Use this tag for question on large deviations theory
Large deviations theory is interested in the rate of decay of rare events in the central limit theorem.
Questions tagged [large-deviation-theory]
191 questions
2
votes
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Having trouble in an exercice on Large Deviations.
In the book "Large deviations" by Frank den Hollander, one reads in pg 30:
Exercise III.10 (Suggested by G. O'Brien.) Let $Z_n$ be a single random variable with a binomial distribution with parameters $n$ and $p_n$. Let…
Conrado Costa
- 6,071
2
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1 answer
Cramer, $P(S_n\geqslant na)\sim e^{-n I(a)}$
Let $(X_i)_{i\in\mathbb{N}}$ be an iid sequence of random variables and $S_n:=\sum_{i=1}^n X_i$. Moreover, let $M_{X_1}$ denote the moment-generating function and $\Lambda:=\log M_{X_1}$. Define
$$
I(x):=\sup_{\theta\geqslant 0}\left\{\theta…
Rhjg
- 2,029
2
votes
1 answer
Strange version of Cramér's Theorem: Why is it necessary that the supremum is obtained at some interior point of the neighborhood?
Recently, I dealt with the Cramér theorem, see here, Theorem 1 on page 1.
In this version of the theorem, it is needed that
(i) the moment-generating function $M$ is finite on a neighborhood $B_0$ of $0$ and, additionally, that
(ii) the supremum…
Rhjg
- 2,029
1
vote
0 answers
why non-gaussian case we do not have $S_n>an$ being rare?
I am taking a course on Large deviation theory but I'm a bit stuck at the first place. The lecturer gave an example to motivate the study of large deviation.
First, if $X_1$ is standard gaussian, and $X_1,\cdots,X_n$ are i.i.d. r.v.s, then…
chloe
- 512
1
vote
1 answer
Large Deviations rate function and Cramérs Theorem
Given $X_1,...$ of iid random variables. We know that if the moment generating function $M(\theta) < \infty, \forall \theta $ from Cramérs Theorem we get:
$\lim_{n\to \infty} \frac{1}{n}\log \mathbb{P}(S_n \ge na) = -I(a)$ where
$I(a) =…
nevergivenaname
- 415
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vote
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Large deviations principle for $X^{\lfloor \alpha N \rfloor}/N$
Let $(X^N/N, N \in \mathbb{N}$) satisfy a large deviations principle in $\mathbb{R}$ with convex rate function $I$. Let $\alpha$ be a positive real number. Show that $X^{\lfloor \alpha N \rfloor}/N, N \in \mathbb{N}$ satisfies a large deviations…
Jon Snow
- 41
1
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1 answer
Proof of Chernoff's bound: last step missing
Let $\left\{X_i\right\}$ be a sequence of i.i.d. random variables and $S_n:=\sum_{i=1}^n X_i$. Then, for $\theta\ge 0$, we have
$$
P(S_n\ge nx)\le e^{-n\sup_{\theta\ge 0}\left\{\theta x-\log M(\theta)\right\}},
$$
where…
M. Meyer
- 639
0
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Equivalence in the formulation of the "Large Deviation Principle"
I'm reading this notes on the Internet http://staff.utia.cas.cz/swart/lecture_notes/LDP4.pdf. And I'm stuck in this proposition:
Let E be a Polish space. A sequence of finites measures $\{\mu_n\}$ with a good rate function…
DonQuixote
- 391
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Large deviation problem
Let $s_1,s_2,\cdots,s_n$ be n i.i.d r.v. drawn from a probability distribution $p$ with bounded support. Show that, to leading exponential order, $$P\{s_1+\cdots+s_n\leq0\}=\{\inf_{z\geq0}E[e^{-zs_1}]\}^n$$
yya
- 1