# Questions tagged [large-deviations]

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
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 <...