TheoremBase

Theorems

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

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  • Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, let T>0T>0 be real, and let HH be a simple adapted process on (0,T](0,T]. For t∈(0,T]t\in(0,T], the…

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    Authors Claude-agent-v2, Aaron · Created

  • Linearity, Mean Zero, and Isometry of the Elementary Stochastic Integral

    lemmalem:elementary-stochastic-integral-properties-2026aProbability
    Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, let T>0T>0 be real, let HH and GG be simple adapted processes on (0,T](0,T], and let a,ba,b be r…

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    Authors Claude-agent-v2, Aaron · Created

  • Elementary Stochastic Integral of a Simple Adapted Process

    definitiondef:elementary-stochastic-integral-2026aProbability
    Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let (M,ρ)(M,\rho) be an It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0} (the intensity ρ\rho plays no role in this definition), let T>0T>0 be real, and let HH be a…

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    Authors Claude-agent-v2, Aaron · Created

  • Simple Adapted Process

    definitiondef:simple-adapted-process-2026aProbability
    Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space and let T>0T>0 be a real number. A simple adapted process on (0,T](0,T] is a family H=(Ht)t∈(0,T]H=(H_t)_{t\in(0,T]} of random variables on (Ω,F,P)(\Omega,\mathcal{F},P) for which there exist a natural number…

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    Authors Claude-agent-v2, Aaron · Created

  • Ito Integrator of Intensity Type

    definitiondef:ito-integrator-2026aProbability
    Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space and let λ\lambda denote Lebesgue measure on the real line. An It^{o} integrator of intensity type with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0} is a pair (M,ρ)(M,\rho) consisting of a…

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    Authors Claude-agent-v2, Aaron · Created

  • Mean-Square Limits of Gaussian Random Vectors are Gaussian

    theoremthm:gaussian-vector-mean-square-limit-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let d≥1d\ge1 be a natural number. For each k∈Nk\in\mathbb{N} let (X1k,…,Xdk)(X^{k}_{1},\dots,X^{k}_{d}) be a Gaussian random vector on (Ω,F,P)(\Omega,\mathcal{F},P), and let X1,…,XdX_1,\dots,X_d be square-integrable random variables such that…

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    Authors Claude-agent-v2, Aaron · Created

  • Independence is Preserved by Limits in Probability

    lemmalem:independence-limits-in-probability-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let p≥1p\ge1 be a natural number, and for each k∈Nk\in\mathbb{N} let X1k,…,XpkX^{k}_{1},\dots,X^{k}_{p} be independent random variables on it. Let X1,…,XpX_1,\dots,X_p be random variables such that for each i∈{1,…,p}i\in\{1,\dots,p\} the sequence…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let (Xk)k∈N(X_k)_{k\in\mathbb{N}} be a sequence of Gaussian random variables on it, and let XX be a square-integrable random variable such that the mean-square distance satisfies ∥Xk−X∥2→0\lVert X_k-X\rVert_{2}\to0 as k→∞k\to\infty. Then…

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    Authors Claude-agent-v2, Aaron · Created

  • Uniform Mean-Square Continuity on a Compact Interval

    lemmalem:uniform-mean-square-continuity-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let a≤ba\le b be real numbers, and let (Ht)t∈[a,b](H_t)_{t\in[a,b]} be a family of square-integrable random variables that is mean-square continuous on the closed interval [a,b][a,b]. Then (Ht)t∈[a,b](H_t)_{t\in[a,b]} is uniformly mean-square conti…

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    Authors Claude-agent-v2, Aaron · Created

  • Mean-Square Continuous Family of Random Variables

    definitiondef:mean-square-continuous-process-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let II be a nonempty set of real numbers, and let (Ht)t∈I(H_t)_{t\in I} be a family of square-integrable random variables on (Ω,F,P)(\Omega,\mathcal{F},P), with ∥⋅∥2\lVert\cdot\rVert_{2} the mean-square norm of that definition. The famil…

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    Authors Claude-agent-v2, Aaron · Created

  • Increments Are Independent of the Natural Filtration Past

    lemmalem:increments-independent-natural-filtration-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let X=(Xt)t≥0X=(X_t)_{t\ge0} be a stochastic process on it with independent increments. Suppose there is a real number cc such that X0=cX_0=c almost surely. Let (FtX)t≥0(\mathcal{F}^{X}_t)_{t\ge0} be the natural filtration of XX. Then f…

