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Category: Probability & Distributions

Probability & Distributions in applied mathematics and stochastic physics explores probability measure theory, continuous and discrete probability distributions, Brownian motion, stochastic differential equations, and functional central limit theorems (Donsker’s theorem).

Foundational Pillars
Principles & Methods
Topic Directory
Frequently Asked Questions

Foundational Pillars (Start Here)

STOCHASTIC CALCULUS

Ito Formula Brownian Cube: Stochastic Calculus Explained

Brownian motion quadratic variation, Ito integration, stochastic differential equations, and higher-order moments of Brownian paths.

LIMIT THEOREMS

Donsker’s Theorem: Weak Convergence and Subsequences

Functional Central Limit Theorem, weak convergence in Skorokhod space D[0,1], and scaled random walks converging to Wiener processes.

STOCHASTIC BRIDGES

Brownian Bridge Rayleigh: Distribution Applications and Theory

Conditioned Wiener processes pinned at start and end points, maximum excursion distributions, and connections to the Rayleigh distribution.

Wiener Processes, Brownian Bridges, and Invariance Principles

A standard Wiener process W_t is characterized by three core axioms: W_0 = 0, almost surely continuous sample paths, and independent Gaussian increments W_t – W_s ~ N(0, t – s). A Brownian Bridge B_t on interval [0,1] conditions the Wiener process to pin at zero at both endpoints: B_t = W_t – t · W_1. The maximum excursion of a Brownian Bridge, M = sup_{t ∈ [0,1]} |B_t|, directly dictates the asymptotic distribution of the Kolmogorov-Smirnov statistic.

Stochastic Process / TheoremMathematical FormulationCore Asymptotic PropertyPhysics / Mathematical Application
Standard Wiener ProcessW_t – W_s ~ N(0, t – s) ; (dW_t)² = dtContinuous nowhere differentiable Gaussian martingaleBrownian diffusion in fluids, thermal noise in resistors (Johnson noise)
Brownian BridgeB_t = W_t – t · W_1, t ∈ [0, 1]Conditioned Gaussian process with covariance s(1 – t)Non-parametric empirical distribution tests and boundary crossing modeling
Donsker’s Invariance TheoremX_n(t) = (1/√n) · Σ ξ_i ⇒ W_t in C[0,1]Functional Central Limit Theorem under weak convergenceProving universality of continuous approximations for discrete random walks
Lognormal Survival Modelsln(T) ~ N(μ, σ²)Multiplicative degradation processes with heavy right skewRadiobiological radiation survival curves and component wear fatigue

In modern computational physics and machine learning, frameworks like JAX accelerate probabilistic sampling via functional PRNG keys. In JAX, pseudo-random number generation is stateless: `key, subkey = jax.random.split(key)` guarantees reproducible, vector-parallel sampling of Bernoulli and Gaussian distributions across GPU cores without race conditions.

Topic Directory & Methodologies

Stochastic Processes & Calculus

Sampling, Distributions & Biophysics

Frequently Asked Questions

What is Donsker’s theorem and why is it called the functional CLT?

The classical Central Limit Theorem states that the normalized sum of i.i.d. random variables converges to a normal random variable. Donsker’s theorem extends this from numbers to continuous functions: the random walk path formed by partial sums converges in distribution to an entire continuous Wiener process W_t.

Why is the quadratic variation of Brownian motion non-zero?

Brownian paths are continuous everywhere but differentiable nowhere, exhibiting infinite total variation. Over any time interval Δt, the sum of squared increments Σ (ΔW_i)² converges in probability to the elapsed time Δt as the partition mesh goes to zero, meaning (dW_t)² = dt in stochastic calculus.

What is a Brownian Bridge used for in statistics?

A Brownian Bridge models the difference between an empirical cumulative distribution function (ECDF) and the true cumulative distribution function (CDF). Because both F_n(x) and F(x) match at -∞ (value 0) and +∞ (value 1), the normalized discrepancy behaves precisely as a Brownian bridge pinned at zero at both boundaries.

Why is JAX pseudo-random number generation stateless?

Traditional Python libraries (like NumPy or random) maintain an implicit global PRNG state. In multi-threaded GPU/TPU environments, parallel access to a mutable global state causes race conditions. JAX requires explicit, immutable PRNG keys that split functionally, guaranteeing identical outputs on parallel hardware.