Questions tagged [limit-theorems]
For questions about limit theorems of probability theory: (functional or not) central limit theorem, law of large numbers, law of iterated logarithm, etc.
109 questions
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Central limit theorem and convergence of means [closed]
If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so ...
CommunityBot
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Lyapunov Exponents for independent-nonidentically distributed matrices?
My question is highlighted in bold at the end.
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <...
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Intuition on Lindeberg condition
I want to know how Lindeberg came up with the condition which is sufficient for CLT to hold ? What is the intuition behind such an expression ?
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Convergence rate of the central limit theorem near the center of the distribution
I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.
Specifically, from the general convergence rates stated in the Berry–Esseen ...
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Stable local limit theorems
Consider a sequence of integer valued indentically distributed centered independent random variables $X_1, X_2, \ldots$ with the additional condition that the support of $X_1$ is aperiodic. Suppose ...
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Generalized central limit theorem
I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X_j)_{j\geqslant 1}$ is a stationary sequence such that $X_i$ ...
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Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
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Asymptotically independent increments random elements: Billingsley Ch:$4$
Let $X_n$ be random elements of $D$ (space of cad lag functions on $[0,1]$ as domain). $X_n$ has asymptotically independents if $0\leq s_1 \leq t_1 \leq s_2 \leq \ldots < s_r \leq t_r \leq 1$, then ...
CommunityBot
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Donsker Theorem Billingsley
Theorems $16.1$ and $16.3$ in Billingsley Convergence of measures.
$16.1$ reads : Random variables $u_1,\ldots$ on $(\Omega,\mathcal{B},\mathbb P)$
and are i.i.d. with $0$ mean and finite variance $\...
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