Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
M5A42 APPLIED STOCHASTIC
PROCESSES
Professor G.A. Pavliotis
Department of Mathematics Imperial College London, UK
LECTURE 1 06/10/2016
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Lectures: Thursdays 14:00-15:00, Huxley 140, Fridays
10:00-12:00, Huxley 130.
Office Hours: Thursdays 15:00-16:00, Fridays 13:00-14:00
or by appointment.
Course webpage:
http://www.ma.imperial.ac.uk/~pavl/M4A42.htm
Text: Lecture notes, available from the course webpage.
Also, recommended reading from various textbooks.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
This is an introductory course on stochastic processes and
their applications, aimed towards students in applied
mathematics.
The emphasis of the course will be on the presentation of
analytical tools that are useful in the study of stochastic
models that appear in various problems in applied
mathematics, physics, chemistry and biology.
Numerical methods for stochastic processes are presented
in the course M5A44 Computational Stochastic Processes
that is offered in Term 2.
This is a year-long introductory graduate level course
on stochastic processes: the analytical techniques that
will be presented in Term I (M5A42) will provide the
necessary theoretical background for the development of
the computational techniques for studying stochastic
processes that will be developed in Term II (M5A44).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Prerequisites
Elementary probability theory.
Ordinary and partial differential equations.Linear algebra.
Some familiarity with analysis (measure theory, linearfunctional analysis) is desirable but not necessary.
Course Objectives
By the end of the course you are expected to be familiar
with the basic concepts of the theory of stochasticprocesses in continuous time and to be able to use various
analytical techniques to study stochastic models that
appear in applications.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Course assessment
Final exam (May/June 2017).There will be no assessed coursework.
Problem Sheets–Feedback
Problem sheets and solutions are already available from
the course webpage.
Problem classes/office hours.
Please do contact me, come to my office during office
hours.
Student Evaluation Forms.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Probability theory and random variables (2 lectures).
Basic definitions, probability spaces, probability measures
etc. Random variables, conditional expectation,
characteristic functions, limits theorems.
Stochastic processes (6 lectures). Basic definitions.
Brownian motion. Stationary processes. Other examples
of stationary processes. The Karhunen-Loeve expansion.
Markov processes (4 lectures). Introduction and
examples. Basic definitions. The Chapman-Kolmogorov
equation. The generator of a Markov process and its
adjoint. Ergodic and stationary Markov processes.
Diffusion processes (4 lectures). Basic definitions and
examples. The backward and forward (Fokker-Planck)
Kolmogorov equations. Connection between diffusion
processes and stochastic differential equations.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Stochastic Differential Equations (6 lectures). Basic
properties of SDEs. Itô’s formula. Linear SDEs. SDEs with
multiplicative noise.
The Fokker-Planck equation (6 lectures). Basic
properties of the FP equation. Examples of diffusion
processes and of the FP equation. The
Ornstein-Uhlenbeck process. Gradient flows and
eigenfunction expansions.
Exit problems for diffusion processes The mean first
passage time. One dimensional examples. Escape from a
potential well. Stochastic resonance.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Lecture notes will be provided for all the material that we
will cover in this course. The notes will be available from
the course webpage.
The notes are based on my book Stochastic processes
and applications : diffusion processes, the Fokker-Planck
and Langevin equations. It is available from the Central
Library, 519.23 PAV. The material relevant for this course
will be available from the course webpage.
There are many excellent textbooks/review articles on
applied stochastic processes, at a level and style similar to
that of this course.
Standard textbooks that cover the material on probability
theory, Markov chains and stochastic processes are:
Grimmett and Stirzaker: Probability and Random
Processes.Karlin and Taylor: A First Course in Stochastic Processes.
Lawler: Introduction to Stochastic Processes.Resnick: Adventures in Stochastic Processes.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Books on stochastic processes with a view towardsapplications, mostly to physics, are:
Horsthemke and Lefever: Noise induced transitions.
