Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

Decomposition of multivariate infinitely divisible characteristic functions with absolutely continuous Poisson spectral measures

We shall consider the decomposition problem of multivariate infinitely divisible characteristic functions which have no Gaussian component and have absolutely continuous Poisson spectral measures. Under the condition that A = {x;f(x) > 0} is open, where f is the density of spectral measure, we shall show that a known sufficient condition for the membership of the class I_0m (i.e., infinitely divisible characteristic functions having only infinitely divisible factors) is also necessary.     

Moment properties of multivariate infinitely divisible laws and criteria for multivariate self-decomposability

Ramachandran (1969) [9, Theorem 8] has shown that for any univariate infinitely divisible distribution and any positive real number α, an absolute moment of order α relative to the distribution exists (as a finite number) if and only if this is so for a certain truncated version of the corresponding Lévy measure. A generalized version of this result in the case of multivariate infinitely divisible distributions, involving the concept of g-moments, was given by Sato (1999) [6, Theorem 25.3]. We extend Ramachandran’s theorem to the multivariate case, keeping in mind the immediate requirements under appropriate assumptions of cumulant studies of the distributions referred to; the format of Sato’s theorem just referred to obviously varies from ours and seems to have a different agenda. Also, appealing to a further criterion based on the Lévy measure, we identify in a certain class of multivariate infinitely divisible distributions the distributions that are self-decomposable; this throws new light on structural aspects of certain multivariate distributions such as the multivariate generalized hyperbolic distributions studied by Barndorff-Nielsen (1977) [12] and others. Various points relevant to the study are also addressed through specific examples.     

Moments of Exit Times from Wedges for Non-homogeneous Random Walks with Asymptotically Zero Drifts

We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time τ _α from a wedge with apex at the origin and interior half-angle α by a non-homogeneous random walk on ?² with mean drift at x of magnitude O(∥x∥^?1) as ∥x∥→∞. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors stated that τ _α <∞ a.s. for any α. Here we study the more difficult problem of the existence and non-existence of moments ${\mathbb{E}}[ \tau_{\alpha}^{s}]$ , s>0. Assuming a uniform bound on the walk’s increments, we show that for α<π/2 there exists s ₀∈(0,∞) such that ${\mathbb{E}}[ \tau_{\alpha}^{s}]$ is finite for s<s ₀ but infinite for s>s ₀; under specific assumptions on the drift field, we show that we can attain ${\mathbb{E}}[ \tau_{\alpha}^{s}] = \infty$ for any s>1/2. We show that there is a phase transition between drifts of magnitude O(∥x∥^?1) (the critical regime) and o(∥x∥^?1) (the subcritical regime). In the subcritical regime, we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.     

A generalization of the results of Pillai

In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1–E (–x ), 0<1; 0x, whereE (x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ).     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

The dual of an infinitely divisible distribution on ? ^d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ _f ? ρ of ρ to μ in the class of infinitely divisible distributions on ? ^d , where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ? ^d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.     

On the infinite divisibility of the product of two г-distributed stochastical variables

The authors give a positive answer to the question “if X_α is г-distributed of order α, and X_β of order β, with X_α and X_β independent, is X_αX_β infinitely divisible?”. This question, posed by Steutel in Ref. 1, has not been answered up to now, so far as they can find in the literature. In addition they show that the distribution of X_αX_β is a generalized г-distribution.     

On infinitely divisible semimartingales

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.     

On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

On the maximal content of a dam and logarithmic concave renewal functions

For the classical dam model the distribution of the supermum of the dam content between two successive downcrossings of level x>0 by the content process is studied. The result generalizes previous results for the M/G/1 queueing system. The derivation which is rather simple is based on some recent results concerning up- and downcrossings. The resulting distribution is a functional of the solution of a renewal type integral equation occurring frequently in applied probability models. It is shown that this solution is logarithmic concave and that its reciprocal is an infinitely divisible p-function, thus leading to a number of hitherto unknown properties of a certain class of renewal functions.     

Lattice point theory in polyhedra

The exact order of the remainder term is determined in the formula for the number of lattice points in the region α₁∥u₁ + b₁∥ + α₂∥u₂ + b₂∥ + … + α_r∥u_r + b_r∥ ≤ x in dependence on the arithmetical properties of the coefficients α₁, α₂,…, α_r.     

