# MATH310 Spring 2026 - Day 23
**Date**: March 11, 2026
**Topic**: From Fourier to Random Walks - Introduction to Stochastic Processes
**Lecture**: 23
## Administrative Notes
- Video example of cross-correlation shown at 1.5x speed initially
- Connection to previous work on cross-correlation and convolution
## Review: Cross-Correlation
Started class with a brief review of cross-correlation from [[MATH310S26-Day22-WorkdayMaterials (Cross-correlation and signal detection)|Day 22]]:
$(f \star g)(\tau) = \int_{-\infty}^{\infty} f(t)g(t+\tau)\,dt$
Where:
- $\tau$ is the lag parameter
- This describes similarity between $f$ and $g$ for lag $\tau$
- When $g = f$, this becomes **autocorrelation** - correlation of data with itself
## Transition: What is Noise?
### Key Observations
1. **Signals often have extra components** whose sources are unclear or not easy to quantify
2. **Challenge**: How can we model these unknown components mathematically?
### Philosophical Approach to Noise
- **Idea**: Some noise could be the consequence of the accumulation of many lower-order effects
- Instead of modeling each individual source (lightning, solar radiation, etc.), treat the combined effect as a **random or stochastic process**
### Important Notes on Noise Modeling
1. **Hope**: The signal is much stronger than the noise
2. **Assumption**: If not truly stochastic, it's "not too bad" to model this way
3. **Practical consideration**: On machines, we often need sequences that act random → [pseudorandom numbers](https://en.wikipedia.org/wiki/Pseudorandom_number_generator)
### Types of Noise (Video Example)
- **White noise**: Equal/uniform probabilities in frequency space
- **Pink noise**: Emphasizes lower frequencies, dampens higher ones (sounds like rain)
- **Brown noise**: Even more aggressive low-frequency emphasis
- Generated by randomly exciting different Fourier modes with changing amplitudes
## Introduction to Random Walks
### Definition: Simple Random Walk
A [random walk](https://en.wikipedia.org/wiki/Random_walk) is a [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process) defined by:
$S_{n+1} = S_n + X_n, \quad n = 0, 1, 2, \ldots$
Where:
- $S_n$ is the random variable representing position at time $n$
- $X_n$ are **i.i.d.r.v.** (independent identically distributed random variables)
- $S_0$ is the starting location
### Understanding i.i.d.r.v.
- **[Independent](https://en.wikipedia.org/wiki/Independence_(probability_theory))**: Each pick of random variable doesn't depend on previous picks
- **[Identically distributed](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables)**: All $X_n$ are drawn from the same distribution
- **Distributed**: Refers to the [probability distribution](https://en.wikipedia.org/wiki/Probability_distribution)
- **[Random Variables](https://en.wikipedia.org/wiki/Random_variable)**: The mathematical objects we're working with
### Physical Interpretation
- Today's position: $S_n$
- Tomorrow's position: $S_{n+1} = S_n + X_n$
- Random step taken today: $X_n$
- Each step is independent of all previous steps
### Closed-Form Representation
By recursively expanding the definition:
$S_{n+1} = S_0 + \sum_{i=0}^{n} X_i$
This shows the position as an **accumulation of i.i.d.r.v.**
## Probability Space for Simple Random Walk
### Sample Space
$\Omega_X = \{\text{go left amount } L_1, \text{ go right amount } L_2\}$
This is the [sample space](https://en.wikipedia.org/wiki/Sample_space) of possible outcomes.
### Random Variable Mapping
The random variable $X$ maps from sample space to event space:
- $X(\text{left}) = -L_1$ (where $L_1 > 0$)
- $X(\text{right}) = L_2$
### Event Space (Support)
$R_X = \{-L_1, L_2\}$
### Probability Function
$P: R_X \to [0, 1]$
- $P(X = -L_1) = p_1$
- $P(X = L_2) = p_2$
- By [law of total probability](https://en.wikipedia.org/wiki/Law_of_total_probability): $p_1 + p_2 = 1$
## First Study: The Steps
### Expected Value of Steps
Computing the [expectation](https://en.wikipedia.org/wiki/Expected_value):
$E[X] = \sum_{i=1}^{2} x_i \cdot P(X = x_i)$
$E[X] = (-L_1) \cdot p_1 + L_2 \cdot p_2$
Substituting $p_1 = 1 - p_2$:
$E[X] = -L_1(1 - p_2) + L_2 \cdot p_2$
$E[X] = -L_1 + p_2(L_1 + L_2)$
### Special Case: Symmetric Random Walk
Let $L_1 = L$ and $L_2 = L$ (equal step lengths):
$E[X] = -L + p_2 \cdot 2L = -L + 2p_2L$
**Key Result**: $E[X] = 0$ when $p_2 = \frac{1}{2}$
This represents:
- Equal probability of going left or right (unbiased)
- Equal step lengths in both directions
- Expected displacement of zero per step
### Connection to Walker Statistics
For the walker's position (using [linearity of expectation](https://en.wikipedia.org/wiki/Expected_value#Linearity)):
$E[S_{n+1}] = E\left[S_0 + \sum_{i=0}^{n} X_i\right]$
Since expectation is linear:
$E[S_{n+1}] = E[S_0] + \sum_{i=0}^{n} E[X_i] = S_0 + (n+1)E[X]$
For symmetric random walk with $p_2 = \frac{1}{2}$:
$E[S_{n+1}] = S_0$
The expected position remains at the starting point!
## Visual Example: Plinko Board
The lecture ended with a video of the game show "The Price is Right" featuring [Plinko](https://en.wikipedia.org/wiki/Plinko) (also known as a [Galton board](https://en.wikipedia.org/wiki/Galton_board)):
- Chip drops and hits pegs
- At each peg: chance to go left or right
- Multiple chips approximate a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution)
- Physical demonstration of [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem)
## Check Your Understanding
1. What happens to the expected value of a random walk when step probabilities are unequal?
2. Why is the variance of sums more complicated than the expectation of sums? (explored in [[MATH310S26-Day24-Notes|Day 24]])
3. How does the accumulation of many random steps lead to normal distribution? (connects to [[MATH310S26-Day25-Notes|Day 25]])
## Key Takeaways
- **Transition from deterministic to stochastic**: Fourier analysis deals with deterministic signals; random walks introduce randomness
- **Noise as accumulation**: Many small, unknown effects can be modeled as random process
- **Mathematical framework**: Random walks provide rigorous way to study stochastic processes
- **Simple walk characteristics**: Even with randomness, we can derive expected values and other statistics
## Mathematical Connections
- Links [[MATH310S26-Day22-WorkdayMaterials (Cross-correlation and signal detection)|Day 22 cross-correlation]] to noise analysis
- Fourier modes used in noise generation (white, pink, brown)
- Sets foundation for understanding [Brownian motion](https://en.wikipedia.org/wiki/Brownian_motion) and [Wiener process](https://en.wikipedia.org/wiki/Wiener_process)
- Prepares for [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem) applications
- Connection to [Markov chains](https://en.wikipedia.org/wiki/Markov_chain) and [Markov processes](https://en.wikipedia.org/wiki/Markov_process)
## Next Class Preview
- [[MATH310S26-Day24-Notes|Day 24]]: Variance calculations for random walks
- Explore distribution of walker positions over time
- Connect to central limit theorem and normal distributions