A tricky integral equation

In the context of reconstruction of climate data from ice cores (see related question at MSE) I came about the following problem. (More background and motivation can be given on demand.) Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be an arbitrary smooth function, and let $p(x)$ be a probability function with $\int_{-\infty}^{\infty} p(x)\text{d}x = 1$. We can write: $$\int_{-\infty}^{\infty} … Read more

Expectation of exponential of a function of independent Rademacher r.v.’s involving the error function

Let $Z,Z’\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.’s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ’}\left[ \exp\left({k\left(e^{\sum_{j=1}^n \mathrm{Erf}(\varepsilon Z_j /\sqrt{d})} -1\right)\left(e^{\sum_{j’=1}^n \mathrm{Erf}(\varepsilon Z’_{j’} /\sqrt{d})} -1\right)}\right) \right] \tag{$\dagger$} $$ where $t\geq 1$, $\varepsilon \in(0,1]$, $k\geq 1$ is an integer, and $\mathrm{Erf}$ … Read more

How sensitive are probability distributions to noise?

I’m trying to prove a result but I’m stuck at the very end of it: I’m having troubles understanding how noise propagates when considering a probability distribution. In other words, if I inject some noise in a vector, how does it change its probability distribution? Let’s give some notation: Let σ:Rd→(0,1)d be the softmax operator … Read more

Bounding an expectation involving i.i.d. standard Gaussians and Rademacher

I have tried to bound the following quantity, but cannot get the “right” (conjectured) bound: ϕ(γ,d,n)=−1+e12nγ2dEX[EZ[∏nj=1(1+γ⟨X(j),Z⟩)]2∏di=1cosh(γ∑nj=1X(j)i)] where γ∈(0,1], d≫1, X=(X(j))1≤j≤n is a collection of i.i.d. standard d-dimensional Gaussian r.v.’s, and Z is uniform on {−1,1}d (Rademacher) independent of the X(j)‘s. Conjecture. Assuming γ2d≤1, as long as nγ4d≪1 we have ϕ(γ,d,n)≪1 I haven’t been able to … Read more

Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). This led to the following question. Let n be a positive integer. Consider the set C_n = \{1,\ldots, 2n^2\}\times\{0,1\}. … Read more

Expectation of multiplied random variables given their individual expectations

Suppose that I have two non-negative real valued random variables x,y∈Z+ that always satisfy x+y≤1. Also suppose that E[x]=1/2 and E[y]=1/4. What is the maximum possible value of E[xy]? Can it be larger than 1/8? More generally, is there a systematic way of analyzing E[xy] for say other values of E[x] and E[y]? (You can … Read more

Сoincidence of discrete random variables

Let ξ,η be a discrete random values and \mathbb E| ξ |, \mathbb E | η | < +\infty, and any value of these values ​​are accepted with a non-zero probability. How to prove that from \mathbb E (ξ \mid η) ≥ η, \mathbb E (η \mid ξ) ≥ ξ follows ξ = η? Answer … Read more

Neighboring number of a permutation

For any positive integer n∈N let Sn denote the set of all bijective maps π:{1,…,n}→{1,…,n}. For n>1 and π∈Sn define the neighboring number Nn(π) as the minimum distance of π-neighbors, or more formally: Nn(π)=min({|π(k)−π(k+1)|:k∈{1,…,n−1}}∪{|π(1)−π(n)|}). For n>1 let En be the expected value of the neighboring number of a member of Sn. Question. Do we have … Read more

Expectation of period length of functions f:{1,…,n}→{1,…,n}f:\{1,\ldots,n\}\to \{1,\ldots,n\}

For n∈N, let [n]:={1,…,n}. Let Fun(n) denote the set of all functions f:[n]→[n]. To f∈Fun(n) associate a sequence seq(f)) defined recursively by seq(f)1=f(1), and seq(f)k+1=f(seq(f)k) for all k∈N. Eventually seq(f) will be periodic, and with per(f) we denote the length of the period of seq(f). By En we denote the expected value of per(f) for … Read more

Approximating E[1/X]\mathbb{E}[1/X]

I am well aware (as for instance discussed here https://math.stackexchange.com/questions/910846/is-it-true-in-general-that-e1-x-1-ex) that for an arbitrary random variable X it does not hold that E[1/X]=1/E[X]. However, when one cannot calculate E[1/X] directly, is there a good (rigorous) way to approximate E[1/X] (ideally, regardless of the distribution of X)? Answer Assume that X≥0. Then, by Jensen’s inequality for … Read more