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 ( U f ) ? 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 ( P 1 ≤ 1) on E denumerable. P considerated as a transition kernel : E x [1 { X 0 = x o , X 1 = x 1 , ... , X t = x t } ] = I ( x , x o ) P ( x o , x 1 ) ... P ( x t − 1 , x t ) When P 1 � = 1, the markov process can die. the potential kernel relative to P by : ∞ � � P t ( x , y ) = E x U ( x , y ) = 1 { X t = y } t =0 t (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 ג o goes from a state to the following state at the same altitude until X cross under X 0 . 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. Last First mininum mininum ρ ' ρ ζ 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 / 4 0 0  K o = 1/4 0 1 / 2 1 / 2   1 0 2 / 3 1 / 3 1/3   1 / 4 3 / 4 0 1/2 3/4 K ′ = 0 1 / 2 1 / 2   0 2 / 3 1 / 3 1/2 2 3   0 3 / 4 0 2/3 K = 0 0 0   0 0 0 Vincent Vigon () LU -factorization and probabilities 6 septembre 2007 7 / 22

K as series Let A ⊂ E . We denote by � P t P A ( x , y ) = P ( x , y )1 { x ∈ A } 1 { y ∈ A } and U A = A t with A = {� x } = { y : y � x } : � P t P � x ( x , y ) = P ( x , y )1 { y � x } and U � x = � x t We have K ( x , y ) = U � x P ( x , y )1 { x ≻ y } K ′ ( x , y ) = PU ≻ x P ( x , y )1 { x � y } + P ( x , y )1 { x � y } K o ( x , y ) = PU ≻ x P ( x , y )1 { x ∼ y } + P ( x , y )1 { x ∼ y } V ( x , y ) = U ≻ y P ( x , y ) + 1 { x = y } V ′ ( x , y ) = U � y ( x , y ) V o ( x , y ) = U � y ( x , y )1 { x ∼ y } Vincent Vigon () LU -factorization and probabilities 6 septembre 2007 8 / 22

Sweveling Kernels K , V ... are functions of P � � P n K [ P ] ( x , y ) = U � x P ( x , y )1 { x ≻ y } = � x ( x , a ) P ( a , y )1 { x ≻ y } a n V [ P ] ( x , y ) = U ≻ y P ( x , y ) + 1 { x = y } We define K � ( x , y ) = K [ P T ] ( y , x ) V � ( x , y ) = V [ P T ] ( y , x ) Those double transpositions give : K � ( x , y ) = PU � y ( 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 : when K � < ∞ ( I − P ) = ( I − K � )( I − K ′ ) The three factors one : when K � < ∞ ( I − P ) = ( I − K � )( I − K o )( I − K ) The inverse one : U = V ′ V � = VV o V � always true Vincent Vigon () LU -factorization and probabilities 6 septembre 2007 10 / 22

Proof of the ”inverse” factorization y S X 0 =x S � V � ( x , y ) = E x 1 { X t = y } t =0 S � � � � V ′ V � ( x , y ) = E x 1 { X t = y } ◦ θ t t ∈• t =0 y x Vincent Vigon () LU -factorization and probabilities 6 septembre 2007 11 / 22

Interpretation of K � t ∈ { oplit on y } ⇔ X t = y ≻ X 0 and X � y on [1 , t ]. y x 1 3 oplits on y K ( x , y ) = PU � y ( x , y )1 { x ≺ y } = E x [ ♯ { 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 .   1 / 2 1 / 2 0 1/2 P = 1 / 3 2 / 3 0   3 / 4 0 1 / 4 1/3 1/2     1 / 2 1 / 2 0 0 0 0 2/3 K ′ =  K � = 0 1 0 2 / 3 0 0 3/4    0 0 1 / 4 3 / 2 ∞ 0 K � K ′ + P = K ′ + K � Yes. 1/4 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 = λ I − Q with ρ ( Q ) ≤ λ If h is an egein vector associate to ρ ( Q ), then 1 h ( x ) Q ( x , y ) is sub-Markovian and so 1 h ( y ) h ( y ) h ( x ) A ( x , y ) is a generator λ λ 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 + R � R ′ 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 0 0 − 1 / 2 1 / 2 0 0 1/2 1/2    =   0 0 1 / 2 − 1 / 2  0 0 − 1 / 2 1 / 2 1/2 1/2     1 0 0 0 1 / 2 − 1 / 2 0 0 − 1 1 0 0 0 0 0 0     1/2 1/2     0 1 0 0 0 1 / 2 − 1 / 2 a     0 − 1 1 0 0 0 0 b 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 ρ ′ ◦ θ 1 = ∞ ] = 0 P x [ X ≻ x on [1 , ∞ [ , X 1 � = † , x Vincent Vigon () LU -factorization and probabilities 6 septembre 2007 18 / 22

h -transform of K � � ( I − K ′ ) h � ( I − P ) h ≥ 0 ⇒ ( I − K � ) ≥ 0 k ′ := ( I − K ′ )1 = E x � � X ≻ x on [1 , ∞ [ . K ( x , y ) := k ′ ( y ) ˇ k ′ ( x ) K � ( x , y ) is sub-Markovian Recall that : K � = K T [ P T ] ... Vincent Vigon () LU -factorization and probabilities 6 septembre 2007 19 / 22

Minima Suppose U ( x , z ) ∈ ]0 , ∞ [. Let E x ⊲ z be the law of X started at x , killed the last time it goes in z . z x X ρ ' We have E x ⊲ z [ X ρ ′ = a ] U ( x , z ) = V ′ ( x , a ) V � ( a , z ) Suppose X dies : E x [ X ρ ′ = a ] = V ′ ( x , a ) � V � ( a , z ) P ( z , † ) 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