# Questions tagged [large-deviations]

The large-deviations tag has no usage guidance.

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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 ...

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**4**answers

164 views

### Concentration of closed random walks

Consider a random walk $S_n=\sum_{i=1}^n X_i$ where $P(X_i=+1)=P(X_i=-1)=1/2$ with $n$ large. By Chernoff's bound we know that, for example, $\sum_{i=1}^{n/2} X_i=O(\sqrt{n})$ with high probability.
...

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72 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 ...

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107 views

### 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^...

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16 views

### 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 ...

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74 views

### 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 ...

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26 views

### Efficiency of importance sampling in terms of the size of the the support of sampling distribution

In importance sampling, one proposes to compute an integral $I:=\mathbb E_{x \sim P}[h(x)]$ by rewritting it as
$$
I=\mathbb E_{x \sim Q}\left[w(x)h(x)\right],\text{ with }w(x):=\frac{p(x)}{q(x)},
$$
...

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votes

**1**answer

83 views

### Tail condition (Varadhan's lemma)

I would like your help with the following tail condition, which arises in the theory of large deviations.
Let $P(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$, $ G:P(\mathbb{...

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21 views

### limit of large time of an ergodic diffusion and differentiability with respect to a parameter : towards Ellis-Gartner theorem

Consider an ergodic diffusion. For instance, we can think of $X_t \in \mathbb{R}^d$ satisfying
$$
d X_t = b(X_t) d t + \sigma d W_t,
$$
here $W$ is a $\mathbb{R}^m$ valued Wiener process, $\sigma$ is ...

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**1**answer

79 views

### Large deviation upper bound for Chi-squared random variable

Let $X \sim \chi^2_n$ random variable. I am looking for a large deviation upper bound for $X$. The answer here, says that
Since you said that you're looking for an upper bound, it should also be ...

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votes

**1**answer

81 views

### Concentration inequalities specialized for log-likelihood / log-density functions

Let $P$ be a probability measure and $f$ be some probability density function (not necessarily related to $P$). Consider the function
$$
L(X_1,\ldots,X_n)
=\frac1n\sum_{i=1}^n\log f(X_i),
\quad
X_i\...

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votes

**2**answers

152 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 ...

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**1**answer

79 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 ...

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187 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 ...

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**1**answer

129 views

### Large deviations for integrands

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_{...

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**0**answers

108 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{...

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90 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 ...

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55 views

### 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.

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196 views

### Large deviation/concentration inequality for submartingale

Let $S_t = M_t + D_t$ be the sum of a martingale $\left(M_t\right)_{t=1,2,\ldots}$ and a predictable process $(D_t)_{t=1,2,\ldots}$ such that the variance of the increments of $M$ is uniformly bounded ...

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227 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 ...

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49 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-...

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42 views

### 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,...

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118 views

### 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. ...

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126 views

### 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 <...

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**1**answer

659 views

### What is “tilting” in the context of large deviations?

I have seen references to the "tilting method" in the theory of large deviations. Is there a simple explanation of what this is, exactly?