A d-DEFORMATION OF FREE GAUSSIAN RANDOM VARIABLES

2005 ◽  
Vol 08 (04) ◽  
pp. 669-680 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Tensor Product ◽  
Hypergeometric Series ◽  
Fock Space ◽  
Recurrence Formula ◽  
Random Variables ◽  
Product Formula ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Gaussian Random Variables ◽  
Special Case

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product [Formula: see text]. Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

scholarly journals Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators

2021 ◽  
Vol 15 (6) ◽  
Author(s):  
Vitonofrio Crismale ◽  
Maria Elena Griseta ◽  
Janusz Wysoczański
Keyword(s):  
Fock Space ◽  
Recursive Formula ◽  
Partial Sums ◽  
Fock Spaces ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Single Operator ◽  
Meixner Class ◽  
Finite Sum ◽  
Poisson Type

AbstractWe study the vacuum distribution, under an appropriate scaling, of a family of partial sums of nonsymmetric position operators on weakly monotone and monotone Fock spaces, respectively. We preliminary treat the case of weakly monotone Fock space, and show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any finite sum. Since the operators are monotone independent, the distribution is the monotone convolution of the free Meixner law above. We also investigate the asymptotic measure for these sums, which can be seen as “Poisson type” limit law. It turns out to belong to the free Meixner class, with an atomic and an absolutely continuous part (w.r.t. the Lebesgue measure). Finally, we briefly apply analogous considerations to the case of monotone Fock space.

Asymptotics for geometric location problems over random samples

1999 ◽  
Vol 31 (3) ◽  
pp. 632-642 ◽  
Author(s):  
K. McGivney ◽  
J. E. Yukich
Keyword(s):  
Finite Number ◽  
Positive Constant ◽  
Complete Convergence ◽  
Location Problem ◽  
Random Variables ◽  
Location Problems ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Random Samples ◽  
Image Position

Consider the basic location problem in which k locations from among n given points X1,…,Xn are to be chosen so as to minimize the sum M(k; X1,…,Xn) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the Xi, i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X1, and c.c. denotes complete convergence.

Asymptotics for geometric location problems over random samples

1999 ◽  
Vol 31 (03) ◽  
pp. 632-642
Author(s):  
K. McGivney ◽  
J. E. Yukich
Keyword(s):  
Finite Number ◽  
Positive Constant ◽  
Complete Convergence ◽  
Location Problem ◽  
Random Variables ◽  
Location Problems ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Random Samples ◽  
Image Position

Consider the basic location problem in which k locations from among n given points X 1,…,X n are to be chosen so as to minimize the sum M(k; X 1,…,X n ) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the X i , i ≥ 1, are i.i.d. random variables with values in [0,1] d and when k = ⌈n/(D+1)⌉ we show that where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X 1, and c.c. denotes complete convergence.

The qq-bit (I): Central limits with left q-Jordan–Wigner embeddings, monotone interacting Fock space, Azema random variable, probabilistic meaning of q

2018 ◽  
Vol 21 (04) ◽  
pp. 1850030 ◽  
Author(s):  
Luigi Accardi ◽  
Yun-Gang Lu
Keyword(s):  
Tensor Product ◽  
Fock Space ◽  
Type Theorem ◽  
Random Variables ◽  
Random Variable ◽  
Asymmetry Index ◽  
Trivial Extension ◽  
Bernoulli Random Variables ◽  
Boson Fock Space ◽  
Interacting Fock Space

The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper1 deals with the functional extension of this CLT, leading to a process. In paper2 the left [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-embeddings. The radical differences between the results of the present paper and those of2 raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper3 for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

scholarly journals Performance Degradation in Cross-Eye Jamming Due to Amplitude/Phase Instability between Jammer Antennas

Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5027
Author(s):  
Je-An Kim ◽  
Joon-Ho Lee
Keyword(s):  
Performance Analysis ◽  
Phase Difference ◽  
Taylor Expansion ◽  
Amplitude Ratio ◽  
Random Variables ◽  
Performance Degradation ◽  
Analytical Performance ◽  
Phase Instability ◽  
Gaussian Random Variables ◽  
First Order

Cross-eye gain in cross-eye jamming systems is highly dependent on amplitude ratio and the phase difference between jammer antennas. It is well known that cross-eye jamming is most effective for the amplitude ratio of unity and phase difference of 180 degrees. It is assumed that the instabilities in the amplitude ratio and phase difference can be modeled as zero-mean Gaussian random variables. In this paper, we not only quantitatively analyze the effect of amplitude ratio instability and phase difference instability on performance degradation in terms of reduction in cross-eye gain but also proceed with analytical performance analysis based on the first order and second-order Taylor expansion.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

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