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Questions tagged [large-deviation-theory]

Use this tag for question on large deviations theory

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Using Girsanov Theorem Backwards?/ Obtaining Radon-Nikodym Derivative

On page 112/133 of Den Hollanders book on Large Deviations he wants to calculate the R.N derivative between two path measures : one is the path measure of the solution to an SDE $dX_t=H(X_t)dt+dW_t$ ...
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8 views

The rate function in large deviations

Why do we use rate functions in the definition of Large Deviations. That is why do we require the function to be lower-semicontinuous?
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46 views

Distance from origin of biased random walk conditioned to be positive at time n

Let $S_n$ be the position of a simple random walk on the integers started from $0$ that moves right with probability $p<1/2$. What is the asymptotic behavior of $$E[ S_n \mid S_n >0 ]$$ as $n \...
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6 views

Heat kernel trace bound

Are there any ways to efficiently bound the sum of the following type: $$\varphi(t) := \sum_{k=1}^{\infty}{\exp(- k^{\frac{2}{d}} \cdot t)}$$ where $d > 0$, $d \in \mathbb{N}$. This sum ...
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1answer
47 views

Asymptotically tight bounds $P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]$

I am looking for assymptotically tight bounds on \begin{align} P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]=P \left[ \left| E[Z^2] - \frac{\sum_{i=1}^n Z_i^2 }{n} ...
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1answer
37 views

Asymptotics of binary call option in Black–Scholes model in the large deviations sense

Consider Black–Scholes model where asset log-price is given by $$ X _ { t } = \sigma W _ { t } + \mu t $$ for $W_t$ – Brownian motion. I want to show that $$ P \left[ X _ { t } > k \right] \...
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68 views

Is $\inf$ of a differentiable function still differentiable?

I'm new on stackexchange, thanks for this fantastic platform! Let's get to the point. Suppose we have a function $I(x,y) \in \mathbb{R}$, $I(x,y) \geq 0$, convex, and differentiable for $x \in X$ ...
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Function of transient markov processes

Suppose that $X_t$ and $Z_t$ are Markov processes, which are $\mathbb{P}\otimes m$-a.e. (here $m$ is the Lebesgue measure on $\mathbb{R}$) not equal. If $X_t$ and $Z_t$ are transient, it seems right ...
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10 views

Reverse Hoeffding Inequalities

Suppose that $X_t$ is a super-martingale, the Hoeffding inquality gives an exponential upper bound on the quatity $$ \mathbb{P}\left( \sup_{0 \leq t\leq T}X_t \geq x \right). $$ When can a lower-...
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34 views

Reference book on large deviations for mean field theory

I am looking for reference material, notes or published books, which study applications of large deviations on mean field theory / interacting particle systems etc.. I am especially interested in ...
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23 views

Determining if a Large Deviation Principle exists for a sequence of probability measures.

Given a sequence $(P_n)_n$ of probability measures on a space satisfies a LDP with rate $r_n$ and rate function $I$ if both the following hold: $$ \limsup_{n}\frac{1}{r_n}\log P_n [F] \le - \inf_{x ...
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1answer
28 views

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 \...
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60 views

Sum of variables of a martingale

I have the sequence $X_1, X_2,...X_n$ as a martingale, each of which is bounded. Now I want to explore some upper bound for the sum $S_n=X_1+X_2+...+X_n$, e.g., the format like Hoeffding inequality or ...
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2answers
35 views

Integrate: $\int_{-\infty}^\infty \exp(-||\vec x||^2) ||\vec x||^{-m}\mathrm{d}\vec{x}$

Let $m,n$ be two positive integers with $0 < m < n$. Can we integrate this: $$I = \int_{-\infty}^\infty \mathrm{d}x_1 \dots \int_{-\infty}^\infty \mathrm{d}x_n \left(\sum_{i=1}^n x_i^2\right)...
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28 views

Derivation of Sanov's theorem for continuous variables

Where can I find a derivation of Sanov's theorem for continuous variables? I am familiar with the derivation for discrete variables. I am looking hopefully for something similarly intuitive.
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95 views

Integrate: $\int_0^1 ||\vec x||^{-m}\mathrm{d}\vec{x}$

Let $m,n$ be two positive integers with $0 < m < n$. Can we integrate this: $$I = \int_0^1 \mathrm{d}x_1 \dots \int_0^1 \mathrm{d}x_n \left(\sum_{i=1}^n x_i^2\right)^{-m/2}$$ If a closed ...
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Simplify $\int_a^b \mathrm{d}x_1 …\mathrm{d}x_m \int_c^d \mathrm{d}y_1 …\mathrm{d}y_n \prod_{i=1}^m \prod_{j=1}^n f(x_i y_j)$

Let $a \le b$ and $c\le d$ be real numbers and $f$ a real function. I am struggling with an integral of the form: $$I = \int_a^b \mathrm{d}x_1 ...\mathrm{d}x_m \int_c^d \mathrm{d}y_1 ...\mathrm{d}...
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107 views

