6.4 The Infinitely Many Alleles Model

Size: px
Start display at page:

Download "6.4 The Infinitely Many Alleles Model"

Transcription

1 6.4. THE INFINITELY MANY ALLELES MODEL 93 NE µ [F (X N 1 ) F (µ)] = + 1 i<j n ( f i f j, µp N f i, µp N f j, µp N ) l:l i,j n Af i, µ f j, µ f j, µp N + O(N 1 ) j:j<i = GF (µ) + o(1) uniformly in µ. The completes the verification of condition (6.7). j:j>i 6.4 The Infinitely Many Alleles Model f l, µp N This is a special case of the Fleming-Viot process which has played a crucial role in modern population biology. It has type space E = [, 1] and type-independent mutation operator with mutation source ν P([, 1]) Af(x) = θ( p(x, dy)f(y) f(x)) = θ( f(y)ν (dy) f(x)). Since A is a bounded operator we can take indicator functions of intervals in D(A). If we have a partition [, 1] = K j=1b j where the B j are intervals, consider the set D(G) of functions (6.1) F (µ) = f 1, µ... f n, µ with n 1 and where the functions f 1,..., f n are finite linear combinations of indicator functions of the intervals {A j }. Then the function GF (µ) can be written in the same form and we can prove that the -valued process {p t (A 1 ),..., p t (A K )} is a version of the K allele process with generator (6.11) G K f(p) = 1 2 i,j=1 p i (δ ij p j ) 2 f(p) + θ p i p j (ν (A i ) p i ) f(p) p i. We will next give an explicit construction of this process that allows us to derive a number of interesting properties of this important model Projective Limit Construction of the Infinitely Many Alleles Model Let µ, ν P(E), C = C [, ) ([, )). Let U denote the collection of finite partitions u = (A u 1,..., A u u ) of E into measurable subsets in B(E) and u denotes

2 94 CHAPTER 6. INFINITELY MANY TYPES MODELS the number of sets in the partition u. We place a partial ordering on U as follows: v u if v is a refinement of u. We can also identify partitions with the finite algebras of subsets of E they generate. Given a partition we define the probability measure, P u on C u as the law of the Wright-Fisher diffusion with generator G (K) f(p) = i,j=1 ν i := ν (A j ) p i (δ ij p j ) 2 f(p) p i p j θ(ν i p i ) f(p) p i and initial measure µ, that is, the law of (p t (A u 1),..., p t (A u u ) (and the additive extension of this to the algebra generated by u). Remark 6.6 Recall that the associated Markov transition function is determined by the joint moments as follows. Since the family of functions p k , p k belong to D(G(K) ) we can apply G (K) and obtain the following system of equations for the joint moments: (6.12) m k1,...,k (t) := E[p k 1 1 (t)... p k (t)], t m k 1,...,k (t) = 1 k i (k i 1)m k1,.,k 2 i 1,..k (t) i 1 k i k j m k1,...k 2 K (t) + θ 2 i j ν i k i m k1,.k j 1,.,k i +1,..k K (t) θ k i m k1,...k 2 (t) Since this system of linear equations is closed, there exists a unique solution which characterizes the K-allele Wright-Fisher diffusion. In a similar way we can apply this to the function corresponding to the coalescence of two partition elements f(p) = f( p) p = ( p 1,..., p ) = (p 1,..., p l 1, p l+1,..., p k 1, p k+1,..., p, (p l + p k ))