Risken: The Fokker-Planck equation.Gardiner: Handbook of stochastic methods.
van Kampen: Stochastic processes in physics and
chemistry.Mazo: Brownian motion: fluctuations, dynamics and
applications.
Chorin and Hald: Stochastic tools for mathematics andscience.
Gillespie; Markov Processes.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
The rigorous mathematical theory of probability andstochastic processes is presented in
Koralov and Sinai: Theory of probability and random
processes.Karatzas and Shreeve: Brownian motion and stochastic
calculus.Revuz and Yor: Continuous martingales and Brownian
motion.
Stroock: Probability theory, an analytic view.
Books on stochastic differential equations and theirnumerical solution are
Oksendal: Stochastic differential equations.
Kloeden and Platen, Numerical Solution of StochasticDifferential Equations.
An excellent book on the theory and the applications ofstochastic processes is
Bhatthacharya and Waymire: Stochastic processes andapplications.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
A stochastic process is used to model systems thatevolve in time and whose laws of evolution are probabilisticin nature.
The state of the system evolves in time and can bedescribed through a state variable x(t).The evolution of the state of the system depends on the
outcome of an experiment. We can write x = x(t , ω), whereω denotes the outcome of the experiment.
Examples:
The random walk in one dimension.
Brownian motion.The exchange rate between the British sterling and the US
dollar.Photon emission.
The spread of the SARS epidemic.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
The One-Dimensional Random Walk
We let time be discrete, i.e. t = 0, 1, . . . . Consider the following
stochastic process Sn:
S0 = 0;
at each time step it moves to ±1 with equal probability 12.
In other words, at each time step we flip a fair coin. If the
outcome is heads, we move one unit to the right. If the outcome
is tails, we move one unit to the left.
Alternatively, we can think of the random walk as a sum of
independent random variables:
Sn =
n∑
j=1
Xj ,
where Xj ∈ −1,1 with P(Xj = ±1) = 12 .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
We can simulate the random walk on a computer:
We need a (pseudo)random number generator to
generate n independent random variables which are
uniformly distributed in the interval [0,1].
If the value of the random variable is >12
then the particle
moves to the left, otherwise it moves to the right.
We then take the sum of all these random moves.
The sequence SnNn=1 indexed by the discrete time
T = 1, 2, . . .N is the path of the random walk. We use a
linear interpolation (i.e. connect the points n,Sn by
straight lines) to generate a continuous path.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
0 5 10 15 20 25 30 35 40 45 50
−6
−4
−2
0
2
4
6
8
50−step random walk
Figure: Three paths of the random walk of length N = 50.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
0 100 200 300 400 500 600 700 800 900 1000
−50
−40
−30
−20
−10
0
10
20
1000−step random walk
Figure: Three paths of the random walk of length N = 1000.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Every path of the random walk is different: it depends on
the outcome of a sequence of independent random
experiments.
We can compute statistics by generating a large number of
paths and computing averages. For example,
E(Sn) = 0, E(S2n) = n.
The paths of the random walk (without the linear
interpolation) are not continuous: the random walk has a
jump of size 1 at each time step.
This is an example of a discrete time, discrete space
stochastic processes.
The random walk is a time-homogeneous (the
probabilistic law of evolution is independent of time)
Markov (the future depends only on the present and not
on the past) process.
If we take a large number of steps, the random walk starts
looking like a continuous time process with continuous
paths.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Consider the sequence of continuous time stochastic
processes
Z nt :=
1√n
Snt .
In the limit as n → ∞, the sequence Z nt converges (in
some appropriate sense) to a Brownian motion with
diffusion coefficient D = ∆x2
2∆t = 12 .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
0 0.2 0.4 0.6 0.8 1−1.5
−1
−0.5
0
0.5
1
1.5
2
t
U(t)
mean of 1000 paths5 individual paths
Figure: Sample Brownian paths.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Brownian motion W (t) is a continuous time stochastic
processes with continuous paths that starts at 0
(W (0) = 0) and has independent, normally. distributed
Gaussian increments.
