# Questions tagged [large-deviations]

The tag has no usage guidance.

57 questions
Filter by
Sorted by
Tagged with
70 views

### 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 ...
66 views

### 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-...
1 vote
189 views

### 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 ...
34 views

### Prove the statistical rate lower bound of a given complicated statistics

Given a i.i.d. sequence of random variables $\{Z_i\}_{i=1}^n$ who has mean zero. Two i.i.d. sequence of random vectors $\{X_i\}_{i=1}^n$, $\{Y_i\}_{i=1}^n$ who have the same covariance matrix $\Sigma$....
1 vote
86 views

### 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 ...
1 vote
91 views

### 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 ...
108 views

### 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 ...
46 views

209 views

### 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 $f(n)=P[\sum_{i=1}^n Z_i > 0]$. ...
1 vote
85 views

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

222 views

### Fast Algorithms for sum of independent random variables

CLT implies the sum of n i.i.d random variables,after property normalized converge to a Normal distribution as n goes to infinity. Furthermore, Linderberg's condition points out not necessarily ...
94 views

### Family of large deviation principles

The following question may be a bit imprecise in its formulation, I guess however the problem I have in mind is clear. Although to me it looks like a fairly standard question, I couldn't find any ...
611 views

### 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 ...
144 views

I am a physicist caught in the following situation: I have two probability measures $\mathbb{P}_1$ and $\mathbb{P}_2$ and have to deal with the following integral where $X_i$ are random iid: $$\int_{... 2 votes 0 answers 116 views ### Modified Wigner semicircle law The Wigner semicircle law states that for a random GOE-matrix M^N \in \mathbb{R}^{N \times N} in the N \rightarrow \infty limit for any f \in C^b(\mathbb{R})$$\lim_{N \rightarrow \infty}\frac{... 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 ...