Let Ω be a countable infinite product $\Omega {₁}^{\mathbb{N}}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*}\left[X₁\right]\le liminfSₙ/n\le limsupSₙ/n\le E*\left[X₁\right]$, where $E_{*}$ and E* are the lower and upper expectations. We ask...

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix $H{ₙ}^{\left(0\right)}$ and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of $H{ₙ}^{\left(0\right)}$ and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

Let $F_n$ be the empirical distribution function (df) pertaining to independent random variables with continuous df $F$. We investigate the minimizing point ${\widehat{\tau }}_n$ of the empirical process $F_n-F_0$, where $F_0$ is another df which differs from $F$. If $F$ and $F_0$ are locally Hölder-continuous of order $\alpha$ at a point $\tau$ our main result states that $n^{1/\alpha }({\widehat{\tau }}_n-\tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...

Let Fn be the empirical distribution function (df) pertaining to independent random variables with continuous df F. We investigate the minimizing point ${\widehat{\tau }}_n$ of the empirical process Fn - F0, where F0 is another df which differs from F. If F and F0 are locally Hölder-continuous of order α at a point τ our main result states that $n^{1/\alpha }({\widehat{\tau }}_n-\tau )$ converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...

In this note, we prove an asymptotic expansion and a central limit theorem for the multiple overlap R1, ..., s of the SK model, defined for given N, s ≥ 1 by R1, ..., s = N-1Σi≤N σ1i ... σsi. These results are obtained by a careful analysis of the terms appearing in the cavity derivation formula, as well as some graph induction procedures. Our method could hopefully be applied to other spin glasses models.

Capital $"O"$ and lower-case $"o"$ approximations of the expected value of a class of smooth functions $(f\in C^r\left(R\right))$ of the normalized random partial sums of dependent random variables by the expectation of the corresponding functions of Gaussian random variables are established. The same types of approximation are also obtained for dependent random vectors. This generalizes and improves previous results of the author (1980) and Rychlik and Szynal (1979).

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C(x,y)$ defining an $\alpha$-determinantal point process (DPP). Assuming absolute integrability of the function $C_0\left(x\right)=C(o,x)$, we show that a stationary $\alpha$-DPP with kernel function $C_0\left(x\right)$ is “strongly” Brillinger-mixing, implying, among others, that its tail-$\sigma$-field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch some applications...

Upper estimates are presented for the universal constant in the Katz-Petrov and Osipov inequalities which do not exceed 3.1905.