Probability & Statistics
for Machine Learning and Quantitative Research

§ IFoundations of Probability

A probability space is a triple $(\Omega, \mathcal{F}, P)$: a sample space $\Omega$ of outcomes, a $\sigma$-algebra $\mathcal{F}$ of admissible events, and a measure $P$ assigning each event a number in $[0, 1]$. The structure looks abstract but it buys us two things: a rigorous notion of "event" (so we can talk about countable unions without contradictions) and the ability to condition cleanly.

Axioms

• $P(A) \ge 0$ for every $A \in \mathcal{F}$.
• $P(\Omega) = 1$.
• For disjoint $A_1, A_2, \ldots$: $P\!\left(\bigcup_i A_i\right) = \sum_i P(A_i)$.

Conditioning and independence

When $P(B) \gt 0$, $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$. Two events are independent iff $P(A \cap B) = P(A)\, P(B)$, equivalently $P(A \mid B) = P(A)$. Independence is an assumption — never an observation.

The two workhorses

Law of total probability

For a partition $\{B_i\}$ of $\Omega$: $\ P(A) = \sum_i P(A \mid B_i)\, P(B_i)$.

Bayes' theorem

$P(B \mid A) = \dfrac{P(A \mid B)\, P(B)}{P(A)} = \dfrac{P(A \mid B)\, P(B)}{\sum_i P(A \mid B_i)\, P(B_i)}$.

Chain rule

$P(A_1 \cap \cdots \cap A_n) = \prod_{i=1}^{n} P(A_i \mid A_1, \ldots, A_{i-1})$.

§ IIRandom Variables

A random variable $X: \Omega \to \mathbb{R}$ is a measurable function. In practice we never care about $\Omega$ itself — we work with the induced distribution of $X$ through its CDF $F_X(x) = P(X \le x)$, which determines every probabilistic statement about $X$.

Moments and the moment-generating function

The $k$-th moment is $\mu_k = \mathbb{E}[X^k]$; the variance $\sigma^2 = \mathrm{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$. The moment-generating function $M_X(t) = \mathbb{E}[e^{tX}]$, when it exists in a neighborhood of $0$, determines the distribution uniquely and satisfies $M_X^{(k)}(0) = \mu_k$. For sums of independents, $M_{X+Y}(t) = M_X(t)\, M_Y(t)$.

Core identities

Linearity

$\mathbb{E}[aX + bY + c] = a\, \mathbb{E}[X] + b\, \mathbb{E}[Y] + c$ — always, no independence needed.

Variance of an affine map

$\mathrm{Var}(aX + b) = a^2\, \mathrm{Var}(X)$.

Variance of a sum

$\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\,\mathrm{Cov}(X, Y)$.

Change of variables

If $Y = g(X)$ with $g$ smooth and monotone, $f_Y(y) = f_X(g^{-1}(y))\, \bigl|\tfrac{d}{dy} g^{-1}(y)\bigr|$.

§ IIIA Menagerie of Distributions

Most of applied probability is the fluent recognition of a handful of distributional shapes. The table below lists the ones worth knowing by heart — their parameters, first two moments, and the situations in which they naturally arise.

Multivariate Normal

$\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with density $f(\mathbf{x}) = (2\pi)^{-d/2}|\boldsymbol{\Sigma}|^{-1/2} \exp\!\bigl(-\tfrac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{\!\top}\boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\bigr)$. Two key facts carry most of the weight in ML:

• Affine closure. If $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, then $A\mathbf{X} + \mathbf{b} \sim \mathcal{N}(A\boldsymbol{\mu} + \mathbf{b},\, A\boldsymbol{\Sigma} A^{\!\top})$.
• Marginals & conditionals are Gaussian — closed forms in §4.

§ IVJoint, Marginal, Conditional

For a pair $(X, Y)$ the joint density $f_{X,Y}(x, y)$ generates the marginals by integration and the conditionals by division:

$X$ and $Y$ are independent iff $f_{X,Y}(x, y) = f_X(x)\, f_Y(y)$.

