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Yes, it's true. You only need basic facts about convergence in distribution (of real rvs). Both can be e.g. be found in Billingsley's book "Convergence of Probability Measures". Let $(Y_n)$ be a s …

(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 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 …

I add two hopefully useful remarks. I consider the general situation. In the sequel $k\geq 1$ and $\lambda=(\lambda_1,\ldots,\lambda_{k+1})$ are fixed, $s:=\sum_{i=1}^{k+1}\lambda_i$ and $n\geq s$. …

Here is a simple description: $$Z\stackrel{d}{=}A\cdot Y_m$$ i.e. $Z$ is distributed as the product of two random variables, where the the factors are independent, $A$ is $\arcsin$-distributed, and $ …

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 completely elementary argument. Let $T_{56}:=\tau_{56}+1, T_{66}:=\tau_{66}+1$ be the first times the patterns have appeared, i.e. $$T_{56}:=\inf\{ n \geq 2\;:\;X_{n-1}=5,\,X_n=6 \}$$ etc. …

Here is an approach via Lagrange inversion. Let $N$ denote the time of the first repeat, and let $T(z)$ (the ``tree function'') be the formal power series satisfying $T(z)=z\,e^{T(z)}$. If $F$ is a …

(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 …

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

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 …

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 …