Lecture 03: Properties of Poisson Process
1
Conditional Distribution of Arrivals
Proposition 1.1. Let cfw_N (t), t > 0 be a Poisson process with cfw_Ai PR+ : i [n] a set of finite
disjoint intervals with B = i[n] Ai , and cfw_ki N : i [n] and k = i[n] ki
Lecture 05: Renewal Theory
1
Renewal Theory
One of the characterization for the Poisson process is of it being a counting process with iid
exponential inter-arrival times. Now we shall relax the exponential part.
Definition 1.1. A counting process cfw_N (
Lecture 09: Equilibrium Renewal Processes and Renewal
Reward Processes
1
Renewal theory Contd. Delayed Renewal processes
1.1
Example:
(Optional not covered in class)
Consider two coins and suppose that each time is coin flipped, it lands tail with some un
Lecture 10: Discrete Time Markov Chains
1
Introduction
Definition 1.1 (DTMC). A stochastic process cfw_Xn , n N0 , where Xn S, where S is
assumed at most countable, is called a DTMC (Discrete Time Markov Chain) if
(n)
P [Xn+1 = j|Xn = i, Xn1 = in1 , , X0
Lecture 06: Key Renewal Theorem and Applications
1
Key Renewal Theorem and Applications
Definition 1.1 (Lattice Random Variable). A non-negative random variable X is said to
be lattice if there exists d 0 such that
X
Prcfw_X = nd = 1.
nN
For a lattice X,
Lecture 04: Compound Poisson Processes
1
Queueing Theory
Consider the scenario of a bus stop or a movie ticket counter. Each person arrives to the queue at
a random time and has to wait another random amount of time before he is serviced. A natural
questi
Lecture 01: Introduction to Stochastic Processes
1
Probability Review
Definition 1.1. A probability space (, F, P ) consists of set of all possible outcomes denoted
by and called a sample space, a collection of subsets F of sample space, and a non-negativ
Lecture 14 : Continuous Time Markov Chains
1
Markov Process
Definition 1.1. For a countable set I a continuous time stochastic process cfw_X(t) I, t > 0
is a Markov process if
Prcfw_X(t + s) = j|X(u), u [0, s] = Prcfw_X(t + s) = j|X(s), for all s, t > 0 a
Lecture 24: Brownian Motion
1
Introduction
Definition 1.1. A continuous time stochastic process cfw_X(t), t > 0 is called standard Brownian motion process if the following hold.
i) X(0) = 0.
ii) Process has stationary and independent increments.
iii) For
Lecture 21: Exchangeability
1
Random Walk
Definition 1.1. Let cfw_Xi : i N be iid random variables with finite E[|X1 |]. Let
Sn =
n
X
Xi , n N0 .
k=1
Then the process cfw_Sn : n N0 is called a random walk.
Definition 1.2. A random walk is called a simple
Lecture 13: Foster-Lyapunov Theorem
1
Fosters Theorem
Theorem 1.1 (Foster,1950). Let cfw_Xn n0 be a irreducible DTMC on N0 if there exist a
function L : N0 R+ with E[L(X0 )] < , such that for some K > k 0, and > 0:
1. |cfw_x N0 : L(x) k| <
2. E[L(Xn )|Xn
Lecture 02: Poisson Process
1
Simple point processes
Definition 1.1. A stochastic process cfw_N (t), t > 0 is a point process if
1. N (0) = 0, and
2. for each , the map t 7 N (t) is non-decreasing, integer valued, and right continuous.
Definition 1.2. A s
Lecture 18: Queueing Networks
1
Migration Processes
Corollary 1.1. Consider an M/M/s queue with Poisson() arrivals and each server having
exponential service time exp() service. If > s, then the output process in steady state is
Poisson().
Proof. Let X(t)
Lecture 12 : Convergence of DTMCs and Coupling theorem
1
Total Variation Distance
Definition 1.1. Given two probability distributions p and q defined on a countable space I,
their total variation distance is defined as
dT V (p, q) ,
1
kp qk1 .