We can simulate the Brownian motion on a computer using
a random number generator that generates normally
distributed, independent random variables.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
We can write an equation for the evolution of the paths of a
Brownian motion Xt with diffusion coefficient D starting at
x:
dXt =√
2DdWt , X0 = x .
This is an example of a stochastic differential equation.
The probability of finding Xt at y at time t , given that it was
at x at time t = 0, the transition probability density
ρ(y , t) satisfies the PDE
∂ρ
∂t= D
∂2ρ
∂y2, ρ(y ,0) = δ(y − x).
This is an example of the Fokker-Planck equation.
The connection between Brownian motion and the
diffusion equation was made by Einstein in 1905.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Why introduce randomness in the description of physical
systems?
To describe outcomes of a repeated set of experiments.
Think of tossing a coin repeatedly or of throwing a dice.
To describe a deterministic system for which we haveincomplete information: we have imprecise knowledge ofinitial and boundary conditions or of model parameters.
ODEs with random initial conditions are equivalent tostochastic processes that can be described using
stochastic differential equations.
To describe systems for which we are not confident about
the validity of our mathematical model (uncertainty
quantification).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
To describe a dynamical system exhibiting very
complicated behavior (chaotic dynamical systems).
Determinism versus predictability.
To describe a high dimensional deterministic system using
a simpler, low dimensional stochastic system. Think of the
physical model for Brownian motion (a heavy particle
colliding with many small particles).
To describe a system that is inherently random. Think of
quantum mechanics.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
ELEMENTS OF PROBABILITY THEORY
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
The set of all possible outcomes of an experiment is called the
sample space and is denoted by Ω.
Example
The possible outcomes of the experiment of tossing a coin
are H and T . The sample space is Ω =
H, T
.
The possible outcomes of the experiment of throwing a die
are 1, 2, 3, 4, 5 and 6. The sample space is
Ω =
1, 2, 3, 4, 5, 6
.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
A collection F of Ω is called a field on Ω if
1 ∅ ∈ F ;
2 if A ∈ F then Ac ∈ F ;
3 If A, B ∈ F then A ∪ B ∈ F .
From the definition of a field we immediately deduce that F is
closed under finite unions and finite intersections:
A1, . . .An ∈ F ⇒ ∪ni=1Ai ∈ F , ∩n
i=1Ai ∈ F .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
When Ω is infinite dimensional then the above definition is not
appropriate since we need to consider countable unions of
events.
Definition
A collection F of Ω is called a σ-field or σ-algebra on Ω if
1 ∅ ∈ F ;
2 if A ∈ F then Ac ∈ F ;
3 If A1, A2, · · · ∈ F then ∪∞i=1Ai ∈ F .
A σ-algebra is closed under the operation of taking countable
intersections.
Example
F =
∅, Ω
.
F =
∅, A, Ac, Ω
where A is a subset of Ω.
The power set of Ω, denoted by 0,1Ω which contains all
subsets of Ω.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Let F be a collection of subsets of Ω. It can be extended to
a σ−algebra (take for example the power set of Ω).
Consider all the σ−algebras that contain F and take their
intersection, denoted by σ(F), i.e. A ⊂ Ω if and only if it is
in every σ−algebra containing F . σ(F) is a σ−algebra. It
is the smallest algebra containing F and it is called the
σ−algebra generated by F .
Example
Let Ω = Rn. The σ-algebra generated by the open subsets of
Rn (or, equivalently, by the open balls of Rn) is called the Borel
σ-algebra of Rn and is denoted by B(Rn).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Let X be a closed subset of Rn. Similarly, we can define
the Borel σ-algebra of X , denoted by B(X ).
