# Questions tagged [large-deviations]

The large-deviations tag has no usage guidance.

72
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### 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 ...

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### 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 $X^\varepsilon$?

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### 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. ...

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### 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^\...

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### 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 ...

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### 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 ...

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### 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,...

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226
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### Concentration of minimum Hamming distance between $N$ points sampled iid from uniform distribution on $n$-dim hypercube $\{0,1\}^n$

Let $n$ be a large positive integer. Sample $N \ge 2$ points $x_1,\ldots,x_N$ iid from the uniform distribution on the $n$-dimensional hypercube $\{0,1\}^n$. Define the gap $\delta_{N,n} := \min_{i \...

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### Asymptotic distribution of counting process when we include higher moments (thus not Gaussian)

Question
Consider a counting process $(N(t))_{t \ge 0}$ such that the time $\tau$ between arrivals is I.I.D with some probability distribution $f$. Call $\langle \tau^n\rangle$ the $n$th moment of $$, ...

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148
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### Small noise limits with irregular drift

Let $W$ be a standard $d$-dimensional Brownian motion.
Suppose $b: \mathbb R^d \to \mathbb R^d$ is measurable and bounded. Consider, for every $\varepsilon > 0$, the solution $X^\varepsilon$ on $[0,...

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286
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### Lower bound on sum of independent heavy-tailed random variables

I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...

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### Extended large deviation principle on product space

My question is how to prove the large deviation property in a product space if at each coordinate there is a large deviation property. I will explain what I mean in detail as below:
Let me start with ...

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### Mutual information in large deviation theory

Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate ...

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662
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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\...

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244
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### Large deviations for sequences that are not sums of iid

Suppose $(S_n)_n$ is a sequence of real random variables. Denote their cumulant generating functions by $K_n(t) = \log\mathbb{E}\left[\mathrm{e}^{t S_n}\right]$, and assume that each $K_n$ is finite ...

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### Endpoint of Brownian motion conditional on high maxima

Note: This question is closely related to an earlier question: A large noise limit.
Let $W$ be a standard one dimensional Brownian motion.
For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...

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### LDP for Marchenko Pastur with k/n tending to 0

I am interested in the determinant of $W = X * X'$, where $X \in \mathbb{R}^{k \times n}$ is a matrix with each row drawn IID from some sub-Gaussian distribution on $\mathbb{R}^{n}$. (I am aware of ...

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### Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator

Let $\{X_i \}_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X_1 = 0$ and $\mathbb{E} X_1 ^2 < \infty$. Assume that $\{S_n \}_{n \in \mathbb{N}}$ is a non-...

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295
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### A large noise limit

Let $f: [0, 1] \to \mathbb R$ be a bounded, continuous function, and $W$ a standard Brownian motion.
Denote $Y := \int_0^1 f(t) \, dW_t$.
For each $\varepsilon > 0$, consider the conditioned random ...

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### Probability distribution of $\sum_i^n X_i - T$ where $\sum_i^nX_i <T<\sum_i^{n+1} X_i $

Let $X_{1}, X_{2}, \ldots, X_{n}$ be IID random variables with mean $\mu$ and variance $\sigma^2$. Let $S_n=\sum_i^{n}X_i.$
Let $T\gg1$ and define $\tau=T-S_n$ where $n$ satisfies the following ...

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158
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### Large deviation for empirical median

I found this exercise while reading some notes on Large Deviation Principle. This exercise is at the end of the very first chapter, including Cramer's Theorem and essentially nothing more (no Sanov ...

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### Determine the affine envelope of a random process's MGF

Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a ...

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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 {...

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### Concentration Inequality for Bounding Lipschitz Empirical Lass

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(X_n:\Omega\rightarrow \mathbb{R}^m)_n$ be a sequence of i.i.d. random variables and let $L:\mathbb{R}^m\rightarrow [0,\infty)$ be ...

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### 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\...

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### Large deviations: Growth of empirical average of iid non-negative random varialbes with infinite expectations?

Let $X_1,X_2,X_3,...$ be iid non-negative random variables with $E[X_i]=\infty$. I am looking for references on the growth in $n$ of the empirical average under assumptions on $X_1.,..,X_n$.
A more ...

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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 ...

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223
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### Upper bound for $P(X \geq x)$, where $X \sim \operatorname{Pois}(\lambda)$

I posted the following question in a comment on CDF of a log-concave discrete random variable. Since it is not related to my main question, I thought of reposting it as separate post.
Question:
Let $X ...