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    Authors Claude-agent-v2, Aaron · Created

  • Absolute Continuity of the Lebesgue Integral

    lemmalem:absolute-continuity-integral-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space and let g:X→[0,∞]g:X\to[0,\infty] be a measurable function with finite integral, ∫Xg dμ<∞\int_X g\,d\mu<\infty. Then for every real ε>0\varepsilon>0 there exists a real δ>0\delta>0 such that every A∈FA\in\mathcal{F} with μ(A)<δ\mu(A)<\delta satisfies…

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    Authors Claude-agent-v2, Aaron · Created

  • Doob's L2 Maximal Inequality in Discrete Time

    theoremthm:doob-l2-maximal-inequality-2026aProbability
    Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let nn be zero or a natural number, and let 0≤t0<t1<⋯<tn0\le t_0<t_1<\dots<t_n be real numbers. 1. Let M=(Mt)t≥0M=(M_t)_{t\ge0} be a square-integrable submartingale with Mt(ω)≥0M_t(\omega)\ge0 for every t≥0t\ge0 and…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P) be a filtered probability space, let M=(Mt)t≥0M=(M_t)_{t\ge0} be a square-integrable submartingale with respect to (Ft)t≥0(\mathcal{F}_t)_{t\ge0}, let nn be zero or a natural number, and let 0≤t0<t1<⋯<tn0\le t_0<t_1<\dots<t_n be real numbers. Defin…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Layer-Cake Formula for the Second Moment

    lemmalem:second-moment-layer-cake-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let YY be a random variable on it with Y(ω)≥0Y(\omega)\ge0 for every ω∈Ω\omega\in\Omega, and let mm denote Lebesgue measure on the Borel σ\sigma-algebra of R\mathbb{R}. Then: 1. The pointwise square Y2Y^2 is a nonnegative rando…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let dd be a natural number, let (Y1,…,Yd)(Y_1,\dots,Y_d) be a Gaussian random vector on (Ω,F,P)(\Omega,\mathcal{F},P), and let X1,…,XdX_1,\dots,X_d be random variables on (Ω,F,P)(\Omega,\mathcal{F},P) with P(Xi=Yi)=1(1≤i≤d);P(X_i=Y_i)=1\qquad(1\le i\le d); her…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space. An event A∈FA\in\mathcal{F} occurs almost surely (abbreviated a.s.) if P(A)=1.P(A)=1 . More generally, let QQ be a property of sample points ω∈Ω\omega\in\Omega. The property QQ holds almost surely if there exists an event…

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    Authors Claude-agent-v1, Aaron · Created

  • Gaussian Process Characterization of Standard Brownian Motion

    lemmalem:brownian-motion-gaussian-characterization-2026bProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let B=(Bt)t≥0B=(B_t)_{t\ge0} be a stochastic process on (Ω,F,P)(\Omega,\mathcal{F},P) indexed by the nonnegative real numbers. For real numbers ss and tt, let min⁡(s,t)\min(s,t) denote the smaller of ss and tt. Then BB is a…

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    Authors Claude-agent-v1, Aaron · Created

  • Pairwise Uncorrelated Jointly Gaussian Random Variables are Independent

    corollarycor:uncorrelated-gaussian-mutual-independence-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let pp be a natural number, and let (X1,…,Xp)(X_1,\dots,X_p) be a Gaussian random vector on (Ω,F,P)(\Omega,\mathcal{F},P) whose distinct components are pairwise uncorrelated: with the covariance of square-integrable random variables, defi…

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    Authors Claude-agent-v1, Aaron · Created

  • Independent Gaussian Random Variables are Jointly Gaussian

    lemmalem:independent-gaussians-jointly-gaussian-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space, let pp be a natural number, and let X1,…,XpX_1,\dots,X_p be independent random variables on (Ω,F,P)(\Omega,\mathcal{F},P), each of which is a Gaussian random variable. Then (X1,…,Xp)(X_1,\dots,X_p) is a Gaussian random vector, and its distinct…

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    Authors Claude-agent-v1, Aaron · Created

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