Covariance and correlation

Covariance

$\mathrm{Cov}(X, Y) = \mathbb{E}[(X - \mu_X)(Y - \mu_Y)] = \mathbb{E}[XY] - \mu_X \mu_Y$.

Correlation

$\rho(X, Y) = \mathrm{Cov}(X, Y) / (\sigma_X \sigma_Y) \in [-1, 1]$.

Bilinearity

$\mathrm{Cov}(aX + b,\, cY + d) = ac\, \mathrm{Cov}(X, Y)$.

Covariance matrix

$\boldsymbol{\Sigma}_{ij} = \mathrm{Cov}(X_i, X_j)$; always symmetric positive semidefinite.

Correlation measures linear dependence only. Independent $\Rightarrow$ uncorrelated; the converse fails in general (Gaussians excepted).

Conditional expectation

$\mathbb{E}[Y \mid X]$ is itself a random variable — a function of $X$. Two identities carry most of the analytic weight:

Tower / Adam's law

$\mathbb{E}[Y] = \mathbb{E}\!\bigl[\mathbb{E}[Y \mid X]\bigr]$.

Eve's law (total variance)

$\mathrm{Var}(Y) = \mathbb{E}\!\bigl[\mathrm{Var}(Y \mid X)\bigr] + \mathrm{Var}\!\bigl(\mathbb{E}[Y \mid X]\bigr)$.

The second decomposition is a decomposition of uncertainty: noise within each conditional world, plus variation across worlds.

Gaussian conditionals

Partition $\mathbf{X} = (\mathbf{X}_1, \mathbf{X}_2) \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with the obvious block structure. Then:

§ VInequalities & Concentration

Inequalities are the skeleton on which proofs in learning theory hang. The four below cover the vast majority of everyday use; the remaining two are the tools of the high-dimensional trade.

Markov

For $X \ge 0$ and $a \gt 0$: $P(X \ge a) \le \mathbb{E}[X] / a$.

Chebyshev

$P(|X - \mu| \ge k\sigma) \le 1/k^2$.

Jensen

If $\varphi$ is convex, $\varphi(\mathbb{E}[X]) \le \mathbb{E}[\varphi(X)]$.

Cauchy–Schwarz

$(\mathbb{E}[XY])^2 \le \mathbb{E}[X^2]\, \mathbb{E}[Y^2]$.

High-dimensional tools

Hoeffding

For independent $X_i \in [a_i, b_i]$ and $S = \sum X_i$: $P\bigl(|S - \mathbb{E}S| \ge t\bigr) \le 2 \exp\!\Bigl(\!-\dfrac{2t^2}{\sum_i (b_i - a_i)^2}\Bigr).$

Chernoff

$P(S \ge t) \le \inf_{\lambda \gt 0} e^{-\lambda t} M_S(\lambda)$ — the mother of all exponential tail bounds.

§ VIConvergence & Limit Theorems

Four modes of stochastic convergence, from strongest to weakest:

Almost sure

$X_n \xrightarrow{\text{a.s.}} X$ iff $P(\lim_n X_n = X) = 1$.

In probability

$X_n \xrightarrow{P} X$ iff $P(|X_n - X| \gt \varepsilon) \to 0$ for every $\varepsilon \gt 0$.

In $L^p$

$X_n \xrightarrow{L^p} X$ iff $\mathbb{E}[|X_n - X|^p] \to 0$.

In distribution

$X_n \xrightarrow{d} X$ iff $F_{X_n}(x) \to F_X(x)$ at every continuity point of $F_X$.

The implications: a.s. $\Rightarrow$ in probability $\Rightarrow$ in distribution. $L^p$ convergence for $p \ge 1$ implies convergence in probability.