A sub-σ–algebra is a collection of subsets of a σ–algebra
which satisfies the axioms of a σ–algebra.
The σ−field F of a sample space Ω contains all possible
outcomes of the experiment that we want to study.
Intuitively, the σ−field contains all the information about the
random experiment that is available to us.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
A probability measure P on the measurable space (Ω, F) is
a function P : F 7→ [0,1] satisfying
1 P(∅) = 0, P(Ω) = 1;
2 For A1, A2, . . . with Ai ∩ Aj = ∅, i 6= j then
P(∪∞i=1Ai) =
∞∑
i=1
P(Ai).
Definition
The triple(
Ω, F , P)
comprising a set Ω, a σ-algebra F of
subsets of Ω and a probability measure P on (Ω, F) is a called
a probability space.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Example
A biased coin is tossed once:
Ω = H, T, F = ∅, H, T , Ω = 0,1, P : F 7→ [0,1] such
that P(∅) = 0, P(H) = p ∈ [0,1], P(T ) = 1 − p, P(Ω) = 1.
Example
Take Ω = [0,1], F = B([0,1]), P = Leb([0,1]). Then (Ω,F ,P)is a probability space.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
A family Ai : i ∈ I of events is called independent if
P(
∩j∈J Aj
)
= Πj∈JP(Aj)
for all finite subsets J of I.
When two events A, B are dependent it is important to
know the probability that the event A will occur, given that
B has already happened. We define this to be conditional
probability, denoted by P(A|B). We know from elementary
probability that
P(A|B) =P(A ∩ B)
P(B).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
A family of events Bi : i ∈ I is called a partition of Ω if
Bi ∩ Bj = ∅, i 6= j and ∪i∈I Bi = Ω.
Theorem
Law of total probability. For any event A and any partition
Bi : i ∈ I we have
P(A) =∑
i∈I
P(A|Bi)P(Bi).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Let (Ω,F ,P) be a probability space and fix B ∈ F . Then
P(·|B) defines a probability measure on F :
P(∅|B) = 0, P(Ω|B) = 1
and (since Ai ∩ Aj = ∅ implies that (Ai ∩ B) ∩ (Aj ∩ B) = ∅)
P(∪∞j=1Ai |B) =
∞∑
j=1
P(Ai |B),
for a countable family of pairwise disjoint sets Aj+∞j=1 .
Consequently, (Ω,F ,P(·|B)) is a probability space for
every B ∈ F .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
The function of the outcome of an experiment is a random
variable, that is, a map from Ω to R.
Definition
A sample space Ω equipped with a σ−field of subsets F is
called a measurable space.
Definition
Let (Ω,F) and (E ,G) be two measurable spaces. A function
X : Ω → E such that the event
ω ∈ Ω : X (ω) ∈ A =: X ∈ A (1)
belongs to F for arbitrary A ∈ G is called a measurable function
or random variable.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
When E is R equipped with its Borel σ-algebra, then (1) can by
replaced with
X 6 x ∈ F ∀x ∈ R.
Let X be a random variable (measurable function) from
(Ω,F , µ) to (E ,G). If E is a metric space then we may define
expectation with respect to the measure µ by
E[X ] =
∫
ΩX (ω)dµ(ω).
More generally, let f : E 7→ R be G–measurable. Then,
E[f (X )] =
∫
Ωf (X (ω))dµ(ω).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Let U be a topological space. We will use the notation B(U) to
denote the Borel σ–algebra of U: the smallest σ–algebra
containing all open sets of U. Every random variable from a
probability space (Ω,F , µ) to a measurable space (E ,B(E))induces a probability measure on E :
µX (B) = PX−1(B) = µ(ω ∈ Ω;X (ω) ∈ B), B ∈ B(E). (2)
The measure µX is called the distribution (or sometimes the
law) of X .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Example
Let I denote a subset of the positive integers. A vector
ρ0 = ρ0,i , i ∈ I is a distribution on I if it has nonnegative
entries and its total mass equals 1:∑
i∈I ρ0,i = 1.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Consider the case where E = R equipped with the Borel
σ−algebra. In this case a random variable is defined to be a
function X : Ω → R such that
ω ∈ Ω : X (ω) 6 x ⊂ F ∀x ∈ R.