3 6.4. THE INFINITELY MANY ALLELES MODEL 95 G (K) f(p) = θ 2 K 2 i,j=1 p i (δ ij p j ) 2 f( p) p i p j K 2 ( ν i p i ) f( p) p i = G () f( p) In other words we have consistency under coalescence of the partition elements. Because of uniqueness this implies that the process p(t) = ( p 1 (t),..., p (t)) coincides with the (K 1) allele Wright-Fisher diffusion. We denote the canonical projections π u : C B(E) C u and π uv : C v C u if v u such that π u = π uv π v, v u. The family {P u } u U forms a projective system of probability laws, that is for every pair, (u, v), v u, {P u } then satisfies (6.13) π uv (P v ) = P u, P u (B) = P v (π 1 uv (B)). Therefore, by Theorem 17.6 (in Appendix I) there exists a projective limit measure, that is, a probability measure P on C B([,1]) such that for any u U, π u P = P u. For fixed t (or any finite set of times) we can identify the projective limit, (6.14) { p t (A) : A B([, 1])} with an element of X ([, 1]), the space of all finitely additive, non-negative, mass one measures on [, 1], equipped with the projective limit topology, i.e., the weakest topology such that for all Borel subset B of [, 1], µ(b) is continuous in µ. Under this topology, X ([, 1]) is Hausdorff. The σ-algebra B of the space X ([, 1]) is the smallest σ-algebra such that for all Borel subset B of [, 1], µ(b) is a measurable function of µ. For fixed t [, ), p t ( ) is a.s. a finitely additive measure, that is, a member of X [, 1] and satisfies the conditions of Theorem 17.8 in the Appendices (conditions 1,2 follow immediately from the construction, 3 follows since for any A B([, 1]) E(p t (A)) max(µ(a), ν (A)) and (4) is automatic since all measures are bounded by 1). Therefore for fixed t this determines a unique countably additive version p t ( ), that is, a random countable additive measure p t P([, 1]) a.s. Similarly, taking two times t 1, t 2 we obtain a the joint distribution of a pair (p t1, p t2 ) of random probability measures. We can then verify that t f(x)p t (dx) is a.s. continuous for a countable convergence determining class of functions so that there is an a.s. continuous version with respect to the topology of weak convergence.

4 96 CHAPTER 6. INFINITELY MANY TYPES MODELS Remark 6.7 We can carry out the same construction assuming that for each u U the Wright-Fisher diffusion starts with the stationary Dirichlet measure and obtain by the projective limit a probability measure on P(E) which for any partition has the associated Dirichlet distribution. 6.5 The Jirina Branching Process In 1964 Jirina [355] gave the first construction of a measure-valued branching process. The state space is the space of finite measures on [, 1], M f ([, 1]). ν M 1 ([, 1]). We will construct a version of this process with immigration by a projective limit construction. Given a partition (A 1,..., A K ) of [, 1] let {X t (A i ) : t, i = 1,..., K} satisfy the SDE (Feller CSB plus immigration): (6.15) dx t (A i ) = c(ν (A i ) X t (A i ))dt + 2γX t (A i )dw A i t X (A i ) = µ(a i ) where ν is in P([, 1]) and for each i, W A i t is a standard Brownian motion and for i j W A i t and W A j t are independent. We can then verify that the processes X t (A i ) : i = 1,..., K are independent and as t, X t (A i ) converges in distribution to a stationary measure X (A i ) with density which satisfies f i (x) = 1 Z xθ i 1 e θx, x > where θ = c γ, θ i = θν (A i ). This can be represented by X (A) = θ 1 G(θν (A)) where θ = c γ and L{(X (A 1 ),..., X (A K ))} = L{ 1 θ [G(θ 1), G(θ 1 + θ 2 ) G(θ 1 ),..., G(θ) G(θ θ K )]} where G(s) is the Moran subordinator - see subsection below. For u = (A u 1,..., A u u ) U (defined as in the last subsection) let {P u = L({(X t (A 1 ),..., X t (A u ) : t, A u})}. Then the collection {P u } u U forms a projective system and as in the previous section there exists a projective limit measure P on (C [, ) ([, ))) B([,1]). Moreover for fixed t [, ), X t ( ) is a.s. a finitely additive measure that is regular (on a countable generating subset of B([, 1])) we obtain a unique countably additive version (recall Theorem 17.8). Thus, {X t ( ) : t } is a measure-valued process and again we can obtain an a.s. continuous M F ([, 1])-valued version. This M F ([, 1])-valued process is called the Jirina process.

5 6.6. INVARIANT MEASURES OF THE IMA AND JIRINA PROCESSES 97 Corollary 6.8 The stationary measure for the Jirina process is given by the random measure (6.16) X (A) = 1 θ 1 1 A (x)dg(θs), A B([, 1]) where G( ) is the Moran gamma subordinator. 6.6 Invariant Measures of the IMA and Jirina Processes The Moran (Gamma) Subordinator We begin by recalling the the Gamma distribution with parameter α > given by the density function g α (u) = u α 1 e u /Γ(α) and Laplace transform of g α is g α (y)e λy dy = 1 (1 + λ) α, λ > 1, The Moran subordinator {G(α) : α } is an increasing process with stationary independent increments G(α 2 ) G(α 1 ), α 1 < α 2 given by g α2 α 1. Lévy representation Lemma 6.9 (6.17) E ( e λg(α)) = exp Proof. Note that λ ( α (1 e λz )z 1 e z dz = ) (1 e uλ ) e u u du. (1 e λz )z 1 e z dz = log(1 + λ) (e λz )e z dz = λ Hence we have the Lévy-Khinchin representation with Lévy measure e z z, z > (6.18) { 1 (1 + λ) = exp α α (1 e λz )z 1 e z dz }.