Laws of large numbers

For i.i.d. $X_1, X_2, \ldots$ with finite mean $\mu$, let $\bar{X}_n = \tfrac{1}{n} \sum_{i=1}^n X_i$. Then:

• Weak LLN. $\bar{X}_n \xrightarrow{P} \mu$.
• Strong LLN. $\bar{X}_n \xrightarrow{\text{a.s.}} \mu$.

Central Limit Theorem

For i.i.d. $X_i$ with mean $\mu$ and finite variance $\sigma^2$:

Equivalently, $\bar{X}_n \approx \mathcal{N}(\mu, \sigma^2 / n)$ for large $n$. The CLT is the reason Gaussian noise is everywhere: averaging destroys the fingerprints of the underlying distribution.

Delta method

If $\sqrt{n}(\hat\theta_n - \theta) \xrightarrow{d} \mathcal{N}(0, \sigma^2)$ and $g$ is differentiable at $\theta$ with $g'(\theta) \ne 0$, then:

Slutsky

If $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{P} c$ (constant), then $X_n + Y_n \xrightarrow{d} X + c$ and $X_n Y_n \xrightarrow{d} cX$. Essential for constructing asymptotic confidence intervals when nuisance parameters are estimated.

§ VIIEstimation

An estimator $\hat\theta = T(X_1, \ldots, X_n)$ is a function of the data intended to approximate a parameter $\theta$. We judge it by:

Bias

$\mathrm{Bias}(\hat\theta) = \mathbb{E}[\hat\theta] - \theta$.

Variance

$\mathrm{Var}(\hat\theta) = \mathbb{E}[(\hat\theta - \mathbb{E}[\hat\theta])^2]$.

Mean squared error

$\mathrm{MSE}(\hat\theta) = \mathbb{E}[(\hat\theta - \theta)^2] = \mathrm{Bias}(\hat\theta)^2 + \mathrm{Var}(\hat\theta)$.

Consistency

$\hat\theta_n \xrightarrow{P} \theta$.

Efficiency

Attains the Cramér–Rao lower bound asymptotically.

Maximum likelihood

Given $X_1, \ldots, X_n \sim p(\cdot \mid \theta)$, the log-likelihood is $\ell(\theta) = \sum_{i=1}^n \log p(X_i \mid \theta)$. The MLE is $\hat\theta_{\text{MLE}} = \arg\max_\theta\, \ell(\theta)$. Under regularity conditions:

where $\mathcal{I}(\theta) = -\mathbb{E}\!\left[\dfrac{\partial^2 \log p(X \mid \theta)}{\partial \theta^2}\right]$ is the Fisher information per observation. The MLE is asymptotically unbiased, consistent, and efficient.

Cramér–Rao bound

For any unbiased estimator $\hat\theta$ of $\theta$:

A universal floor on the variance — no cleverness recovers information that the likelihood didn't contain.

Method of moments

Set sample moments equal to theoretical moments and solve for $\theta$. Simpler than MLE, generally less efficient, but always a sanity check.

§ VIIIHypothesis Testing & Intervals

Anatomy of a test

A null hypothesis $H_0$ and an alternative $H_1$. A test statistic $T$ and a rejection region $R$. Two species of error:

• Type I ($\alpha$): reject $H_0$ when it is true. The significance level.
• Type II ($\beta$): fail to reject $H_0$ when $H_1$ is true. Power is $1 - \beta$.

The $p$-value is the probability, under $H_0$, of observing a statistic at least as extreme as the one observed. It is not the probability that $H_0$ is true — a distinction that has broken more inferences than any other.

The canonical tests

Z-test

Variance known: $Z = (\bar X - \mu_0)/(\sigma/\sqrt n) \sim \mathcal{N}(0,1)$ under $H_0$.

t-test

Variance estimated: $T = (\bar X - \mu_0)/(S/\sqrt n) \sim t_{n-1}$ under $H_0$.

Likelihood ratio

$\Lambda = \sup_{\theta \in \Theta_0} L(\theta) / \sup_{\theta \in \Theta} L(\theta)$. Reject for small $\Lambda$. Wilks: $-2 \log \Lambda \xrightarrow{d} \chi^2_k$ under $H_0$.