We can now define the probability distribution function of X ,
FX : R → [0,1] as
FX (x) = P(
ω ∈ Ω∣
∣X (ω) 6 x)
=: P(X 6 x). (3)
In this case, (R,B(R),FX ) becomes a probability space.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
The distribution function FX (x) of a random variable has the
properties that limx→−∞ FX (x) = 0, limx→+∞ F (x) = 1 and is
right continuous.
Definition
A random variable X with values on R is called discrete if it
takes values in some countable subset x0, x1, x2, . . . of R.
i.e.: P(X = x) 6= x only for x = x0, x1, . . . .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
With a random variable we can associate the probability mass
function pk = P(X = xk ). We will consider nonnegative integer
valued discrete random variables. In this case
pk = P(X = k), k = 0,1,2, . . . .
Example
The Poisson random variable is the nonnegative integer valued
random variable with probability mass function
pk = P(X = k) =λk
k!e−λ, k = 0,1,2, . . . ,
where λ > 0.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Example
The binomial random variable is the nonnegative integer valued
random variable with probability mass function
pk = P(X = k) =N!
n!(N − n)!pnqN−n k = 0,1,2, . . .N,
where p ∈ (0,1), q = 1 − p.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
A random variable X with values on R is called continuous if
P(X = x) = 0 ∀x ∈ R.
Let (Ω,F ,P) be a probability space and let X : Ω → R be a
random variable with distribution FX . This is a probability
measure on B(R). We will assume that it is absolutely
continuous with respect to the Lebesgue measure with density
ρX : FX (dx) = ρ(x)dx . We will call the density ρ(x) the
probability density function (PDF) of the random variable X .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Example
1 The exponential random variable has PDF
f (x) =
λe−λx x > 0,0 x < 0,
with λ > 0.
2 The uniform random variable has PDF
f (x) =
1b−a a < x < b,
0 x /∈ (a,b),
with a < b.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
Two random variables X and Y are independent if the events
ω ∈ Ω |X (ω) 6 x and ω ∈ Ω |Y (ω) 6 y are independent for
all x , y ∈ R.
Let X , Y be two continuous random variables. We can view
them as a random vector, i.e. a random variable from Ω to R2.
We can then define the joint distribution function
F (x , y) = P(X 6 x , Y 6 y).
The mixed derivative of the distribution function
fX ,Y (x , y) :=∂2F∂x∂y (x , y), if it exists, is called the joint PDF of the
random vector X , Y:
FX ,Y (x , y) =
∫ x
−∞
∫ y
−∞fX ,Y (x , y)dxdy .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
If the random variables X and Y are independent, then
FX ,Y (x , y) = FX (x)FY (y)
and
fX ,Y (x , y) = fX (x)fY (y).
The joint distribution function has the properties
FX ,Y (x , y) = FY ,X (y , x),
FX ,Y (+∞, y) = FY (y), fY (y) =
∫ +∞
−∞fX ,Y (x , y)dx .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
We can extend the above definition to random vectors of
arbitrary finite dimensions. Let X be a random variable from
(Ω,F , µ) to (Rd ,B(Rd )). The (joint) distribution function
FXRd → [0,1] is defined as
FX (x) = P(X 6 x).
Let X be a random variable in Rd with distribution function
f (xN) where xN = x1, . . . xN. We define the marginal or
reduced distribution function f N−1(xN−1) by
f N−1(xN−1) =
∫
R
f N(xN)dxN .
