# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
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### What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
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### Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
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### A roadmap to Hairer's theory for taming infinities

Background Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum ...
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### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...
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### Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
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### Mean minimum distance for N random points on a unit square (plane)

A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random ...
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### Expected supremum of average?

Is there either a closed form (in terms of the moments of $X_1$, say) or good bounds on $$\mathbb{E} \sup_{k \leq n} \frac{1}{k} \sum_{i=1}^k X_i,$$ where $X_i$ are iid and arbitrarily nice? (In my ...
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### Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
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### Distribution of ratio between complex Gaussian and Chi-square R.V.s

What would be the distribution (p.d.f.) of the following ratio? $$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$ where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...
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### What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
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### Why polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
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### What is the order of the lower tail of a Chi-Squared distribution?

Let X be a random variable with having a chi-squared distribution with n degrees of freedom and let y be some real number at most n. Is it known how P (X < y) behaves at least in some reasonable ...
Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ and the function $$h_p(x):=e^{-\vert x \vert^p}.$$ My goal is to analyze $$F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*... 1answer 659 views ### Reference request: a conjecture of Rota on positive functions of a random variable Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows: Let p \in \mathbb{R}[x_1, x_2, ...] be a polynomial such that, for any sequence X_1, X_2, ... 1answer 535 views ### Twisted random walks Suppose the points of two random walks in \mathbb{R}^2 are given the step number (or time) as a third coordinate, so that they become paths in \mathbb{R}^3. Here are several pairs of walks of n=... 3answers 2k views ### Packing density of randomly deposited circles on a plane Let's say that I have a rectangular two-dimensional surface of bounded dimensions, [0,A] and [0,B]: Under "no overlap" constraints, I sequentially deposit circles of radii r_c on this surface,... 1answer 347 views ### Reformulation - Construction of thermodynamic limit for GFF I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ... 2answers 247 views ### Weak convergence for discrete-time processes using characteristic functions I am looking for a good reference about the analogues of the Bochner Theorem and the Lévy Continuity Theorem for probability measures on \mathbb{R}^{\mathbb{N}} with the product topology. ... 1answer 453 views ### Formula for U(N) integration wanted Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group. What I would like is a formula ... 1answer 421 views ### A set of questions on continuous Gaussian Free Fields (GFF) As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in \varphi^{4}, but I know that the absense of ... 1answer 376 views ### Can one divide algebraic manifolds ? Make sense: Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2} Let's start from a little bit far. Basic probability theory - chain rule reads:$$ P(AB) = P(A)P(B|A)$$Example: consider n+m balls, where n - white balls, m - black balls, consider A - first ... 1answer 1k views ### Location of maximum of Brownian motion with rough drift I am interested in the distribution of the \text{argmax}_{t \in [0,1]} \{B(t) + f(t)\}, where B is a Brownian motion (or Brownian bridge) and f:[0,1] \to \mathbb{R} is a continuous function. ... 2answers 333 views ### Brownian motion and hitting a Quadrilateral I want to compute the hitting probability of a bounded plane by a Brownian motion starting at the origin. In other words, given the coordinates of a quadrilateral A , can we compute P(T_{A}<\... 1answer 231 views ### Universal decay rate of the Fisher information along the heat flow I'm looking for a reference for the following fact: In the torus \mathbb T^d let me denote by u_t=u(t,x) the (unique, distributional) solution of the heat equation$$ \partial_t u=\Delta u  ...
Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If \$...