6 98 CHAPTER 6. INFINITELY MANY TYPES MODELS Poisson representation The Poisson random field with intensity measure µ is a random counting measure Π on a space S. Π(A i ), Π(A j ) are independent if i j and Π(A) is Poisson with parameter µ(a). Theorem 6.1 (Campbell s Theorem.) Let Π be a Poisson random field with intensity µ M(S) and f : S R, Σ = x Π f(x) = f(x)π(dx) converges a.s. if and only if min( f(x), 1)µ(dx) < S and then E(e s f(x)π(dx) ) = exp( (e sf(x) 1)µ(dx)), s R provided the integral on the right exists. Now consider the Poisson random measure on [, 1] (, ) (6.19) Ξ θ = δ {x,u} with intensity measure θν (dx) e u u du. Let X (A) := A uξ θ (dx, du). Then by Campbell s Theorem (6.2) E(e λ X (A) ) = E(e λ A uξ θ(dx,du) ) = e θν (A) (1 e λu ) e u u du. Hence we can represent equilibrium of the Jirina process by the random measure with Poisson representation {X (A) : A B([, 1])} by (6.21) X (A) = θ 1 u Ξ θ (dx, du) A and this can be obtained as the projective limit of the finite systems. If ν is Lebesgue measure on [, 1] then have that the {X ([, s)} s 1 = {G(s)} s 1 where G(s) is the Moran subordinator with with increments G(s 2 ) G(s 1 ) having the Gamma θ(s 2 s 1 ) distribution θ = c γ.

7 6.6. INVARIANT MEASURES OF THE IMA AND JIRINA PROCESSES Representation of the Infinitely Many Alleles Equilibrium Recall (Theorem 5.9) that the Dirichlet distribution Dirichlet(θ 1,..., θ n ) has the joint density on relative to (n 1)-dimensional Lebesgue measure on n 1 given by f(p 1,..., p n 1 ) = Γ(θ θ n ) Γ(θ 1 )... Γ(θ n ) pθ p θ p θn 1 n. Recall that if the θ are large the measure concentrates away from the boundary whereas is the θ are small things concentrate near the boundary corresponding to highly disparate p with a few large p j and the others small. For example if the θ j are small but equal there is a high probability that at least one of the p j is much greater than average; and which value or values of j have large p j is a matter of chance. Proposition 6.11 Let X denote the equilibrium random measure for the Jirina process and consider a partition [, 1] = n A i and define (6.22) Y (A i ) := X (A i ) X ([, 1]) = G(θ A i ). G(θ) Then the family (Y (A 1 ),..., Y (A K )) is independent of X ([, 1]) and has as distribution the Dirichlet(θ 1,... θ K ) where θ j = θν (A j ). Proof. Let Y be Gamma(θ) and (P 1,..., P K ) Dirichlet (θ 1,..., θ K ) with Y and (P 1,..., P K ) independent, and define (Y 1,..., Y K ) by (6.23) Y i := Y P i. We will verify that (Y 1,..., Y K ) has the joint probability density function (6.24) g(y 1,..., y K ) = K u θ i 1 i e u i /Γ(θ i ). Consider the 1-1 transformation (Y 1, Y 2,..., Y K ) (Y, P 2,..., P K ) with Jacobian { (6.25) J = x } 1,..., x K, x 1 = y, x 2 = p 2,..., x K = p K = 1 y 1,..., y K y. By independence of Y and (P 1,..., P K ), we obtain the joint density of (Y 1,..., Y K ) as