We can define other reduced distribution functions:
f N−2(xN−2) =
∫
R
f N−1(xN−1)dxN−1 =
∫
R
∫
R
f (xN)dxN−1dxN .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
We can use the distribution of a random variable to compute
expectations and probabilities:
E[f (X )] =
∫
R
f (x)dFX (x) (4)
and
P[X ∈ G] =
∫
G
dFX (x), G ∈ B(E). (5)
The above formulas apply to both discrete and continuous
random variables, provided that we define the integrals in (4)
and (5) appropriately.
When E = Rd and a PDF exists, dFX (x) = fX (x)dx , we have
FX (x) := P(X 6 x) =
∫ x1
−∞. . .
∫ xd
−∞fX (x)dx ..
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Example (Normal Random Variables)
Consider the random variable X : Ω 7→ R with pdf
γσ,m(x) := (2πσ)−12 exp
(
−(x − m)2
2σ
)
.
Such an X is termed a Gaussian or normal random
variable. The mean is
EX =
∫
R
xγσ,m(x)dx = m
and the variance is
E(X − m)2 =
∫
R
(x − m)2γσ,m(x)dx = σ.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Example (Normal Random Variables contd.)
Let m ∈ Rd and Σ ∈ R
d×d be symmetric and positive
definite. The random variable X : Ω 7→ Rd with pdf
γΣ,m(x) :=(
(2π)d detΣ)− 1
2exp
(
−1
2〈Σ−1(x − m), (x − m)〉
)
is termed a multivariate Gaussian or normal random
variable. The mean is
E(X ) = m (6)
and the covariance matrix is
E
(
(X − m)⊗ (X − m))
= Σ. (7)
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Let X , Y be random variables we want to know whether they
are correlated and, if they are, to calculate how correlated they
are. We define the covariance of the two random variables as
cov(X ,Y ) = E[
(X − EX )(Y − EY )]
= E(XY )− EXEY .
The correlation coefficient is
ρ(X ,Y ) =cov(X ,Y )
√
var(X )√
var(X )(8)
The Cauchy-Schwarz inequality yields that ρ(X ,Y ) ∈ [−1,1].We will say that two random variables X and Y are
uncorrelated provided that ρ(X ,Y ) = 0. It is not true in general
that two uncorrelated random variables are independent. This
is true, however, for Gaussian random variables.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Assume that E|X | < ∞ and let G be a sub–σ–algebra of F . The
conditional expectation of X with respect to G is defined to be
the function E[X |G] : Ω 7→ E which is G–measurable and
satisfies∫
G
E[X |G]dµ =
∫
G
X dµ ∀G ∈ G.
We can define E[f (X )|G] and the conditional probability
P[X ∈ F |G] = E[IF (X )|G], where IF is the indicator function of F ,
in a similar manner.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
φ(t) =
∫
R
eitλ dF (λ) = E(eitX ). (9)
For a continuous random variable for which the distribution
function F has a density, dF (λ) = p(λ)dλ, (9) gives
φ(t) =
∫
R
eitλp(λ)dλ.
For a discrete random variable for which P(X = λk ) = αk , (9)
gives
φ(t) =
∞∑
k=0
eitλk ak .
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
The characteristic function determines uniquely the distribution
function of the random variable, in the sense that there is a
one-to-one correspondance between F (λ) and φ(t).
Lemma
Let X1,X2, . . .Xn be independent random variables with
characteristic functions φj(t), j = 1, . . . n and let Y =∑n
j=1 Xj
with characteristic function φY (t). Then
φY (t) = Πnj=1φj(t).
Lemma
Let X be a random variable with characteristic function φ(t) and
assume that it has finite moments. Then
E(X k) =1
ikφ(k)(0).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Theorem
Let b ∈ Rn and Σ ∈ R
n×n a symmetric and positive definite
matrix. Let X be the multivariate Gaussian random variable with
probability density function
γ(x) =1
Zexp
(
−1
2〈Σ−1(x − b),x − b〉
)
.
