# 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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### An expansion from Ramanujan related to birthday problem

A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has ...
0answers
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### Optimal tail bounds for the sample mean

Given a bounded random variable $X$ such that $|X| < M$, I want to know how many iid samples $x_1, \dots, x_n \sim X$ have to be drawn such that the sample mean $\bar{x} = (x_1 + \dots + x_n)/n$ ...
1answer
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### The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook): Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w....
0answers
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### Bridging between Rosethal Inequalities and log convex tails

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $\|X\|_p = (E|X|^p)^{1/p}$. Then we have the classical "Rosenthal-type ...
1answer
259 views

### Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
0answers
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1answer
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### Moment generating function of random unit vector

Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?