8 1 CHAPTER 6. INFINITELY MANY TYPES MODELS g(y 1,..., y K ) = f(p 1,..., p K Y )f Y (y) J 1 = f(p 1,..., p K ) Γ(θ) yθ 1 e y 1 y Γ(θ) = Γ(θ 1 )... Γ(θ K ) ( ) θ1 1 ( ) θk 1 y1 yk yi yi Γ(θ) ( y i ) (θ 1) e yi.( y i ) () K 1 = Γ(θ i ) yθ i 1 i e y i Note that this coincides with the Dirichlet(θ 1,..., θ K ) distribution. Corollary 6.12 The invariant measure of the infinitely many alleles model can be represented by the random probability measure (6.26) Y (A) = X (A),, A B([, 1]). X ([, 1]) where X ( ) is the equilibrium of the above Jirina process and Y ( ) and X ([, 1]) are independent. Reversibility Recall that the Dirichlet distribution is a reversible stationary measure for the K type Wright-Fisher model with house of cards mutation (Theorem 5.9). From this and the projective limit construction it can be verified that L(Y ( )) is a reversible stationary measure for the infinitely many alleles process. Note that reversibility actually characterizes the IMA model among neutral Fleming-Viot processes with mutation, that is, any mutation mechanism other than the typeindependent or house of cards mutation leads to a stationary measure that is not reversible (see Li-Shiga-Yau (1999) [431]) The Poisson-Dirichlet Distribution Without loss of generality we can assume that ν is Lebesgue measure on [, 1]. This implies that the IMA equilibrium is given by a random probability measure which is pure atomic (6.27) p = a i δ xi, a i = 1, x i [, 1]

9 Supplementary EXERCISES FOR LECTURE 9 1. Consider two Feller CSB with immigration as in Equation 6.15 in the notes (with Wt Ai, W Aj t independent Brownian motions, i j). Show that Y t := X t (A i ) + X t (A j ) is a Feller CSB with immigration. 2. Prove Campbell s Theorem. 3. Consider the Gamma subordinator {G(s) : s } and let a < b. (a) Prove that P (G(b) G(a) = ) = (b) Calcuate the probability that there is no jump in (a, b) larger than 1. 1

yk 1 y K 1 .( y i ) (K 1) Note that this coincides with the Dirichlet(θ 1,..., θ K ) distribution.

yk 1 y K 1 .( y i ) (K 1) Note that this coincides with the Dirichlet(θ 1,..., θ K ) distribution. 1 CHAPTER 6. INFINITELY MANY TYPES MODELS g(y 1,..., y K ) = f(p 1,..., p K Y )f Y (y) J 1 = f(p 1,..., p K ) Γ(θ) yθ 1 e y 1 y K 1 Γ(θ) = Γ(θ 1 )... Γ(θ K ) ( ) θ1 1 ( ) θk 1 y1 yk 1.... yi yi Γ(θ) (

More information

How To Find Out How To Calculate A Premeasure On A Set Of Two-Dimensional Algebra

How To Find Out How To Calculate A Premeasure On A Set Of Two-Dimensional Algebra 54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM TORONTO THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

More information

CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

Hydrodynamic Limits of Randomized Load Balancing Networks

Hydrodynamic Limits of Randomized Load Balancing Networks Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let H and J be as in the above lemma. The result of the lemma shows that the integral Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Grey Brownian motion and local times

Grey Brownian motion and local times Grey Brownian motion and local times José Luís da Silva 1,2 (Joint work with: M. Erraoui 3 ) 2 CCM - Centro de Ciências Matemáticas, University of Madeira, Portugal 3 University Cadi Ayyad, Faculty of

More information

Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

More information

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

More information

THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE. Contents THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

More information

Math 526: Brownian Motion Notes

Math 526: Brownian Motion Notes Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t

More information

Characterizing Option Prices by Linear Programs

Characterizing Option Prices by Linear Programs Contemporary Mathematics Characterizing Option Prices by Linear Programs Richard H. Stockbridge Abstract. The price of various options on a risky asset are characterized via a linear program involving

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Finite dimensional topological vector spaces

Finite dimensional topological vector spaces Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

More information

Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

Separation Properties for Locally Convex Cones

Separation Properties for Locally Convex Cones Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

More information

How To Prove The Dirichlet Unit Theorem

How To Prove The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Extension of measure

Extension of measure 1 Extension of measure Sayan Mukherjee Dynkin s π λ theorem We will soon need to define probability measures on infinite and possible uncountable sets, like the power set of the naturals. This is hard.