Then
1 The normalization constant is Z = (2π)n/2√
det(Σ).
2 The mean vector and covariance matrix of X are given by
EX = b and E((X − EX)⊗ (X − EX)) = Σ.
3 The characteristic function of X is
φ(t) = ei〈b,t〉− 12〈t,Σt〉.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
One of the most important aspects of the theory of random
variables is the study of limit theorems for sums of random
variables.
The most well known limit theorems in probability theory
are the law of large numbers and the central limit
theorem.
There are various different types of convergence for
sequences or random variables.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Definition
Let Zn∞n=1 be a sequence of random variables. We will say
that
(a) Zn converges to Z with probability one if
P(
limn→+∞ Zn = Z)
= 1.
(b) Zn converges to Z in probability if for every ε > 0
limn→+∞ P(
|Zn − Z | > ε)
= 0.
(c) Zn converges to Z in Lp if limn→+∞ E[∣
∣Zn − Z∣
∣
p]= 0.
(d) Let Fn(λ),n = 1, · · ·+∞, F (λ) be the distribution functions
of Zn n = 1, · · ·+∞ and Z , respectively. Then Zn converges
to Z in distribution if limn→+∞ Fn(λ) = F (λ) for all λ ∈ R at
which F is continuous.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
The distribution function FX of a random variable from a
probability space (Ω,F ,P) to R induces a probability measure
on R and that (R,B(R),FX ) is a probability space. We can
show that the convergence in distribution is equivalent to the
weak convergence of the probability measures induced by the
distribution functions.
Definition
Let (E ,d) be a metric space, B(E) the σ−algebra of its Borel
sets, Pn a sequence of probability measures on (E ,B(E)) and
let Cb(E) denote the space of bounded continuous functions on
E . We will say that the sequence of Pn converges weakly to the
probability measure P if, for each f ∈ Cb(E),
limn→+∞
∫
E
f (x)dPn(x) =
∫
E
f (x)dP(x).
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Theorem
Let Fn(λ),n = 1, · · ·+∞, F (λ) be the distribution functions of
Zn n = 1, · · · +∞ and Z , respectively. Then Zn converges to Z
in distribution if and only if, for all g ∈ Cb(R)
limn→+∞
∫
X
g(x)dFn(x) =
∫
X
g(x)dF (x). (10)
Remark
(10) is equivalent to
En(g) = E(g),
where En and E denote the expectations with respect to Fn and
F, respectively.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
When the sequence of random variables whose
convergence we are interested in takes values in Rn or,
more generally, a metric space space (E ,d) then we can
use weak convergence of the sequence of probability
measures induced by the sequence of random variables to
define convergence in distribution.
Definition
A sequence of real valued random variables Xn defined on a
probability spaces (Ωn,Fn,Pn) and taking values on a metric
space (E ,d) is said to converge in distribution if the indued
measures Fn(B) = Pn(Xn ∈ B) for B ∈ B(E) converge weakly
to a probability measure P.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
Let Xn∞n=1 be iid random variables with EXn = V . Then,
the strong law of large numbers states that average of
the sum of the iid converges to V with probability one:
P(
limn→+∞
1
N
N∑
n=1
Xn = V)
= 1.
The strong law of large numbers provides us with
information about the behavior of a sum of random
variables (or, a large number or repetitions of the same
experiment) on average.
We can also study fluctuations around the average
behavior.
Course Overview Course Outline Bibliography Introduction Elements of Probability Theory
let E(Xn − V )2 = σ2. Define the centered iid random
variables Yn = Xn − V . Then, the sequence of random
variables 1
σ√
N
∑Nn=1 Yn converges in distribution to a
N (0,1) random variable:
limn→+∞
P
(
1
σ√
N
N∑
n=1
Yn 6 a
)
=
∫ a
−∞
1√2π
e− 12
x2
dx .
This is the central limit theorem.