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Here is an "arbirarily nice" example with closed form results. Let $X_1,X_2,\ldots$ be i.i.d. real random variables with partial sums $S_k:=\sum_{i=1}^kX_i$ and let $M_n:=\sup_{k\leq n} \frac{S_k}{k} …

(1) Simple bounds: For $m=1$ (see here (or here and here)) the inequalities \begin{align*} \sqrt{\frac{\pi}{2}}{1\over \lVert p\rVert_2}&\leq \mathbb{E}(M_1)\leq \sqrt{\frac{\pi}{2}}{1 \over \lVer …

Here is a (surprising) proof using Cauchy-Schwarz and "rearrangement". The following lemma will be the key. Lemma : Let $X,Y$ be independent integer-valued rvs, then \begin{align*} (a)\; &\mbox{ for …

Here's an alternative proof based on probabilistic arguments (showing different aspects). Let $$f_n(x):=\sum_{j=0}^n { x \choose j}=[t^n]\,\frac{(1+t)^x}{1-t}\;\;,$$ and let $^\prime$ denote deriv …

Here is a another approach. For convenience I write $n$ instead of $N$, and $A_n$ for $A$. By definition $$\det(A_n) = \sum_{\pi\in S_n} \operatorname{sign}(\pi) \prod_{i=1}^n a_{i,\pi(i)}$$ By …

Here is a solution for even $k\leq d$. I. A lower bound for even $k$. Simple lower bound (for $k$ even) follows from standard combinatorics of events and Bonferroni's inequalities. We need the followi …

Here is another method. Since the uniform distribution on the unit disk is rotationally symmetric (invariant under orthogonal transformations), the problem can be reduced to a random walk problem in $ …

This question was completely settled by J.A. Fill here: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.641

(I change notation from $N,C$ to $n,c$ since I use capitals throughout to denote rvs). Let $X_i$ be the random variable "number of the box the $i$-th coin", then $X_1,X_2,\ldots$ is an i.i.d. sequenc …

(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.) A simple way is to use …

$\lambda(f):=\kappa_f-1$ is called "the coefficient of coalescence of $f$" here: https://msp.org/pjm/1982/103-2/pjm-v103-n2-p03-p.pdf (note the typo on p.269, the correct definition appears on p.27 …

General solution: assume there are $n$ passwords, $k$ hashes and $x_i$ passwords hashing to $i$. The expected time (drawing without replacement) for the drawing of the first password with hash $i$ is …

I reformulate slightly, please check. You are considering a sequence $X_1,X_2,\ldots$ of (discrete) i.i.d random variables and want an upper bound for the probability $\mathbb{P}(R>n)$ in terms of $ …

Here is a proof using generating functions. Let $c\geq 2$ and $k\geq 2$ be fixed. Let $X=(X_1(n),\ldots,X_k(n))$ be the $k$-tuple of occupancy numbers at time $n$, i.e. $X_i(n)$ = number of bins of ty …

The following inequality holds: $$\mathbb{P}(C_k(m)\geq 1)\geq \mathbb{P}(\mathrm{Bin}(m, \lVert A\rVert_k)\geq k)$$ where here and in the sequel $\mathrm{Bin}(n,p)$ denotes a binomially distributed r …