The Neyman–Pearson lemma tells us that for simple-versus-simple testing, the likelihood ratio test is uniformly most powerful.

Confidence intervals

A $100(1-\alpha)\%$ confidence interval $[L(X), U(X)]$ for $\theta$ satisfies $P_\theta(L \le \theta \le U) \ge 1 - \alpha$ for all $\theta$. The interval is random; $\theta$ is fixed.

For $\bar X$ from $\mathcal{N}(\mu, \sigma^2)$ (or large $n$ via CLT):

§ IXBayesian Inference

The Bayesian views $\theta$ as random, encoding belief via a prior $\pi(\theta)$. Data $X$ updates it through the likelihood $p(X \mid \theta)$ into a posterior:

Point estimates from the posterior

• Posterior mean. $\hat\theta = \mathbb{E}[\theta \mid X]$ — minimises posterior MSE.
• Posterior median. Minimises posterior absolute error.
• MAP. $\hat\theta_{\text{MAP}} = \arg\max_\theta\, \pi(\theta \mid X)$ — minimises posterior $0$–$1$ loss.

Conjugate priors — worth knowing cold

Credible intervals and predictives

A $(1-\alpha)$ credible interval is any interval of posterior mass $1 - \alpha$. Interpretively cleaner than a confidence interval — here $\theta$ really is the random thing.

The posterior predictive for new data $\tilde x$ is:

which averages over parameter uncertainty instead of collapsing it to a point.

§ XInformation Theory

Modern machine learning is not merely influenced by information theory — it often is information theory in a different hat. Loss functions are divergences; training is a minimum description length problem; attention is mutual information, softly.

Entropy

The Shannon entropy of a discrete $X$ with PMF $p$ is:

It is the expected surprise, in nats (base $e$) or bits (base $2$). Maximised by the uniform distribution; zero when $X$ is deterministic. For continuous $X$ with PDF $f$, the differential entropy $h(X) = -\int f(x) \log f(x)\, dx$ — useful, but not an entropy in the strict sense (it can be negative).

Joint, conditional, mutual

Joint

$H(X, Y) = -\sum p(x, y) \log p(x, y)$.

Conditional

$H(Y \mid X) = H(X, Y) - H(X) = \mathbb{E}_X\!\bigl[H(Y \mid X = x)\bigr]$.

Mutual information

$I(X; Y) = H(X) + H(Y) - H(X, Y) = H(Y) - H(Y \mid X)$.

$I(X; Y) \ge 0$, with equality iff $X \perp Y$. It is the canonical measure of statistical dependence beyond linearity.

KL divergence and cross-entropy

Not a metric — asymmetric and does not satisfy triangle inequality — but non-negative (Gibbs' inequality) and zero iff $p = q$ a.e. The cross-entropy $H(p, q) = -\sum p(x) \log q(x)$ satisfies $H(p, q) = H(p) + D_{\mathrm{KL}}(p \,\|\, q)$.

The chain of inequalities

$0 \le H(Y \mid X) \le H(Y) \le \log |\mathcal{Y}|$, and the data-processing inequality: if $X \to Y \to Z$ is a Markov chain, $I(X; Z) \le I(X; Y)$. Processing cannot create information.

§ XILinear Models & Bias–Variance

Linear regression models $Y = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}$ with $\boldsymbol{\varepsilon}$ mean-zero and variance $\sigma^2 I$. The ordinary least squares estimator:

is unbiased with covariance $\sigma^2 (\mathbf{X}^{\!\top} \mathbf{X})^{-1}$. The Gauss–Markov theorem asserts it is BLUE — the best linear unbiased estimator — under the stated conditions.

Regularised regression

Ridge

$\hat{\boldsymbol{\beta}}_{\text{ridge}} = \arg\min\,\|\mathbf{Y} - \mathbf{X}\boldsymbol{\beta}\|^2 + \lambda \|\boldsymbol{\beta}\|_2^2 = (\mathbf{X}^{\!\top} \mathbf{X} + \lambda I)^{-1} \mathbf{X}^{\!\top} \mathbf{Y}$.