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### CDF of a log-concave discrete random variable

In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave.
My questions:
What can we say about this in the discrete setting?. For ex: Is the ...

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### 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(...

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### Sample average L1 convergence speed

Say $X_1, \cdots, X_n$ are i.i.d random variables with mean zero, let $S_n = \sum_{i=1}^n X_i$, we know by SLLN $$\frac{S_n}{n}\rightarrow 0\text{ a.s}$$
We could further know that the sequence of ...

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506
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### Probability of a deviation when Jensen’s inequality is almost tight

This is a cross-post to a yet unanswered question in Math StackExchange
https://math.stackexchange.com/questions/3906767/probability-of-a-deviation-when-jensen-s-inequality-is-almost-tight
Let $X>0$...

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### Concentration inequalities for very rare events on a multiplicative scale

Let $E_1, \dots, E_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A_N = \frac{1}{N} ( 1_{E_1} + \dots + 1_{E_N} )$ be the proportion of the events $E_i$ that ...

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### Large deviation for Brownian occupation time

I am asking for reference about the large deviation principle (LDP) for the occupation time of a Brownian motion/bridge. Let $f:\mathbb{R} \to \mathbb{R}$ be smooth and compactly supported. My ...

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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 ...

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96
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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 ...

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### Tail bounds for the absolute difference of a coupled pair of sub-Gaussian random variables

Let $P$ and $Q$ are sub-Gaussian distributions on $\mathbb R$, and $(X,X')$ be a coupling of $P$ and $Q$, i.e $(X,X') \sim \pi$ for some distribution on $\mathbb R^2$ with marginals $P$ and $Q$.
...

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### Joint typicality of sequences

I know that for two i.i.d distributions $P$ and $Q$ the probability that $Q$ will produce a length $n$ sequence that is $\epsilon$-typical according to $P$ is bounded by
$$Q(T_{P,\epsilon})\leq e^{-nD(...

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### Anti-concentration inequalities: lower bound on realized second moment

Let $X:(\Omega,\Sigma)\rightarrow (\mathbb{R}^d;\mathbb{B}(\mathbb{R}^d))$ be a Borel-measurable random vector. What are some general classes of such random vector for which one can give a "lower ...

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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 $x \in \mathbb{R}^d$. ...

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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 + \...

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### Bound for Large deviations of sums of independent (not identical) variables

I am working with a sum of variables $X_i$; they are all independent, but not identically distributed. For any $i$, I can show the bound $$\Lambda^*_{X_i}(t) := \sup_t \langle t, x \rangle - \Lambda_X(...

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### Reference: hitting time of Gaussian process

Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by
$$
Y_t = y+\int_0^t X_s ds + W_t,
$$
for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...

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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 ...

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### Large deviations for discrete uniform distribution

(Not sure if this belongs on stack-exchange or overflow; let me know if I should switch it).
Given a sum of $n$ IID random variables $\{X_i\}_{i=1}^n$, each uniform on the integers $0,1,...,r$ for ...

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184
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### Local central limit theorem far from the center

Let $X_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$.
Classical local CLT says that the density function $f_n$ of $\frac1{\sqrt n}...

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### Estimating probability that a large sum of i.i.d variables is positive

Let $X$ and $Y$ be i.i.d. random variables with exponential distribution with mean $1$, and let $Z=(X-1)(Y-X)$. Let $Z_1,...,Z_n$ are i.i.d. copies of $Z$, and let $f(n)=P[\sum_{i=1}^n Z_i > 0]$. ...

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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 $\hat{P}_n := (1/n)\sum_{i=1}^n\delta_{a_i}$.
...

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### Large Deviations for Self-Normalized Sums

I am trying to understand the main result (Theorem 1.1) in this paper by Shao, which gives a large deviation bound for the self-normalized sum of iid variables
$$
\frac{\sum X_i}{\sqrt{n}\sqrt{\sum ...

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358
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### Large deviation of random walk

1) Let $\{X_i\}_{i=1}^n$ be i.i.d. such that $\Pr(X_i=1 )=1-\Pr(X_i=-1)=p$. Define the random walk
$$
S_i = \sum_{j=1}^iX_j
$$
for $i=1,2,\ldots,n$.
I am looking for "good" exponential upper bounds ...