More information

Lecture 2: Universality

Lecture 2: Universality CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

More information

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1]. Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3. 5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

More information

Conductance, the Normalized Laplacian, and Cheeger s Inequality

Conductance, the Normalized Laplacian, and Cheeger s Inequality Spectral Graph Theory Lecture 6 Conductance, the Normalized Laplacian, and Cheeger s Inequality Daniel A. Spielman September 21, 2015 Disclaimer These notes are not necessarily an accurate representation

More information

The Ergodic Theorem and randomness

The Ergodic Theorem and randomness The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction

More information

Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion. Ernie Croot. February 3, 2010 Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Duality of linear conic problems

Duality of linear conic problems Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

More information

Some Research Problems in Uncertainty Theory

Some Research Problems in Uncertainty Theory Journal of Uncertain Systems Vol.3, No.1, pp.3-10, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences

More information

Convex analysis and profit/cost/support functions

Convex analysis and profit/cost/support functions CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m

More information

4. Expanding dynamical systems

4. Expanding dynamical systems 4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

More information

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

More information

Introduction to Probability

Introduction to Probability Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

More information

The Exponential Distribution

The Exponential Distribution 21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 425, PRACTICE FINAL EXAM SOLUTIONS. MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but

Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but Chapter 7 Homotopy 7. Basic concepts of homotopy Example: but γ z dz = γ z dz γ 2 z dz γ 3 z dz. Why? The domain of /z is C 0}. We can deform γ continuously into γ 2 without leaving C 0}. Intuitively,

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

More information

Probability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T..

Probability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Probability Theory A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = {Ω, F,

More information

Probability Generating Functions

Probability Generating Functions page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More information

Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

More information

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12 CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

More information

False-Alarm and Non-Detection Probabilities for On-line Quality Control via HMM

False-Alarm and Non-Detection Probabilities for On-line Quality Control via HMM Int. Journal of Math. Analysis, Vol. 6, 2012, no. 24, 1153-1162 False-Alarm and Non-Detection Probabilities for On-line Quality Control via HMM C.C.Y. Dorea a,1, C.R. Gonçalves a, P.G. Medeiros b and W.B.

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

Tail inequalities for order statistics of log-concave vectors and applications

Tail inequalities for order statistics of log-concave vectors and applications Tail inequalities for order statistics of log-concave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.Tomczak-Jaegermann Banff, May 2011 Basic

More information

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

More information

On strong fairness in UNITY

On strong fairness in UNITY On strong fairness in UNITY H.P.Gumm, D.Zhukov Fachbereich Mathematik und Informatik Philipps Universität Marburg {gumm,shukov}@mathematik.uni-marburg.de Abstract. In [6] Tsay and Bagrodia present a correct

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

16.3 Fredholm Operators

16.3 Fredholm Operators Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this

More information

Markov random fields and Gibbs measures

Markov random fields and Gibbs measures Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).

More information

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

More information

On the D-Stability of Linear and Nonlinear Positive Switched Systems

On the D-Stability of Linear and Nonlinear Positive Switched Systems On the D-Stability of Linear and Nonlinear Positive Switched Systems V. S. Bokharaie, O. Mason and F. Wirth Abstract We present a number of results on D-stability of positive switched systems. Different

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

More information

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

More information

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu

Follow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part

More information

Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

More information

Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

More information

Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

More information

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Monte Carlo Methods in Finance

Monte Carlo Methods in Finance Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

More information

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

More information

Stochastic Processes LECTURE 5

Stochastic Processes LECTURE 5 128 LECTURE 5 Stochastic Processes We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that

More information

Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

More information

Ideal Class Group and Units

Ideal Class Group and Units Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

LECTURE 4. Last time: Lecture outline

LECTURE 4. Last time: Lecture outline LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random

More information

An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

UNIVERSITETET I OSLO

UNIVERSITETET I OSLO NIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:

More information

Galois Theory III. 3.1. Splitting fields.

Galois Theory III. 3.1. Splitting fields. Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

1 Norms and Vector Spaces

1 Norms and Vector Spaces 008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic.

Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Martina Fedel joint work with K.Keimel,F.Montagna,W.Roth Martina Fedel (UNISI) 1 / 32 Goal The goal of this talk is to

More information

Methods for Finding Bases

Methods for Finding Bases Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

More information

A spot price model feasible for electricity forward pricing Part II

A spot price model feasible for electricity forward pricing Part II A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 17-18

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

DISINTEGRATION OF MEASURES

DISINTEGRATION OF MEASURES DISINTEGRTION OF MESURES BEN HES Definition 1. Let (, M, λ), (, N, µ) be sigma-finite measure spaces and let T : be a measurable map. (T, µ)-disintegration is a collection {λ y } y of measures on M such

More information

ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS 1. INTRODUCTION ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

5.3 Improper Integrals Involving Rational and Exponential Functions

5.3 Improper Integrals Involving Rational and Exponential Functions Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

More information