2
Lemma 1.2
Lecture 22: Random Walks
1
Duality in Random Walks
Essentially, if X is an exchangeable sequence of random variables, then (X1 , X2 , , Xn ) has the
same joint distribution as (Xn , Xn1 , , X1 ). In particular, an iid sequence of random variables
is excha
Lecture 15 : Limiting Probabilities and Uniformization
1
Limiting Probabilities
We denote by cfw_Sn : n N0 the jump times of CTMC and the probability transition of the
embedded Markov chain is denoted by P = cfw_pij : i 6= j I.
Definition 1.1. If the emb
Lecture 08: Branching Processes and Delayed Renewal
Process
1
Age-dependent Branching Process
Suppose an organism lives upto a time period of X F and produces N P number of offspring.
Let X(t) denote the number of organisms alive at time t. The stochastic
Lecture 16: Reversibility
1
Reversibility
Definition 1.1. A stochastic process X(t) is reversible if (X(ti ) : i [n]) has the same
distribution as (X( ti ) : i [n]) for all ti , I, i [n].
Lemma 1.2. A reversible process is stationary.
Proof. Since X(t) is
Lecture 19 : Martingales
1
Martingales
Definition 1.1. A stochastic process cfw_Zn , n N is said to be a martingale if
1. E[|Zn |] < ,
for all n.
2. E[Zn+1 |Z1 , Z2 , . . . Zn ] = Zn .
If the equality in second condition is replaced by or , then the proce
Lecture 20: Polyas Urn Scheme
The gambling interpretation of the stochastic integral suggests that it is natural
to let the amount bet at time n depend on the outcomes of the first n 1 flips but
not on the flip we are betting on, or on later flips. The ne
Lecture 11 : Discrete Time Markov Chains
1
Discrete Time Markov Chains Contd
Theorem 1.1. An irreducible, aperiodic Markov Chain with countable state space I is of one of
the following types:
i) All the states are either transient or null recurrent. That
Lecture 23: Martingale Concentration Inequalities
1
Introduction
Lemma 1.1. If cfw_Xn : n N is a submartingale and N is a stopping time such that Prcfw_N
n = 1 then
EX1 EXN EXn .
Proof. It follows from optional stopping theorem that since N is bounded, E
Lecture 07: Inspection Paradox and Limiting Mean Excess
Time
1
The Inspection Paradox
Define XN (t)+1 = A(t) + Y (t) as the length of the renewal interval containing t, in other words,
the length of current renewal interval. Inspection paradox says that P
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Sketch of Solution to problem 9:
Given that a0 = 0, a1 = 1, and
an1 + an2
, n 2.
an =
2
From the above recurrence relation, we have n 2
2an = an1 + an2 ,
which implies that
2(an an1 ) = (an1 an2 ).
In other words
(an1 an2 )
.
2
Doing it successively, we g
Department of Mathematics, Indian Institute of Science
Analysis I (MA 221) Test 1
August 2011
Instructor: A.K.Nandakumaran September 24, 9.30 - 11.30 am.
Maximum 25 marks. Write your answers clearly, DO NOT use short notations for English
words.
1.
Recall
GENERAL
I ARTICLE
The Meaning of Integration
A](
1. Riemann
- II
Nandakumaran
Integration
In Part I, we defined the area under a curve, which is
given by a continuous function, as the integral of such
a function.
In this part, we ~iscuss the definition of
GENERAL
I ARTICLE
The Meaning of Integration - I
A K Nandakumaran
1. Introduction
Integration is the 'inverse of differentiation' is what we
are told in our beginning calculus course. It is indeed the
case when the given function is continuousl. Augustin
Department of Mathematics, IISc., Bangalore 560012
ANALYSIS I (MA 221, 3:0, August 2011)
Exercise 3
A. K. Nandakumaran
Submit before
1. Let
f (x) =
2
xn e1/x , if x > 0
0 at x = 0
1
2
e1/x exists for any polynomial Pn .
x0 Pn (x)
Show that lim f (x) exist
Department of Mathematics, IISc., Bangalore 560012
ANALYSIS I (MA 221, 3:0, August 2011)
Exercise 12
A. K. Nandakumaran
Submit before
P
P
(1) Define d1 (x, y) = ni=1 |xi yi |, d2 (x, y) = ( ni=1 |xi yi |2 )1/2 ,
d (x, y) = max |xi yi |.
i=1, ,n
(2)
(3)
(4