# Random walk, Markov chain and Brownian motion

Trang Le, PhD

2020-04-15

Guest lecture, Fundamentals of AI

## Concepts

• Random walk
• Stochastic processes
• Markov chain
• Martingale

# A stochastic process

## is a collection of random variables indexed by time.

$$X_0, X_1, X_2, X_3, \cdots$$      discrete time

$$\{X_t\}_{t\geq 0}$$      continuous time

## Alternatively, ...

a stochastic process is a probability distribution over a space of paths; this path often describes the evolution of some random value/system over time.

• Dependencies in a sequence of values?
• Long term behavior?
• Boundary events (if any)?

## Simple random walk

$$Y_i$$    i.i.d random variables $$\begin{cases} 1 & (\textrm{prob} \frac{1}{2}) \\ -1 & (\textrm{prob} \frac{1}{2}) \end{cases}$$

X_t = \sum_{i = 1}^tY_i
\forall t > 0
X_0 = 0

$$X_0, X_1, X_2, ...$$ is called simple random walk.

hard to visualize the time component

## Some properties

If      $$0 = t_0\leq t_1 \leq \cdots \leq t_k$$

then $$X_{t_{i+1}}- X_{t_i}$$ are mutally independent.

• $$E(X_k) = 0$$
• Independent increment
• Stationary

For all $$h \geq 1, t \geq 0$$, the distribution of $$X_{t+h} - X_t$$ is the same as the distribution of $$X_h$$.

## Example. A coin toss game...

At each turn, my balance goes up by $1 or down by$1.

My balance = Simple random walk

If I play until I win $100 or lose$100,

what is the probability of me winning?

If I play until I win $100 or lose$50,

what is the probability of me winning?

# Brownian motion

## is the "limit" of simple random walks.

Let $$Y_0, Y_1, ..., Y_n$$ be a simple random walk.

$Z\left(\frac{t}{n}\right) = Y_t$

Interpolate linearly and take $$n \to \infty$$.

Then, the resulting distribution is the Brownian motion.

# A Markov chain

## Formally, ...

A discrete-time stochastic process $$X_0, X_1, X_2, ...$$ with state space $$S$$ has Markov property if

P(X_{t+1}=s|X_t, \cdots, X_1, X_0)= P(X_{t+1}=s|X_t)
\forall t \geq 0, \forall s \in S

If the state space $$S$$ ($$X_t \in S$$) is finite,
all the elements of a Markov chain model
can be encoded in a transition probability matrix.

p_{ij} = P(X_{t+1}=j|X_t = i) \quad \forall i, j \in S
\sum_{j\in S}p_{ij} = 1

Why?

\begin{pmatrix} p_{11} & p_{12} & \cdots & p_{1m}\\ p_{21} & p_{22} & \cdots & p_{2m}\\ \vdots & \vdots & \ddots & \\ p_{m1} & p_{m2} & & p_{mm} \end{pmatrix}

transition probability
(jumping from state i to state j)

## Does simple random walk have a transition probability matrix?

P = \begin{pmatrix} p_{11} & p_{12} & \cdots & p_{1m}\\ p_{21} & p_{22} & \cdots & p_{2m}\\ \vdots & \vdots & \ddots & \\ p_{m1} & p_{m2} & & p_{mm} \end{pmatrix}

This matrix contains all the information about the stochastic process.

Why?

What is the probability of going from state 7 to state 9 in one steps?

What is the probability of going from state 7 to state 9 in two steps?

$P_{79}$

$P^2_{79}$

What is the probability of going from state 7 to state 9 in $$n$$ steps?

## An unrealistic example...

What is our transition probability matrix?

state

state

P = \begin{pmatrix} 0.9& 0.1 \\ 0.3 & 0.7\\ \end{pmatrix}

## Stationary distribution

\pi P = \pi \begin{pmatrix} 0.9& 0.1 \\ 0.3 & 0.7\\ \end{pmatrix} = \pi

## Hidden Markov Model (HMM)

Speech recognition

• Observation: sounds
• hidden states: words

Face detection

• Observation: overlapping rectangles of pixel intensities
• hidden states: facial features

DNA sequence region classification

• Observation: nucleotides
• Hidden states: introns, exons

# A martingale

## Formally, ...

A discrete-time stochastic process $$X_0, X_1, X_2, ...$$ is a martingale if

X_t = E[X_{t+1}|\{X_0, X_1,\cdots, X_t\}]
\forall t \geq 0.

i.e., expected gain in the process is zero at all times.

## Is a random walk a martingale?

A random variable $$X$$ is distributed according to either $$f$$ or $$g$$. Consider a random sample $$X_1, \cdots, X_n$$. Let $$Y_n$$ be the likelihood ratio
$Y_n = \prod_{i = 1}^n \frac {g(X_i)}{f(X_i)}$
If X is actually distributed according to the density $$f$$ rather than according to $$g$$, then $\{Y_n: n = 1, 2, 3, \cdots\}$ is a martingale with respect to$$\{X_n: n = 1, 2, 3, \cdots\}$$.

## Review

• Random walk
• Stochastic processes
• Markov chain
• Hidden Markov model
• MCMC
• Martingale

By Trang Le

# Random walk, Markov chain and Brownian motion

Guest lecture for Penn BMIN 520-401 course, Spring 2020

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