On sums of independent random variables with unbounded variance, and estimating the average degree in a graph

We prove the following inequality: for every positive integer $n$ and every collection $X_1, \ldots, X_n$ of nonnegative independent random variables, each with expectation 1, the probability that their sum remains below $n+1$ is at least $\alpha > 0$. ...

Let {x₍₁₎≤···≤x_(n)} denote the increasing arrangement of the components of a vector x=(x₁, …, x_n). A vector x∈Rⁿ majorizes another vector y (written <inline-graphic mime-subtype="gif" xlink="S0269964810000240_inline1"><alt-text>$\bf{x} \mathop{\succeq}\...

The random variable simulation problem consists in using a $\boldsymbol {k}$ -dimensional i.i.d. random vector $\boldsymbol {X}^{\boldsymbol {k}}$ with distribution $\boldsymbol {P}_{\boldsymbol {X}}^{\boldsymbol {k}}$ to simulate an $\boldsymbol {n}$ -dimensional i.i.d. random vector $\boldsymbol {Y}^{\boldsymbol {n}}$ so that its distribution is approximately $\boldsymbol {Q}_{\boldsymbol {Y}}^{\boldsymbol {n}}$ . In contrast to previous works, in this paper, ...