Cramer's theorem and central limit theorem

I just came across Cramer's theorem for large deviations. If $X_1,..,X_n$ are i.i.d and $S_n$ is the mean of the first $n$, then: $$\lim_{n \rightarrow \infty} \frac{\log P(S_n >x)}{n}=-I(x)\...
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Probability relating the sum of tilted distributions

Say I have $iid$ Bernoulli random variables with probability $p$ $$P(X_i = 1) = p$$ I derive the tilted probability $X_i^{(\lambda)}$ such that: $$P(X_i^{(\lambda)} = 1) = \frac{e^{\lambda}p}{e^{\...
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Limit of derivative of moment generating function

Suppose $X>0$ is an integer-valued with $P(X=k)=q_ke^{-t_0k}$, where $t_0>0$ and $\{q_k\}$ satisfies $\frac{1}{k}\log q_k \rightarrow 0$. Let $\phi(t)$ be the MGF of $X$. Let $t_{\text{max}}=\...
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1answer
47 views

Large deviation 2.6 Durrett's book

I'm reading 2.6 of Durrett's book. If we define $\phi(t)=\mathbb{E}[e^{tX}]$ and for $a>\mu$, where $\mu$ is the expectation of corresponding random variable, and we define $$I(a)=\sup_{t>0} \ (...
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76 views

Laplace Method - Estimation of an integral

I am just working on a paper by Shinzo Watanabe on "Asymptotic Evaluations of Wiener Functional expactations": $μ(dx)$ is the $d$-dimensional Gaussian distributions So that is the interesting ...
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2answers
174 views

Why Large deviation theory?

I am trying to understand why we need Large deviation theory/principle. Here is what I understand so far based on the Wikipedia. Let $S_n$ be a random variable which depends on $n$. We are ...
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1answer
71 views

Is there any intuitive way to see $\frac{e^{\delta}}{(1+\delta)^{1+\delta}}\leq e^{-\delta^2/3}$ $0<\delta<1$

I am reading the proof of Chernoff bound ,there's one step here : $$\frac{e^{\delta}}{(1+\delta)^{1+\delta}}\leq e^{-\delta^2/3}$$ where $0<\delta<1$ the book prove that by using the second ...
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1answer
39 views

Large Deviation Principle

I have been reading Amir Dembo's book, and at the very beginning, I found this result that came across and unfortunately, I cannot derive it by myself. So, I'm looking for some help. It happens that ...
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2answers
249 views

Hoeffding inequality for conditional probability (conditioned on event)

Suppose I have independent $X_1\sim\text{Bin}(n,\theta_1)$, $X_2\sim\text{Bin}(n,\theta_2)$ with $X=X_1+X_2$. Suppose that $\theta_1,\theta_2\in(0,1)$. Define the constant (but still depends on $n$) $...
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14 views

Equivalence of two large deviation statements

Let $\{X_n\} \subset \mathcal{X}$ be a sequence of random variables on a compact metric space $(\mathcal{X},d)$. The sequence is Cauchy, i.e. $d(X_n, X_{n+1}) \rightarrow 0$ and it satisfies the ...
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40 views

Reconciling definitions of large deviations principle

I am reading some notes on the Large deviations principle and I want to reconcile the definitions I've seen in the abstract measure theoretic framework and one that is used on an introduction to the ...
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33 views

Limit of a mixture of binomial distributions

Let $0 < r \leq 1/2$ and let $B(n,p)$ be the notation for binomial distribution. Consider the random variable $Z_n$ defined by the recursive equation $$ Z_{n+1} = Z_n + X_n I(Z_{n} \geq r n^2) + ...
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1answer
78 views

Why is the rate function given by Schilder's theorem infinite outside of CM space? Can we understand Schilder's theorem through CM theorem?

Schilder's theorem from large deviations theory tells us that scaled Brownian motion $\sqrt{\varepsilon} W_t$ on Wiener space $C_0([0,T],\Bbb R^d)$ satisfies a large deviation principle with good rate ...
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41 views

Gärtner-Ellis theorem on Markov chains

Let $Z_n \in \mathcal{X}$ be a sequence of independent random variables where $\mathcal{X}$ is a topological vector space and let $\mu_n$ the probability measures associated with $Z_n$. Suppose that $...
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1answer
80 views

What does it mean that “the central limit theorem does not hold far away from the peak”?

So I know nothing about large deviations theory, and I'm reading some notes. They claim that: The CLT does not hold far away from the peak I am not sure how to parse this statement. There are many ...
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1answer
133 views

Good book on large deviations theory

I am interested in reading about large deviations theory. Can anybody please suggest me any good book regarding this. Thanks in advance.
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94 views

Finite Exponential Moment

Consider $X_1, X_2, X_3$ ... random variables i.i.d. such that $P(X_i=1)=p$ and $P(X_i=-1)=1-p$. Consider the random walk $(S_n)_{n\ge 0} $ with $S_0=0$ and for $n\ge 1 $, $S_n = \displaystyle\sum^{...
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1answer
149 views

Large deviation principle for Gaussian random variables with mean $0$ and variance $1/n$.

