : If it is inversible [Fiedler and Ptak, 1962] If it is irreductible [Kuo, 1977] Iff no state is : recurrent and undescendable and reacheable from

• below. [Varga, Cai, 1981]

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 14 / 22

Known theorem on stochastics matrices

Theorems : For generator I − P, LU-factorization is possible : if P is irreductible, recurrent, on finite E [Grassman 1987] if P is irreductible, recurrent, on N [Heyman 1995]

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 15 / 22

Commutativity

Let E be a semigroup, P and be invariant by translation : P(x, y) = P(x + z, y + z) x y ⇔ x + z y + z K ′ and K are also invariant by translation and I − P = (I − K )(I − K ′) = (I − K ′)(I − K ) Case E = Z is better known under the name of ”Wiener-Hopf factorization”

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 16 / 22

Unicity

Theorem E = N. Fix P. Suppose R′ + R = P + RR′ with .... Then R′ = K ′ and :

If j is transient, descendable Then R

ij = K ij ∈ R+

If j is rec., undescendable, reachable from i < j Then R

ij = K ij = ∞

If j is rec., undescendable, not reach. from i < j Then R

ij can be anythink 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2

    1/2 −1/2 −1/2 1/2 1/2 −1/2 −1/2 1/2     =     1 −1 1 a 1 b −1 1         1/2 −1/2 1/2 −1/2    

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 17 / 22

Associativity

Suppose K < ∞. [(I − K )(I − K ′)]1(x) ≤ (I − K )[(I − K ′)1](x) The difference between them is Px[X ≻ x on [1, ∞[ , X1 = † , ρ′ ◦ θ1 = ∞] = 0

x

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 18 / 22

h-transform of K

(I − P)h ≥ 0 ⇒ (I − K )

• (I − K ′)h
• ≥ 0

k′ := (I − K ′)1 = Ex

• X ≻ x on [1, ∞[
• .

ˇ K(x, y) := k′(y) k′(x)K (x, y) is sub-Markovian Recall that : K = K T

[PT ]...

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 19 / 22

Minima

Suppose U(x, z) ∈]0, ∞[. Let Ex⊲z be the law of X started at x, killed the last time it goes in z.

z x Xρ'

We have Ex⊲z[Xρ′ = a]U(x, z) = V ′(x, a)V (a, z) Suppose X dies : Ex[Xρ′ = a] = V ′(x, a)

• z

V (a, z)P(z, †)

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 20 / 22

Algorithm

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 21 / 22

Vincent Vigon () LU-factorization and probabilities 6 septembre 2007 22 / 22