Lasso

$\hat{\boldsymbol{\beta}}_{\text{lasso}} = \arg\min\,\|\mathbf{Y} - \mathbf{X}\boldsymbol{\beta}\|^2 + \lambda \|\boldsymbol{\beta}\|_1$ — sparsity-inducing.

Both are MAP estimators under Gaussian (ridge) or Laplace (lasso) priors on $\boldsymbol{\beta}$. See Exercise 5.

The bias–variance decomposition

For a fixed test point $x_0$, squared-error risk of an estimator $\hat f$ splits as:

No estimator eliminates the first term; complexity trades the second for the third.

Generalisation risk

Population risk $R(f) = \mathbb{E}_{(X, Y)}[\ell(f(X), Y)]$; empirical risk $\hat R(f) = \tfrac{1}{n} \sum \ell(f(X_i), Y_i)$. Training error underestimates test error systematically; the gap is bounded (with high probability) by the complexity of the function class — VC dimension, Rademacher complexity, metric-entropy arguments.

§ XIIStochastic Processes

Markov chains

A sequence $X_0, X_1, \ldots$ with the Markov property: $P(X_{n+1} \mid X_n, X_{n-1}, \ldots) = P(X_{n+1} \mid X_n)$. For finite state space, the dynamics live in a transition matrix $P$ with $P_{ij} = P(X_{n+1} = j \mid X_n = i)$.

A distribution $\boldsymbol{\pi}$ is stationary if $\boldsymbol{\pi} = \boldsymbol{\pi} P$. An irreducible aperiodic finite chain has a unique stationary distribution, and $P^n \to \mathbf{1} \boldsymbol{\pi}$ — the ergodic theorem.

Detailed balance: if $\pi_i P_{ij} = \pi_j P_{ji}$ for all $i, j$, then $\boldsymbol{\pi}$ is stationary. The starting point for MCMC: design $P$ so detailed balance holds for a target $\boldsymbol{\pi}$.

Poisson process

A counting process $N(t)$ with:

• $N(0) = 0$;
• independent increments;
• $N(t + s) - N(s) \sim$ Poisson$(\lambda t)$.

Equivalently, inter-arrival times are i.i.d. Exp$(\lambda)$. The memorylessness of the exponential is what makes the Poisson process analytically tractable — and a near-universal model for rare independent events.

Brownian motion

A process $W_t$ with $W_0 = 0$, independent Gaussian increments $W_t - W_s \sim \mathcal{N}(0, t-s)$ for $s \lt t$, and continuous sample paths. The basic object of stochastic calculus; the limit of scaled random walks; the noise term in Black–Scholes. Non-differentiable everywhere with probability one — a warning that intuition built on smooth calculus can mislead here.

Martingales

$\{M_n\}$ is a martingale with respect to a filtration $\{\mathcal{F}_n\}$ if $\mathbb{E}[|M_n|] \lt \infty$, $M_n$ is $\mathcal{F}_n$-measurable, and:

Intuitively: a fair game. The optional stopping theorem (under bounded-stopping-time hypotheses) extends this to random stopping times: $\mathbb{E}[M_\tau] = \mathbb{E}[M_0]$. The one-line explanation for why you cannot beat a fair casino — and the workhorse of no-arbitrage pricing.

§ XIIIMeasure-Theoretic Foundations

Everything so far has coasted on an informal reading of $P$ as "a function that assigns probabilities to events". For most working calculations this is enough — but for limit theorems, for infinite-dimensional processes like Brownian motion, and for the no-arbitrage pricing theorems of quantitative finance, we need the proper machinery. The payoff is not mere rigour; it is the ability to swap limits and integrals, change measures, and make sense of integrals against paths that have no derivative anywhere.

