Let U 1 , U 2 , . . . be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as n → ∞, the f -vector of the (d + 1)-dimensional convex cone C n generated by U 1 , . . . , U n weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f -vector of C n and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of C n can be expressed through the expected f -vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone C n weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to x −(d+γ) , where γ = 1. We compute the expected number of facets, the expected intrinsic volumes and the expected T -functional of this random convex hull for arbitrary γ > 0.

We consider shot noise processes (X(t)) t≥0 with deterministic response function h and the shots occurring at the renewal epochs 0 = S 0 < S 1 < S 2 . . . of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut)) u≥0 as t → ∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut)) u≥0 converge weakly as t → ∞. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S 1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index −β (with 0 ≤ β < 1/α or 0 ≤ β ≤ α, depending on whether E S 1 < ∞ or E S 1 = ∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut)) u≥0 . The limits are fractionally integrated stable Lévy motions if E S 1 < ∞ and fractionally integrated inverse stable subordinators if E S 1 = ∞.

The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals n, we prove several functional limit theorems (FLTs) in the Skorohod space D[0, 1] endowed with the J 1 -or M 1 -topology for the number K * n (t) of boxes containing at most [n t ] balls, t ∈ [0, 1], and the random distribution function K * n (t)/K * n (1), as n → ∞. The limit processes for K * n (t) are of the form (X(1) − X((1 − t)−)) t∈[0,1] , where X is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small values probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for K * n (t)/K * n (1) is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of K * n (1). First, for any Karlin occupancy scheme with deterministic probabilities (p k ) k≥1 , we obtain an approximation, uniformly in t ∈ [0, 1], of the number of boxes with at most [n t ] balls by a counting function defined in terms of (p k ) k≥1 . Second, we prove several FLTs for the number of visits to the interval [0, nt] by a perturbed random walk, as n → ∞. If the stick-breaking factor has a beta distribution with parameters θ > 0 and 1, the process (K * n (t)) t∈[0,1] has the same distribution as a similar process defined by the number of cycles of length at most [n t ] in a θ-biased random permutation a.k.a. a Ewens permutation with parameter θ. As a consequence, our FLT with Brownian limit forms a generalization of a FLT obtained earlier in the context of Ewens permutations by DeLaurentis and Pittel [5], Hansen [17], Donnelly, Kurtz and Tavaré [6], and Arratia and Tavaré [3].

Let (X1, ξ1), (X2, ξ2), . . . be i.i.d. copies of a pair (X, ξ) where X is a random process with paths in the Skorokhod space D[0, ∞) and ξ is a positive random variable. DefineWe call the process (Y (t)) t≥0 random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of (Y (ut))u>0 as t → ∞. Under the assumptions that the covariance function of X is regularly varying in (0, ∞) × (0, ∞) in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable Lévy motions and their sums when the law of ξ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of ξ belongs to the domain of attraction of a stable law with infinite mean.