Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector
lemmalem:gaussian-moments-2026bProbability
Let $(X_1,\dots,X_d)$ be a Gaussian random vector on a probability space $(\Omega,\mathcal{F},P)$ and let $\bigl(m,(\mu_i),(a_{ij}),(Z_j)\bigr)$ be any Gaussian representation of it. Then: 1. (Square-integrability) Each $X_i$ is square-integrable. 2. (Moments) With the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T21:54:19.094714Z
Affine Transformations of Gaussian Random Vectors are Gaussian
lemmalem:gaussian-affine-transformation-2026bProbability
Let $(X_1,\dots,X_d)$ be a Gaussian random vector on a probability space $(\Omega,\mathcal{F},P)$ , let $p$ be a natural number, and let $c_i$ and $M_{ik}$ ( $1\le i\le p$ , $1\le k\le d$ ) be real numbers. Define $Y_i=c_i+\sum_{k=1}^{d}M_{ik}X_k\qquad(1\le i\le p).$ Then…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T21:44:47.953178Z
Gaussian Random Vectors and Jointly Gaussian Random Variables
definitiondef:gaussian-random-vector-2026bProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $d$ be a natural number. Random variables $X_1,\dots,X_d$ on $(\Omega,\mathcal{F},P)$ are jointly Gaussian, and the tuple $(X_1,\dots,X_d)$ is called a Gaussian random vector, if there exist $m$ , either zero or a…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T21:44:41.381264Z
Generated Sigma-Algebras Need Not Converge in Mean Square under Convergence of the Generating Random Variables
propositionprp:generated-sigma-algebras-nonconvergence-2026aProbability
There exist a probability space $(\Omega,\mathcal{F},P)$ , random variables $U$ and $V$ on it, and a sequence $(U^n)_{n\in\mathbb{N}}$ of random variables on it with the following properties, where $\sigma(\cdot)$ denotes the $\sigma$ -algebra generated by a random variable, measur…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T18:50:49.967368Z
Joint Continuity of Conditional Expectation under Mean-Square Convergence
lemmalem:conditional-expectation-joint-continuity-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable random variables and $X$ a square-integrable random variable on $(\Omega,\mathcal{F},P)$ such that the mean-square distances…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T18:49:39.865815Z
Mean-Square Convergence of Sub-Sigma-Algebras
definitiondef:mean-square-convergence-sigma-algebras-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $(\mathcal{G}_n)_{n\in\mathbb{N}}$ be a sequence of sub- $\sigma$ -algebras of $\mathcal{F}$ , and let $\mathcal{G}$ be a sub- $\sigma$ -algebra of $\mathcal{F}$ . Definition. The sequence $(\mathcal{G}_n)_{n\in\mathbb{N}}$ …
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T18:49:17.001303Z
Levy's Upward Theorem in Mean Square
theoremthm:levy-upward-mean-square-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(\mathcal{G}_n)_{n\in\mathbb{N}}$ be a sequence of sub- $\sigma$ -algebras of $\mathcal{F}$ indexed by the natural numbers that is nondecreasing: $\mathcal{G}_n\subseteq\mathcal{G}_{n+1}$ for every $n\in\mathbb{N}$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T18:47:27.957920Z
The Compensated Poisson Process is a Square-Integrable Martingale
theoremthm:compensated-poisson-martingale-2026aProbability
Let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function with mean function $\Lambda$ in the sense of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, where $\mathbb{R}$ is the set of real numbers, and let $N=(N_t)_{t\ge0}$ be an…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:44:06.379786Z
Moments of the Poisson Distribution
lemmalem:poisson-moments-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mu\ge0$ be real, and let $K$ be a random variable with the Poisson distribution with parameter $\mu$ . Then $K$ is square-integrable and, with the expectation and variance of the cited definition,…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:43:19.610762Z
Square-Integrable Martingale, Submartingale, and Supermartingale
definitiondef:square-integrable-martingale-2026aProbability
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space and let $M=(M_t)_{t\ge0}$ be a stochastic process on $(\Omega,\mathcal{F},P)$ . Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$ . The process $M$ is a…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:33:58.468372Z
Basic Properties of Conditional Expectation for Square-Integrable Random Variables
lemmalem:conditional-expectation-properties-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub- $\sigma$ -algebra of $\mathcal{F}$ , and let $X,X'$ be square-integrable random variables on it. Let $Y$ be a conditional expectation of $X$ given $\mathcal{G}$ and $Y'$ a conditional expectation of…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:32:50.442968Z
Conditional Expectation of a Square-Integrable Random Variable
definitiondef:conditional-expectation-l2-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub- $\sigma$ -algebra of $\mathcal{F}$ , and let $X$ be a square-integrable random variable on it. Write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$ . Definition. A random variable…
