We consider the persistence probabilities of an autoregressive chain of order one with continuous innovations. In the case of positive drifts, we show that these persistence probabilities are compound-geometric and satisfy a Baxter-Spitzer factorization generalizing that of the random walk. In the case of negative drifts, we exhibit a discrete Van Dantzig problem, which implies that the Baxter-Spitzer factorization never happens, except in a degenerate case. For positive drifts and log-concave innovations, we show that the first passage time in (-∞,0) has a log-convex distribution, whereas in the case of negative drifts and log-convex innovations on ℝ^+, it has a log-concave distribution. The case of the bi-exponential innovations is studied in detail, which leads for positive drifts to an additive factorization of the exponential law.

When the memory parameter of the elephant random walk is above a critical threshold, the process becomes superdiffusive and, once suitably normalised, converges to a non-Gaussian random variable. In a recent paper by the first three authors, it was shown that this limit variable has a density and that the associated moments satisfy a nonlinear recurrence relation. In this work, we exploit this recurrence to derive an asymptotic expansion of the moments and the asymptotic behaviour of the density at infinity. In particular, we show that an asymmetry in the distribution of the first step of the random walk leads to an asymmetry of the tails of the limit variable. These results follow from a new, explicit expression of the Stieltjes transformation of the moments in terms of special functions such as hypergeometric series and incomplete beta integrals. We also obtain other results about the random variable, such as unimodality and, for certain values of the memory parameter, log-concavity.

For the two-parameter Mittag-Leffler function E-alpha,E-beta with alpha > 0 and beta >= 0, we consider the question whether |E-alpha,E-beta(z)| and E alpha,beta(Rz) are comparable on the whole complex plane. We show that the inequality |E-alpha,E-beta(z)| <= E-alpha,E-beta(Rz) holds globally if and only if E-alpha,E-beta(-x) is completely monotone on (0,infinity). For alpha is an element of[1,2) we prove that the complete monotonicity of 1/E-alpha,E-beta(x) on (0,infinity) is necessary for the global inequality |E-alpha,E-beta(z)|>= E-alpha,E-beta(Rz), and also sufficient for alpha = 1. For alpha >= 2 we show that the absence of non-real zeros for E-alpha,E-beta is sufficient for the global inequality |E-alpha,E-beta(z)| >= E-alpha,E-beta(Rz), and also necessary for alpha = 2. All these results have an explicit description in terms of the values of the parameters alpha,beta. Along the way, several inequalities for E-alpha,E-beta on the half-plane {Rz >= 0} are established, and a characterization of its log-convexity and log-concavity on the positive half-line is obtained. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

The monotonicity of the Mittag-Leffler function E_α with respect to the parameter α is investigated, via some convex ordering properties for related random variables. In particular, it is shown that the mapping α↦ E_α(x^α) decreases on (0,2) for all x> 0, that the mapping α↦ E_α(-x^α) decreases on (0,1) for all x≥ 1 and that the mapping α↦ E_α(Γ(1+α)x) decreases on (0,1) for all x∈ℝ^∗. Analogous results are presented for the two parameter Mittag-Leffler functions E_α, β with β≥ α, with an emphasis on the extremal case β=α. Several applications of these results are discussed for Abelian integral equations and subdiffusions.

We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of ℒ_p -spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical q -Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution.

We investigate the log-concavity on the half-line of the Wright function ϕ (-α ,β ,-x), in the probabilistic setting α∈ (0,1) and β≥ 0. Applications are given to the construction of generalized entropies associated to the corresponding Mittag-Leffler function. A natural conjecture for the equivalence between the log-concavity of the Wright function and the existence of such generalized entropies is formulated. The problem is solved for β≥α and in the classical case β = 1-α of the Mittag-Leffler distribution, which exhibits a certain critical parameter α _*= 0.771667... defined implicitly on the Gamma function and characterizing the log-concavity. We also prove that the probabilistic Wright functions are always unimodal, and that they are multiplicatively strongly unimodal if and only if β≥α or α≤ 1/2 and β = 0.

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae’s relationships for 3 F 2 ( 1 ) {}_3F_2(1) . Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán’s inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill’s ratio is discussed in detail.

We prove two inequalities for the Mittag-Leffler function, namely that the function log E_α (x^α ) is sub-additive for 0<α <1, and super-additive for α >1. These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull London Math Soc 2010). For 0<α <2, we also show that E_α (x^α ) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.

We investigate the analytical properties of the $\alpha-$Sun random variable, which arises from the domain of attraction of certain storage models involving a maximum and a sum. In the Fr\'echet case we show that this random variable is infinitely divisible, and we give the exact behaviour of the density at zero. In the Weibull case we give the exact behaviour of the density at infinity, and we show that the behaviour at zero is neither polynomial nor exponential. This answers the open questions in the recent paper Witte and Greenwood (2020).

