Filtrations and the Central Limit Theorem

Worked Solution

How to Think About It: These are two foundational concepts that come up in almost every quant interview at some level. For filtrations, the key is the idea of "information growing over time" -- think of it as watching a stock price tick by tick, where at each moment you know everything that has happened so far but nothing about the future. For the CLT, the key is understanding exactly which assumptions you need and what breaks when you relax them.

Key Insight: A filtration formalizes the concept of "what you know at time $t$," and it is the backbone of stochastic calculus and options pricing. The CLT tells you when sums of random variables look Gaussian, and knowing its assumptions tells you when you can and cannot rely on normal approximations.

The Method:

Part 1: Filtration

A filtration on a probability space $(\Omega, \mathcal{F}, P)$ is a family of sigma-algebras $\{\mathcal{F}_t\}_{t \geq 0}$ satisfying:

$\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} \quad \text{for all } s \leq t$

Intuition: $\mathcal{F}_t$ is the collection of all events whose occurrence or non-occurrence you can determine at time $t$. As time progresses, you learn more, so the sigma-algebra grows. You never "forget" information.

Key related concepts:

• Adapted process: A stochastic process $X_t$ is adapted to $\{\mathcal{F}_t\}$ if $X_t$ is $\mathcal{F}_t$-measurable for every $t$. In plain language: the value of $X_t$ is knowable at time $t$.

• Natural filtration: $\mathcal{F}_t = \sigma(X_s : s \leq t)$ -- the smallest sigma-algebra that makes all past values of $X$ measurable. This is the "no extra information" filtration.

• Martingale: An adapted, integrable process satisfying $E[X_t \mid \mathcal{F}_s] = X_s$ for $s < t$. The best forecast of the future value, given current information, is the current value.

Part 2: Central Limit Theorem

Classical (Lindeberg-Levy) CLT:

Let $X_1, X_2, \dots$ be i.i.d. random variables with mean $\mu$ and finite variance $\sigma^2 < \infty$. Then:

$\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1) \quad \text{as } n \to \infty$

where $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$.

Three required assumptions: 1. Independence of the $X_i$ 2. Identical distribution (same mean and variance) 3. Finite variance ($\sigma^2 < \infty$)

Lindeberg-Feller generalization: Drops the "identically distributed" requirement. Instead, for independent (not necessarily identically distributed) $X_i$ with $E[X_i] = \mu_i$ and $\text{Var}(X_i) = \sigma_i^2$, let $s_n^2 = \sum_{i=1}^n \sigma_i^2$. The CLT holds if the Lindeberg condition is satisfied: for every $\epsilon > 0$,

$\frac{1}{s_n^2} \sum_{i=1}^n E\left[(X_i - \mu_i)^2 \, \mathbf{1}\{|X_i - \mu_i| > \epsilon \, s_n\}\right] \to 0$

This ensures no single variable dominates the sum.

Practical Considerations:

• Filtrations are essential for pricing derivatives: the price of an option at time $t$ must be $\mathcal{F}_t$-measurable (you can only use information available now).
• The CLT justifies the widespread use of normal approximations in risk management, but it requires finite variance. For heavy-tailed data (e.g., Cauchy), the CLT does not apply and stable distributions arise instead.
• In finite samples, the CLT is an approximation. Berry-Esseen bounds quantify the approximation error: $O(1/\sqrt{n})$ for i.i.d. variables with finite third moment.

Answer: A filtration is an increasing family of sigma-algebras representing growing information over time; it underpins the definitions of adapted processes and martingales. The CLT requires independent, identically distributed random variables with finite variance, and states that the standardized sample mean converges in distribution to $N(0,1)$. The Lindeberg-Feller version relaxes identical distribution but imposes the Lindeberg condition to prevent any single variable from dominating.