Questions tagged [large-deviations]
The large-deviations tag has no usage guidance.
35
questions with no upvoted or accepted answers
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votes
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Calculate the expectation of the maximum of averaged random walks
Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$
Is ...
- 753
6
votes
0
answers
274
views
universality for large deviations?
This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
- 3,507
5
votes
0
answers
1k
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What exactly is the relationship between Donsker-Varadhan variational formula and the Laplace principle?
Given a nice real valued functional $C$ on some probability space $(\Omega, \mathcal F, P_0)$ we have the following Donsker-Varadhan variational representation
$$\log E_{P_0}\left[e^C\right]=\sup_{P\...
- 1,302
5
votes
1
answer
325
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Bounding the sensitivity of a posterior mean to changes in a single data point
There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently ...
- 1,058
5
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0
answers
1k
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Asymptotic behavior of row sums in 2-d array of random variables
Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables:
$B^m_{1,1}$ $B^...
- 61
3
votes
0
answers
155
views
A variant of the Laplace principle
$\newcommand{\R}{\mathbb R}\newcommand{\eps}{\varepsilon}$In $\R^d$ I am given a sequence of smooth functions $f_\eps(x)$ that converges uniformly to some $f(x)$, which is assumed to be a good rate ...
- 5,163
3
votes
0
answers
78
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References on precise Large Deviations Principle/Laplace method for binomial sum
I am looking for an estimate of the following sum/expectation:
\begin{align*}%$
J_n & = \mathbb{E}\left( e^{n f(X_n) + \log(n) g(X_n) + h(X_n)} \right) \\
& = \frac{1}{2^n} \sum_{k = 0}^n {...
- 549
3
votes
0
answers
59
views
Existence, Uniqueness, and "ODE Characterization" of Minimizers for Variational Functionals from Large Deviations
A [classical result][1] of E. Lieb is that the functional
$$\mathcal E(\phi):=\int_{\mathbb R^3}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^3)^2}\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}~dx~dy$$
for $\phi\in W^1(...
- 891
3
votes
0
answers
162
views
Asymptotic behaviour of principal eigenfunctions and large deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
- 5,163
3
votes
0
answers
175
views
Most probable path for stochastic Hamiltonian systems
It is known that for a real valued stochastic process $X_t$ satisfying
$$
d X_t = b(X_t) d t + \sigma d W_t
$$
where $W$ is real valued Wiener process, the equation for the most probable path from ...
- 353
2
votes
0
answers
90
views
How much is known about the action functional for small noise diffusions with general volatility coefficients?
Let $W$ be a d-dimensional Brownian motion, and for every $\varepsilon > 0$, let $X^\varepsilon$ be the solution to the SDE
$$dX^\varepsilon_t = b(X^\varepsilon_t) \, dt + \varepsilon \sigma (X^\...
- 4,832
2
votes
0
answers
48
views
How to determine speed (rate) in large deviation principle for geometric Brownian motion
By reading Asymptotics for volatility derivatives in multi-factor rough
volatility models by Lacombe, Muguruza and Stone, I am not familiar with the way they deduce the speed (or rate) when showing ...
- 21
2
votes
0
answers
107
views
Large deviation principle for product of iid bounded symmetric random variables
Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,...
- 6,726
2
votes
0
answers
261
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Schilder's theorem for brownian bridges
I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.
A bit of context: usually, Schilder's theorem tells us that the ...
- 5,163
2
votes
0
answers
122
views
Modified Wigner semicircle law
The Wigner semicircle law states that for a random GOE-matrix $M^N \in \mathbb{R}^{N \times N}$ in the $N \rightarrow \infty$ limit for any $f \in C^b(\mathbb{R})$
$$\lim_{N \rightarrow \infty}\frac{...
user118408
2
votes
0
answers
63
views
tail bounds for sum of n iid variables divided by power of n
Let $X_i, 1\leq i\leq n$ be i.i.d. random variables with finite moments. Then $Y_n :=\frac{1}{n^{1+\delta}}\sum_{i=1}^nX_i$ goes to 0 almost surely for any $\delta >0$. What are some good non-...
- 121
2
votes
0
answers
52
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LDP respectively almost sure convergence in the context of randomly weighted trees
I am currently working on the following Problem:
Imagine you are given a $d$-ary tree $T_d$, which means an infinite tree with one vertex $x_0$ on top and in which each vertex has $d$ children.
Next,...
- 21
2
votes
0
answers
151
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A generalized Ballot theorem
Let $\{X_n\}_{n \in \mathbb{N}}$ be i.i.d. real random variables with $\mathbb{E}[X_i] = \mu \in \mathbb{R}$. Let $S_n = X_1 + X_2 + \cdots + X_n$.