Problem: Let $\mu_n$ be the Gaussian distribution with mean zero and variance $1/n$ on $\mathbb{R}$. Show that $\mu_n$ satisfies a large deviation principle with rate $1/n$ and rate function $x^2/2$. ...
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69 views

Prove that MGF is differentiable at everywhere? and $\lim_{s \rightarrow \infty} \frac{\log M(s)}{s} = \infty$?

I am reading the some notes on probability theory and the notes(Chernoff lower bound) left some gap that I should fill in, however I don't even know how to start yet. Here is the assumption the gives ...
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54 views

Pontryagin principle, Optimal control or Numerical scheme ? logical constraint?

I have the following optimal control problem : $ \textit{ Minimize } \qquad J(u,x,T) =\displaystyle \int_0^T \Big[ u_1(t)\log\frac{u_1(t)}{ x_1(t)x_2(t)} - u_1(t) +x_1(t)x_2(t) \qquad \qquad \...
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2answers
49 views

Polynomial term for random walk large deviations

Let $S_n = \sum_1^n X_i$ with $P(X_i =1) = p>1/2$ and $P(X_i = -1) = 1-p$ be a biased random walk. Large deviations tell us that $p_0=P(S_n \leq 0) \leq n^a(2 \sqrt{p(1-p)})^n$. We are curious what ...
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27 views

WKB form, large deviation expansion of the stationary PDF

I am currently reading the book theory and applications of stochastic processes an analytical approach, on page 304, we have the equality: $$ \int_{\mathbb{R}} [p_{\varepsilon}(y-\varepsilon \xi) w(\...
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66 views

an Optimal control problem : infinite time horizon and free end point

i've never worked on optimal control problem before and have an issue with this problem : Let $x_0\in \mathbb{R}^2_+$ and a constant $\lambda \neq0$ be fixed. I want to minimize, $\qquad$ over ...
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52 views

Connection between large deviation principle and weak convergence/

What is the relationship between large deviation and weak convergence? Consider a sequence of random variables {X_n}, does the LDP of sample mean imply the the distribution of X_n converge? If so, ...
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1answer
92 views

Understanding Legendre-fenchel Transform, looking for an easy example and intuition

Looking for help in understanding this transform. I have no background in real analysis but need this stuff for my research. I hope someone can give me some light on the intuition behind this ...
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1answer
70 views

some questions on large deviations

the following excerpt is from den hollander: i just don't get the point. what is meant by cheapest realization? is it, loosely speaking, the realization occuring with highest probability because of ...
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1answer
31 views

Alternate Characterization of Rate of Convergence

Let $\{x_n\}$ be a sequence converging to $L$. According to Wikipedia, if there exists a $\mu\in(0,1)$ satisfying $$\lim_{k→∞}\frac{|x_{k+1}−L|}{|x_k−L|}=μ$$ then we say $\{x_n\}$ converges ...
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1answer
105 views

Doubt in Proof of lower bound in Cramer's theorem

I read proof of Cramer's theorem from the book : 'Large Deviations and Applications' by A.Dembo and Ofer Zeitouni. I have a doubt in the proof of the lower bound. Cramer's theorem(Lower bound) : ...
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1answer
670 views

What does Cramer's Theorem tell us?

Cramer's Theorem States, Let $(Y_i)_{i\geq 1}$ be a sequence of i.i.d. random variables, ${S_n=\frac{1}{n}\sum_{i=1}^n Y_i}$ be their average sum and $M_{Y_1}(u):=\mathrm{E}[e^{uY_1}]<\infty$ be ...
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1answer
377 views

Difference between “large deviation estimate” and “moderate deviation estimate” in probability theory

I am from physics background. Recently, I am reading a book on "limit theorems in probability theory". My question is, What are the fundamental differences between "large deviation estimate" and "...
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1answer
91 views

Understanding a proof in Varadhan's large deviations and applications

I am currently writing a thesis that serves as an introduction to the theory of large deviations for people on about my level. The task is to closely follow Varadhan's booklet "Large Deviations and ...
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61 views

How do we conclude that $h_c$ is non decreasing? A question concernig a paper from Varadhan (2003)

In the paper Large Deviations for Random Walks in a Random Environment (2003) One reads on page 1226: $h_c$ is obtained as the following limit: $$h_c(y) = \lim_n \frac{1}{n} -\log \sup_m \big(\pi^m(...
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88 views

Reference request: Large deviations for a conditional probability

Suppose a sequence of probability measures $(\mathbb P_n)_{n\in\mathbb N}$ on a Polish space $X$ satisfies the large deviations principle with a good rate function $I$ and rate $n$. Informally ...