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Here is a proof which also shows the exponential decay in $n$ for $p\neq q$ Let $0<p,q<1$ and $B_{n,p},B_{n,q}$ independent $\mathrm{Binomial}(n,p)$ resp. $\mathrm{Binomial}(n,q)$ distributed random v …

Here is a way. Start from the exact form of your integral expression (i.e. supply the "Dirichlet factor" $\frac{\Gamma(\alpha_1+\ldots+\alpha_n)}{\Gamma(\alpha_1)\cdots\Gamma(\alpha_n)})$. (1) Writing …

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

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 $ …

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 …

(Since you are looking for a reference, I turn my comment above into an answer:) A proof using classical fluctuation theory is given my answer to Expected supremum of average? (I'm not aware that this …

There a well known generating function methods (the ''symbolic method'' and ''Poissonization'') which can be used to deal with this kind of question. However, I am unable to point to a reference for …

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} …

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 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 …

This is the simplest (discrete) case of Kingman's coalescence. See e.g. https://arxiv.org/abs/0809.4233 and the explanations and references there. The relation to the process considered there can be …

$\lambda(f):=\kappa_f-1$ 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 …

(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 …

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 …

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 …