A new characterization of the Poisson distribution

In this note we show that an infinitely divisible (i.d.) distribution function F is Poisson if and only if it satisfies the conditions F(+0) > 0, for any 0 < ∈ < 1 $$\int_{ - \infty }^{I - E} {\frac{{\left| x \right|}}{{1 + \left| x \right|}}} dF = 0$$ and for any 0 < β < 1 $$\int_0^\infty {e^{\alpha xln(x + 1)} } dF< \infty $$      

Recurrence and Transience Criteria for Two Cases of Stable-Like Markov Chains

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel p(x,dy)=f _x (y?x)?dy, where f _x (y) are probability densities of symmetric distributions and, for large |y|, have a power-law decay with exponent α(x)+1, with α(x)∈(0,2). If f _x (y) is the density of a symmetric α-stable distribution for negative x and the density of a symmetric β-stable distribution for non-negative x, where α,β∈(0,2), then the chain is recurrent if and only if α+β≥2. If the function x?f _x is periodic and if the set {x:α(x)=α ₀:=inf _x∈? α(x)} has positive Lebesgue measure, then, under a uniformity condition on the densities f _x (y) and some mild technical conditions, the chain is recurrent if and only if α ₀≥1.     

Operator Ideals Generalizing the Ideal of Strictly Singular Operators

It is well-known that an operator T ∈ L(E, F) is strictly singular if ∥T_x∥≧λ∥x∥ on a subspace Z ? E implies dim Z < + ∞. The present paper deals with ideals of operators defined by a condition — ∥T_x∥≧λ∥x∥ on an infinite-dimensional subspace Z ? E implies Z ? F — F being a ?quasi-injective”? class of BANACH spaces.     

Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

On the structure of the Wishart distribution

In this paper it is shown that every nonnegative definite symmetric random matrix with independent diagonal elements and at least one nondegenerate nondiagonal element has a noninfinitely divisible distribution. Using this result it is established that every Wishart distribution W_p(k, Σ, M) with both p and rank (Σ) ≥ 2 is noninfinitely divisible. The paper also establishes that any Wishart matrix having distribution W_p(k, Σ, 0) has the joint distribution of its elements in the rth row and rth column to be infinitely divisible for every r = 1,2,…,p.     

A multipoint method of third order

LetF be a mapping of the Banach spaceX into itself. A convergence theorem for the iterative solution ofF(x)=0 is proved for the multipoint algorithmx _n+1=x _n ?ø(x _n ), where $$\phi (x) = F\prime_x^{ - 1} \left[ {F(x) + F\lgroup {x - F\prime_x^{ - 1} F(x)} \rgroup} \right]$$ andF′x is the Frechet derivative ofF. The theorem guarantees that, under appropriate conditions onF, the multipoint sequence {x _n } generated by ø converges cubically to a zero ofF. The algorithm is applied to the nonlinear Chandrasekhar integral equation $$\frac{1}{2}\omega _0 x(t)\int_0^1 {\frac{{tx(s)}}{{s + t}}ds - x(t) + 1 = 0}$$ where ω₀>0. A discretization of the equations of iteration is discussed, and some numerical results are given.     

Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

The minimal number of basic elements in a multiset antichain

Let M be a finite set consisting of k_i elements of type i, i = 1, 2,…, n and let S denote the set of subsets of M or, equivalently, the set of all vectors x = (x₁, x₂,…,x_n) with integral coefficients x_i satisfying 0 ? x_i ? k_i, i = 1, 2,…, n. An antichain

is a subset of S in which there is no pair of distinct vectors x and y such that x is contained in y (that is, there is no pair of distinct vectors x and y such that the inequalities x_i ? y_i, i = 1, 2,…, n all hold). Let $\text{∥Y}\textcolor{red}{}\text{∥}$ denote the number of vectors in S which are contained in at least one vector in

and let $\text{∥B}\textcolor{red}{}\text{∥=∑}{}_{\mathrm{x\in }}\text{(X}{}_1\text{+X}{}_2\text{+?+X}{}_{\mathrm{n}}\text{)}$, the number of basic elements in

. For given m we give procedures for calculating min $\text{∥Y}\textcolor{red}{}\text{∥}$ and min $\text{∥B}\textcolor{red}{}\text{∥}$, where the minima are taken over all m-element antichains

in S.