This work establishes exact formulae for the persistence probabilities pk(θ)=P[Y1⩾0,…,Yk⩾0] of an AR(1) sequence Yn=θYn−1+Xn, n=1,2,… with parameter θ∈R﹨(12,2) and symmetric uniform innovations Xn. The formulae are in terms of certain polynomials, most notably a family that arises in the case −1<θ<12 and was introduced by Mallows and Riordan in the very different context of counting finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results establish convolution-type factorizations in terms of the pk(θ) and their involutive conjugates pk(1/θ) for k=1,…,n and n⩾1. Regarding exact formulae for the pn(θ), these results are used for the cases θ<−1 and θ>2, but they are actually derived under more general conditions and therefore of independent interest, namely, one for AR(1) models with negative θ and continuous innovations, and a second one for AR(1) models with positive θ and continuous and symmetric innovations, the latter extending a classical universal formula of Sparre Andersen for symmetric random walks. We further explain why the case 12<θ<2 does not allow exact formulae for the pn(θ) as in the other cases and show that our results also lead to explicit asymptotic estimates for these probabilities.

A non-negative function $f$ is said to be 'bell-shaped' if $f$ tends to zero at $\pm \infty$ and the $n$-th derivative of $f$ changes its sign $n$ times for every $n = 0, 1, 2, \ldots$ We provide a complete characterisation of the class of bell-shaped functions: we prove that every bell-shaped function is a convolution of a 'P\'olya frequency function' and an *absolutely monotone-then-completely monotone* function. An equivalent condition in terms of the holomorphic extension of the Fourier transform is also given. As a corollary, various properties of bell-shaped functions follow. In particular, we prove that bell-shaped probability distributions are infinitely divisible, and that the zeroes of the $n$-th derivative of a bell-shaped function grow at a linear rate as $n \to \infty$.

We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag-Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible.

We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

We consider three classes of linear differential equations on distribution functions, with a fractional order $\alpha\in [0,1].$ The integer case $\alpha =1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $\alpha-$stable subordinator. From the analytical viewpoint, this law is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fr\'echet cases, and with a Le Roy function for the Gumbel case. By the stochastic representation, we can derive several analytical properties for the latter special functions, extending known features of the classical Mittag-Leffler function, and dealing with monotonicity, complete monotonicity, infinite divisibility, asymptotic behaviour at infinity, uniform hyperbolic bounds.

We investigate certain positive random variables having moments of Gamma type. We give some necessary conditions and some sufficient conditions for their existence. In particular, we observe that the Weber–Schafheitlin formula for the Bessel function allows one to construct non-trivial moments of Gamma type having a signed spectral measure.

We give a simple proof of the moment-indeterminacy of the sequence $(n!)^t$ for $t > 2,$ using Lin's condition. Under a logarithmic self-decomposability assumption, the method conveys to power sequences defined as the rising factorials of a given Bernstein function, and to more general infinitely divisible moment sequences. We also provide a very short proof of the infinite divisibility of all the moment sequences recently investigated in Lin (2017), including Fuss-Catalan's.

We investigate certain analytical properties of the free $\alpha-$stable densities on the line. We prove that they are all classically infinitely divisible when $\alpha\le 1$, and that they belong to the extended Thorin class when $\alpha \leq 3/4.$ The L\'evy measure is explicitly computed for $\alpha =1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $\alpha > 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.

There exist countable groups G with ergodic invariant random subgroups ν such that ν( {H ∈ SubG | H ∼= K } ) = 0 for every subgroup K 6 G. 1. Properly ergodic invariant random subgroups Let G be a countable discrete group and let SubG be the compact space of subgroups H 6 G. Then a Borel probability measure ν on SubG which is invariant under the conjugation action ofG on SubG is called an invariant random subgroup or IRS. If ν is an ergodic IRS of a countable group G, then we obtain a corresponding zero-one law on SubG for the class of group-theoretic properties Φ such that the set {H ∈ SubG | H has property Φ } is ν-measurable. These include those properties that can be expressed using the infinitary language Lω1,ω and thus ν concentrates on a collection of subgroups which are quite difficult to distinguish between. In fact, it seems that all of the examples in the literature have the property that ν concentrates on the subgroups of G of a fixed isomorphism type. For example, the results of Vershik[7], Thomas and Tucker-Drob [6], and Bowen, Grigorchuk and Kravchenko [1] imply that if G is either the group Fin(N) of finitary permutations of N, a diagonal limit of finite alternating groups, or a lamplighter group and ν is an ergodic IRS of G, then there exists a subgroup Kν 6 G such that ν( {H ∈ SubG | H ∼= Kν } ) = 1. Definition 1.1. An ergodic IRS ν of a countable group G is said to be properly ergodic if ν( {H ∈ SubG | H ∼= K } ) = 0 for every subgroup K 6 G. Theorem 1.2. There exist countable groups with properly ergodic IRSs. 1There is a slight inaccuracy in Vershik’s classification [7] of the ergodic IRSs of Fin(N). A corrected statement can be found in Thomas [5]. 1

We investigate the upper tail probabilities of the all-time maximum of a stable Levy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Levy process. We also study the lower tail probabilities of the integrated stable Levy process in the presence of a power positive drift.