Questions tagged [probability-theory]
For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.
46,249 questions
0 votes
1 answer
35 views
A claim in Durrett's proof of Glivenko-Cantelli
In Durrett's Probability: Theory and Examples, Theorem 2.4.9 (the Glivenko-Cantelli Theorem), there's the following claim: In general, if $F_n$ is a sequence of nondecreasing functions that converges ...
- 1,596
0 votes
0 answers
44 views
Function is continuous except on a set of null Wiener measure
Let $C$ the space of real continuous functions defined on $[0,1]$. With the topology induced by the uniform norm on $C$, let $\mathcal{C}$ the borelians induced by this topology. Let $\mathcal{B}$ the ...
- 503
2 votes
1 answer
66 views
Equality related to random variable and its conditional expectation.
Let $(\Omega,\mathcal F,\Bbb P)$ be a probability space and $X$ be a Banach space and $r_1,\cdots, r_N:\omega\to\Bbb R$ be random variables on the probability space, and $x_1,\cdots,x_N\in X.$ Suppose ...
- 340
0 votes
0 answers
22 views
If a Stochastic Differential Equation is bounded [0,1], can we assume that models probability failure; [closed]
I want to model a system in terms of probability of failure. If I use a stochastic differential equation that is bounded [0,1], can I assume that it models probability failure? I know that failure ...
6 votes
2 answers
100 views
Intuition behind using $\pi$-system in Measure Theory
Many proofs in measure theory rely on the π-λ theorem: if a π-system 𝑃 generates a σ-algebra, then it suffices to verify a property on 𝑃 and extend it to the whole σ-algebra. Examples include ...
- 471
1 vote
1 answer
36 views
Family of i.i.d. random variables is ergodic under shift
Let $(S, \mathcal{S})$ be some measurable state space, $\Omega := S^{\mathbb{N}_0}$, $X_i$ the coordinate maps, $\mathcal{A} := \sigma(X_0, X_1, \dots)$ and $\theta: \Omega \rightarrow \Omega, (x_0, ...
- 323
-3 votes
0 answers
78 views
Uniformly integrable [closed]
Assume $\mu_n$ is a family of probability measures and $f_m\to 0$ with $|f_m(x)|\le V(x)$. If we have $\sup_{n\ge 1}\int_{E} V(x)^2\mu_n(dx)<\infty$. Prove that $\lim\limits_{m\to\infty}\sup\...
- 139
0 votes
0 answers
22 views
Convergence of transform of stochastic processes to transform of Brownian bridges
I have two sequences of independent stochastic processes $X^n$ and $Y^n$ that are known to converge weakly to the Brownian bridges $\mathcal X$ and $\mathcal Y$, respectively. For a continuous ...
- 348