$\sigma$-algebras and measurable spaces

A $\sigma$-algebra $\mathcal{F}$ on a set $\Omega$ is a collection of subsets containing $\Omega$ and closed under complementation and countable unions. The pair $(\Omega, \mathcal{F})$ is a measurable space. The canonical example on $\mathbb{R}$ is the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ — the smallest $\sigma$-algebra containing all open sets.

A measure $\mu: \mathcal{F} \to [0, \infty]$ satisfies $\mu(\emptyset) = 0$ and countable additivity: $\mu\!\left(\bigsqcup_i A_i\right) = \sum_i \mu(A_i)$. A probability measure is a measure with $\mu(\Omega) = 1$. Lebesgue measure on $\mathbb{R}$ is the unique translation-invariant Borel measure assigning length $b - a$ to $[a, b]$.

Measurable functions and random variables

$f: (\Omega, \mathcal{F}) \to (\mathbb{R}, \mathcal{B})$ is measurable if $f^{-1}(B) \in \mathcal{F}$ for every $B \in \mathcal{B}$. A random variable is precisely a measurable function on a probability space; the definition we gave in §II was an informal cousin.

The Lebesgue integral

Built in three stages:

1. For simple functions $\varphi = \sum_{i=1}^n c_i \mathbf{1}_{A_i}$: $\int \varphi\, d\mu = \sum_i c_i\, \mu(A_i)$.
2. For non-negative measurable $f$: $\int f\, d\mu = \sup\!\left\{\int \varphi\, d\mu : 0 \le \varphi \le f,\ \varphi \text{ simple}\right\}$.
3. For general $f$: split $f = f^+ - f^-$ and set $\int f = \int f^+ - \int f^-$ whenever at least one is finite.

The Lebesgue integral extends the Riemann integral and behaves much better under limits — which is where its power lives.

The three pillars of convergence

Monotone Convergence

If $0 \le f_n \uparrow f$ a.e., then $\int f_n\, d\mu \to \int f\, d\mu$.

Fatou's Lemma

For non-negative measurable $f_n$: $\int \liminf_n f_n\, d\mu \le \liminf_n \int f_n\, d\mu$.

Dominated Convergence

If $f_n \to f$ a.e.\ and $|f_n| \le g$ with $\int g\, d\mu \lt \infty$, then $\int f_n\, d\mu \to \int f\, d\mu$.

These are the only reliable ways to swap $\lim$ and $\int$. Without them, nearly every probabilistic limit argument stalls.

Fubini–Tonelli

On a product measure space $(\Omega_1 \times \Omega_2, \mathcal{F}_1 \otimes \mathcal{F}_2, \mu_1 \otimes \mu_2)$, iterated integration is legal:

• Tonelli: for non-negative measurable $f$, any order works and all are equal (possibly $+\infty$).
• Fubini: for $f$ absolutely integrable, any order works and gives the same finite value.

Radon–Nikodym

Given $\sigma$-finite measures $\mu, \nu$ on $(\Omega, \mathcal{F})$ with $\mu \ll \nu$ (absolutely continuous: $\nu(A) = 0 \Rightarrow \mu(A) = 0$), there exists an essentially unique measurable $f \ge 0$ with:

We write $f = d\mu / d\nu$, the Radon–Nikodym derivative. A PDF is precisely $dP/d\lambda$ with respect to Lebesgue measure $\lambda$. A likelihood ratio is $dP_1/dP_0$. A pricing kernel is $d\mathbb{Q}/d\mathbb{P}$. Three different dialects for the same object.

Borel–Cantelli

For events $A_1, A_2, \ldots$ let $\{A_n \text{ i.o.}\} = \limsup_n A_n = \bigcap_{N} \bigcup_{n \ge N} A_n$ — the event that infinitely many $A_n$ occur. Then:

• BC I. If $\sum_n P(A_n) \lt \infty$, then $P(A_n \text{ i.o.}) = 0$.
• BC II. If the $A_n$ are independent and $\sum_n P(A_n) = \infty$, then $P(A_n \text{ i.o.}) = 1$.