Let $\nu \leq \mu$ be such that $\mathbb{P}[S_n <...
- 1,210
1
vote
0
answers
42
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From large deviations to finite time probability tails
Cross-Post from Math.SE
Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that
$$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$
One possibility to ...
- 235
1
vote
0
answers
127
views
Large-deviation inequalities for a class of simple random multivariate polynomials
Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...
- 6,726
1
vote
0
answers
95
views
Freidlin Wentzell for stochastic differential inclusions
Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
- 1,302
1
vote
0
answers
82
views
Large deviation inequalities for number of coupon types collected by a coupon collector with fixed budget
In the generalized Coupon Collector's Problem, there are $N$ types of coupon, and for any $i \in [N] := \{1,2,\ldots,N\}$, $p_i \ge 0$ is the probability of obtaining a type-i coupon on any trial. ...
- 6,726
1
vote
0
answers
77
views
Problem on definition of large deviation principle
As in many classic textbooks, the definition of large deviation principle is as follows:
$\{\mu_n\}$ has LDP with speed $a_n$ and rate $I(x)$ if the following holds for any measurable $A$:
$$\limsup\...
- 665
1
vote
0
answers
80
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Large deviations estimate for arbitrary continuous function
Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
- 4,969
1
vote
0
answers
87
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Large Deviations Principle for First Exit time of a Diffusion Process
Let $b:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be a smooth Lipschitz function, $x \in \mathbb{R}^d$, $\sigma >0$, and consider the solution to the SDE $X_t^x$ defined by
$$
dX_t^x = b(X_t^x)dt + \...
- 4,969
1
vote
0
answers
113
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Sanov-type finite-sample bound on $KL(P\|\hat{P}_n)$
Let $P$ be a distribution on an alphabet of size $k$ and let $\hat{P}_n$ be an empirical version of $P$ via $n$ i.i.d samples $a_1,\ldots,a_n \sim P$, i.e $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.
...
- 6,726
1
vote
0
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36
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Large Deviations Rate of Convergence and Robbins Monro
I am looking for a result/paper (if there is any) on the large deviations rate of convergence of the Robbins-Monro (RM) algorithm. Specifically, given $X_k \rightarrow X$ a.s. in the RM algorithm, I ...
- 39
1
vote
0
answers
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Reference: Varadhan's lemma for Finsler Geometry?
Is there a version of Varadhan's lemma for heat-kernels on Finsler manifolds? I expect this to exist but I cannot seem to find any papers on the topic. References would be greatly appreciated.
- 4,969
1
vote
0
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125
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Large deviations type results for sum of i.i.d. random functions
Assume that $f_1, f_2, f_3,\ldots$ are i.i.d. random functions $[0,1]\mapsto \mathbb{R}$ such that
(1) random variables $M_k=\sup_{x\in[0,1]}f_k(x)$ have exponential tails,
(2) $f$'s are a.s. ...
- 1,897
0
votes
0
answers
28
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Moment generating function for product states
In the sequel $B=M_\ell(\mathbb{C})$.
For $M\in\mathbb{N}$ fixed and $N\geq M$ I consider the symmetrizer $\pi_{M,N}(x_M)\in B^{\otimes N}$, which is the symmetrized tensor product of $a_1$,...,$a_M$ ...
- 63
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0
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37
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Weak convergence of Gibbs measures with converging energy functions
Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$
and let $Q$ be a fixed probability measure on $A$.
For each $\theta>0$, define the probability ...
- 483
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votes
0
answers
63
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On the asymptotic behaviour of the kernel of a Toeplitz operator
Consider the following Berezin Topelitz operator on the 2-sphere:
$$Q_N(f)=\frac{N+1}{4\pi}\int_{\mathbb{S}^2}d\Omega \, f(\Omega)|\Omega\rangle\langle\Omega|_N,$$
where $f\in C^\infty(\mathbb{S}^2)$,...
- 63
0
votes
0
answers
637
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sub-exponential type upper bound on the Poisson probability
I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.
Question:
For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
- 11
0
votes
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answers
103
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Comparing Euclidean norm of two normal vectors
Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...
- 101
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0
answers
51
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Probabilistic Approximation of non-linear Dynamical System by Diffusion Process
Setting
Suppose I have a discrete dynamical system given by:
$$
X^{n+1} = f(X^{n})
\qquad X^0 =x
,
$$
where $f$ is some diffeomorphism from $\mathbb{R}^{d}$ to itself, and some $x \in \mathbb{R}^d$. ...
- 4,969