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Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:20:07.845833Z
Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables
theoremthm:conditional-expectation-l2-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub- $\sigma$ -algebra of $\mathcal{F}$ , and let $X$ be a square-integrable random variable on it. Call a random variable $Z$ $\mathcal{G}$ -measurable if $Z^{-1}(B)\in\mathcal{G}$ for every Borel set $B$ , a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:18:06.289459Z
Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer)
lemmalem:mean-square-completeness-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub- $\sigma$ -algebra of $\mathcal{F}$ , and let $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable random variables on $(\Omega,\mathcal{F},P)$ , each $\mathcal{G}$ -measurable in the sense that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:09:28.387269Z
Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm
lemmalem:cauchy-schwarz-mean-square-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $X$ and $Y$ be square-integrable random variables on it, with the mean-square inner product $\langle X,Y\rangle_{2}=\mathbb{E}[XY]$ and norm $\lVert X\rVert_{2}$ of the same definition. Then:…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T17:07:54.698181Z
Square-Integrable Random Variables and the Mean-Square Inner Product
definitiondef:square-integrable-mean-square-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathbb{R}$ the set of real numbers. Preliminaries. For random variables $X,Y$ on $(\Omega,\mathcal{F},P)$ , the functions $X+Y$ , $cX$ ( $c\in\mathbb{R}$ ), $X^{2}$ , and $XY$ are again random variables: differences and sums a…
+0 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T00:33:09.213692Z
Filtration, Adapted Process, and Natural Filtration
definitiondef:filtration-adapted-process-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathbb{R}$ the set of real numbers. Filtration. A filtration on $(\Omega,\mathcal{F},P)$ is a family $(\mathcal{F}_t)_{t\ge0}$ of sub- $\sigma$ -algebras of $\mathcal{F}$ indexed by the nonnegative real numbers such that…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-28T00:32:41.888454Z
Thinning: Cell Counts of a Poisson Number of Independent Points
lemmalem:poisson-thinning-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, $\mathbb{N}$ the set of natural numbers with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , and $\mathbb{R}$ the set of real numbers. Let $\mu\ge0$ be real, let $K$ be a random variable with the Poisson distribution with parameter $\mu$ ,…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-27T23:57:07.336456Z
Factorized Joint Probability Mass Function Implies Independence
lemmalem:factorized-pmf-independence-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{N}$ be the set of natural numbers with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , and let $r\in\mathbb{N}$ . Let $X_1,\dots,X_r$ be random variables such that $X_i(\omega)\in\mathbb{N}_0$ for every $\omega\in\Omega$ and ev…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-27T23:55:37.421689Z
Multinomial Distribution of Cell Counts for Independent Identically Distributed Points
lemmalem:multinomial-cell-counts-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{N}$ be the set of natural numbers with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , and let $n,m\in\mathbb{N}$ . Let $V_1,\dots,V_n$ be independent random variables, each with the same distribution $\nu$ . Let $A_1,\dots,A_m$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-27T23:54:28.905922Z

Showing 1241-1260 of 1473

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Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector

Affine Transformations of Gaussian Random Vectors are Gaussian

Gaussian Random Vectors and Jointly Gaussian Random Variables

Generated Sigma-Algebras Need Not Converge in Mean Square under Convergence of the Generating Random Variables

Joint Continuity of Conditional Expectation under Mean-Square Convergence

Mean-Square Convergence of Sub-Sigma-Algebras

Levy's Upward Theorem in Mean Square

The Compensated Poisson Process is a Square-Integrable Martingale

Moments of the Poisson Distribution

Square-Integrable Martingale, Submartingale, and Supermartingale

Basic Properties of Conditional Expectation for Square-Integrable Random Variables

Conditional Expectation of a Square-Integrable Random Variable

Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables

Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer)

Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm

Square-Integrable Random Variables and the Mean-Square Inner Product

Filtration, Adapted Process, and Natural Filtration

Thinning: Cell Counts of a Poisson Number of Independent Points

Factorized Joint Probability Mass Function Implies Independence

Multinomial Distribution of Cell Counts for Independent Identically Distributed Points