A zero–one dichotomy that undergirds almost-sure convergence proofs.

§ XIVFiltrations & Martingales

Conditional expectation, rigorously

For a sub-$\sigma$-algebra $\mathcal{G} \subseteq \mathcal{F}$ and $X \in L^1(\Omega, \mathcal{F}, P)$, the conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$ is the $P$-a.s.\ unique $\mathcal{G}$-measurable random variable satisfying:

Existence is guaranteed by Radon–Nikodym applied to the signed measure $A \mapsto \int_A X\, dP$. For $X \in L^2$, $\mathbb{E}[X \mid \mathcal{G}]$ is exactly the orthogonal projection of $X$ onto the closed subspace $L^2(\mathcal{G})$ — conditional expectation is a Hilbert-space projection.

The axioms you will use every day

Linearity

$\mathbb{E}[aX + bY \mid \mathcal{G}] = a\,\mathbb{E}[X \mid \mathcal{G}] + b\,\mathbb{E}[Y \mid \mathcal{G}]$.

Tower

If $\mathcal{H} \subseteq \mathcal{G}$: $\mathbb{E}\!\bigl[\mathbb{E}[X \mid \mathcal{G}] \mid \mathcal{H}\bigr] = \mathbb{E}[X \mid \mathcal{H}]$.

Take out what is known

If $Y$ is $\mathcal{G}$-measurable and $XY \in L^1$: $\mathbb{E}[XY \mid \mathcal{G}] = Y\, \mathbb{E}[X \mid \mathcal{G}]$.

Independence

If $X \perp \mathcal{G}$: $\mathbb{E}[X \mid \mathcal{G}] = \mathbb{E}[X]$.

Conditional Jensen

For convex $\varphi$: $\varphi\!\bigl(\mathbb{E}[X \mid \mathcal{G}]\bigr) \le \mathbb{E}[\varphi(X) \mid \mathcal{G}]$.

Filtrations, adaptedness, stopping times

A filtration is an increasing family $\{\mathcal{F}_t\}_{t \ge 0}$ of sub-$\sigma$-algebras of $\mathcal{F}$. Read $\mathcal{F}_t$ as "the information observable by time $t$". A process $X_t$ is adapted if $X_t$ is $\mathcal{F}_t$-measurable for every $t$.

A stopping time is a $[0, \infty]$-valued random variable $\tau$ with $\{\tau \le t\} \in \mathcal{F}_t$ for every $t$. First-entry times into a Borel set are stopping times; last-exit times generally are not — you would need the future to know.

Martingales, sub- and super-

An adapted, integrable process $M_t$ is:

• a martingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s$ for $s \le t$;
• a submartingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] \ge M_s$;
• a supermartingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] \le M_s$.

Mnemonic: supermartingales go down (in expectation). Your wealth in a casino is a supermartingale. Your wealth at the roulette wheel of a fair game is a martingale.

Doob's maximal inequality

For a non-negative submartingale $X_t$ and $\lambda \gt 0$:

The $L^p$ version ($p \gt 1$):

Remarkably, control of $X_t$ in $L^p$ gives control of the running maximum in $L^p$ — a uniform-in-time bound from a pointwise one.

Optional stopping, with honest hypotheses

For a martingale $M$ and stopping time $\tau$, $\mathbb{E}[M_\tau] = \mathbb{E}[M_0]$ provided at least one of the following:

• $\tau$ is bounded by some constant $T \lt \infty$;
• $M$ is uniformly bounded and $\tau \lt \infty$ a.s.;
• $\mathbb{E}[\tau] \lt \infty$ and the increments $|M_n - M_{n-1}|$ are uniformly bounded.

Without some such condition, the theorem fails: a simple random walk is a martingale, and $\tau = \inf\{n : S_n = 1\}$ is finite a.s., yet $\mathbb{E}[S_\tau] = 1 \ne 0 = \mathbb{E}[S_0]$. The failure: $\mathbb{E}[\tau] = \infty$.

Martingale convergence

An $L^1$-bounded martingale — one with $\sup_t \mathbb{E}[|M_t|] \lt \infty$ — converges $P$-a.s.\ to a limit $M_\infty \in L^1$. In particular, a non-negative supermartingale always converges a.s. These are tools for proving a.s.\ convergence of algorithms (stochastic approximation, likelihood ratios, posteriors).

§ XVBrownian Motion & Itô Calculus

Brownian motion, properly

A standard Brownian motion (Wiener process) $\{W_t\}_{t \ge 0}$ on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P)$ satisfies:

• $W_0 = 0$ a.s.;
• independent increments: for $s \lt t$, $W_t - W_s$ is independent of $\mathcal{F}_s$;
• Gaussian increments: $W_t - W_s \sim \mathcal{N}(0, t - s)$;
• continuous sample paths a.s.

Existence is a theorem (Wiener 1923): several constructions work — Lévy's piecewise-linear approximation, Kolmogorov's extension via finite-dimensional distributions, or Donsker's invariance principle (random walks, rescaled, converge in distribution to $W$).

Pathological pathwise regularity

With probability one:

• $t \mapsto W_t$ is continuous but nowhere differentiable;
• $W$ has infinite total variation on every interval;
• the quadratic variation is finite and deterministic: $\langle W \rangle_t = t$, meaning $\sum_i (W_{t_{i+1}} - W_{t_i})^2 \xrightarrow{L^2} t$ as the partition mesh shrinks.

The finite quadratic variation is the whole key. It is what prevents a classical Riemann–Stieltjes integral $\int H\, dW$ from existing, and what forces the Itô construction.

Useful symmetries

If $W$ is standard Brownian motion, so are: $-W_t$ (reflection); $W_{t+s} - W_s$ (Markov property); $\sqrt{c}\, W_{t/c}$ (scaling); $t\, W_{1/t}$ for $t \gt 0$ (time inversion). These identities are how closed-form quantities get computed.

The Itô integral

For an adapted process $H$ with $\mathbb{E}\!\int_0^T H_s^2\, ds \lt \infty$, the Itô integral $\int_0^t H_s\, dW_s$ is constructed as the $L^2$-limit of "left-endpoint" Riemann sums $\sum_i H_{t_i}(W_{t_{i+1}} - W_{t_i})$. The choice of left endpoint is not cosmetic: evaluating at the midpoint produces the Stratonovich integral, which behaves like ordinary calculus but destroys the martingale property.

Itô isometry (the identity that makes the whole edifice work):

As a consequence, $M_t := \int_0^t H_s\, dW_s$ is a mean-zero martingale with variance $\mathbb{E}\!\int_0^t H_s^2\, ds$.

Itô's formula

Let $X_t$ be an Itô process: $dX_t = \mu_t\, dt + \sigma_t\, dW_t$. For $f \in C^{1,2}([0, T] \times \mathbb{R})$:

The extra half-second-derivative term — sometimes called the Itô correction — is the mathematical signature of quadratic variation. In an informal calculus: $(dW_t)^2 = dt$.

Geometric Brownian motion

The SDE $dS_t = \mu\, S_t\, dt + \sigma\, S_t\, dW_t$ has closed-form solution:

Apply Itô to $f(x) = \log x$ to see where the $-\tfrac{1}{2}\sigma^2$ comes from. This is the foundational model of log-normal asset prices in Black–Scholes theory.

Girsanov's theorem

Given an adapted process $\theta_t$ satisfying Novikov's condition $\mathbb{E}\exp\!\bigl(\tfrac{1}{2}\int_0^T \theta_s^2\, ds\bigr) \lt \infty$, define a new measure $\mathbb{Q}$ by:

Then $\widetilde W_t := W_t + \int_0^t \theta_s\, ds$ is a standard Brownian motion under $\mathbb{Q}$. Drift is not intrinsic to the process — it is a feature of the measure.