Abstract
We present some properties of measures (-Gaussian) that orthogonalize the set of -Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having -Gaussian distribution.
1. Introduction
The paper is devoted to recollection of known and presentation of some new properties of a distribution called -Gaussian. We propose also a method of simulation of i.i.d. sequences drawn from it.
-Gaussian is in fact a family of distributions indexed by a parameter . It is defined as follows.
For it is a discrete -point distribution, which assigns values to and .
For it has density given by for . In particular for Hence it is Wigner distribution with radius .
For -Gaussian distribution is the Normal distribution with parameters and
In Figure 1, we present plots of in blue, in orange, in red, and standard normal density in black.
This family of distributions was defined first in the paper of Boejko et al., in 1997 [1] in noncommutative probability context. Later ([2, 3]) it appeared in quite classical context namely as a stationary distribution of discrete time random field defined by the following relationships: where . It turns out that parameters are related to one another in such a way that there are two parameters and and all others can be expressed through them: Then, one proves that where , (similarly one defines ) and are -Hermite polynomials defined below. It turns out that for the one-dimensional distribution of the process is not defined by moments. This case is treated separately (e.g., in [4]).
As mentioned earlier, here we will consider only the case We will denote family of -Gaussian distributions by or simply
It turns out that there is quite large literature where this distribution appears and is used to model different phenomena. See, for example, [5–9]. Random field defined above is a model of notions that first appeared in noncommutative context and hence establishes a link between noncommutative and classical probability theories.
Remark 1.1. In literature there exists another family of distributions under the same name. It appears in the context of (Boltzmann-Gibbs)-statistical mechanics. See, for example, [10] for applications and review.
Both families are indexed by basically one parameter and for both include ordinary distribution.
In the sequel we will use the following traditional notation used in so-called “-series theory” , for and otherwise, and for (so-called Pochhammer symbol). Sometimes will be abbreviated to , if it will not cause misunderstanding. Notice that and that and tend to , and (Newton's symbol), respectively, as
Remark 1.2. Introducing new variable defined by the relationship we can express -Gaussian density through Jacobi functions defined, for example, in [11]. Namely, we have for with where and are so-called third and second Jacobi Theta functions.
Let us introduce family of polynomials (called -Hermite) satisfying the following three-term recurrence relationship: with Notice that where are Chebyshev polynomials of the second kind defined by
and that where are (“probabilist”) Hermite polynomials, that is, polynomials orthogonal with respect to Gaussian measure.
It turns out that -Gaussian is the distribution with respect to which -Hermite polynomials are orthogonal. This fact can be easily deduced from (1.4).
Thus in particular using the condition we can get all moments of -Gaussian distribution. Hence in particular we have if only
The aim of this paper is to make -Gaussian distribution more friendly by presenting an alternative form of the density for , more easy to deal with (in particular we find the c.d.f. of ), and suggest a method of simulation of i.i.d. sequences having density
2. Expansion of
In this section we will prove the following expansion theorem.
Theorem 2.1. For all one has where
As a corollary we get expression for the c.d.f. function of
Corollary 2.2. The distribution function of -Gaussian distribution is given by
Identity (2.1) can be a source of many interesting identities, which may not be widely known outside the circle of researchers working in special functions.
Corollary 2.3. For all (i) where the polynomials are defined by (1.7).In particular, one has(ii) a particular case of so-called Jacobi's “triple product identity”.(iii)(iv)(v)
Corollary 2.4. is bimodal for where is the largest real root of the equation .
Lemma 2.5. For all and one has(i)(ii)
Remark 2.6. Using the assertion of the above corollary one can approximate the density as well as function of by expressions of the type with great accuracy. This expression is simple to analyze, simulate, and calculate interesting characteristics. Of course one should be aware that for small values of is not nonnegative for all To give a scent of how many 's are needed to obtain the given accuracy we solved numerically (using program Mathematica) the equation for several values and Let us denote by the solution of this equation: We also performed similar calculations for equation
3. Simulation
There is an interesting problem of quick simulation of i.i.d. sequences drawn from - distribution, using few realizations of i.i.d. standard uniform variates. One possibility is the rejection method (see, e.g., [12]). It is not optimal in the sense that it uses least realizations of independent, uniform on variates. But, as one can see below, it works.
To apply this method one has to compare density of the generated variates with another density that has the property of being “easy generated” or another words i.i.d. sequences of variables having this control density are easily obtainable. In the case of density such natural candidate is However this density is unimodal, while the densities for below certain negative value are bimodal. This would lead to inefficient simulation method requiring many trial observations to be generated from to obtain one observation from for sufficiently small That is why we decided to take as “easy generated” density the following one: defined for . However to be sure that this distribution can be used one has to prove the following inequalities presented by the following Lemma.
Lemma 3.1. For and one has where
Function has the plot in Figure 2
Now following [12] we can simulate sequences of independent random variables with -Gaussian distribution. If then such simulation is trivial.
For we use Lemma 3.1 and program Mathematica. We generated sequence of independent random variables from density by inversion method (see [12]), since can be integrated leading to cumulative distribution function (c.d.f.): for where denotes quadratic polynomial in with coefficients depending on while the constants and are known functions of Recall that the inversion method requires solving numerically the sequence of equations where are observations drawn from standard uniform distribution.
Since the function is strictly increasing on its support and its derivative is known, there are no numerical problems in solving this equation. Due to efficient procedure “FindRoot” of Mathematica solving this equation is quick.
Now let us recall how rejection method works in case
Applying algorithm described in [12], the rule to get one realization of random variable having density is as follows.
(1)We generate two variables: and (2)Set (3)If then set otherwise repeat (1) and (2).For details of Mathematica program realizing the above mentioned algorithm, see Appendix.
To see how this algorithm works, we present two simulation results performed (consisting of simulations) with (red dots) and (green dots) in Figure 3.
Unfortunately this algorithm turns out to be very inefficient for close to , more practically less than say One can see this by examining Figure 2. Values of are very large then, showing that one needs very large number of observations from density to obtain one observation from Thus there is still an open question to generate efficiently observations from for values close to
One might be inclined to use formula (2.2) and inversion method applied to its finite approximation and again use procedure “FindRoot”. Well we applied this idea to simulate observations from for It worked giving the results in Figure 4:
We used procedure “Findroot” of Mathematica. It worked, as one can see, however it lasted quite a time to get the result.
Besides, when we tried to get observations from for numerical errors seemed to play an important role as one can notice judging from black dots that appeared between levels and on the picture in Figure 5.
4. Proofs
Proof of Theorem 2.1. Let us denote Hence We have Now let us notice that Now notice that since we see that Thus we can write where and also Hence we can write and consequently We will now use so-called “triple product identity” (see [13, Theorem ., page 497]) that states in our setting, that Now notice that Hence, To return to variable we have to recall definition of Chebyshev polynomials. Namely, we have where is the Chebyshev polynomial of the second kind. More precisely we have here It is well known, that sequence satisfies three-term recurrence equation with and can be calculated directly (see [14, Theorem , page 188]) as in (1.7). Thus we have shown that or equivalently
Proof of Corollary 2.2. We have Now we change variable to and use (2.1). Thus We use now the following, easy to prove, formulae: and and after rearranging terms get (2.2).
Proof of Corollary 2.3. Let for Assertion (i) is obtained directly after noting that
Following (2.1), we get
and (iii) are obtained by inserting and in (2.1) and canceling out common factors. From (2.1) it follows also that values and will be needed. Keeping in mind (4.9) we see that and
On the other hand we see that
Putting in (2.3) we get
Now recall that
(v) To see this notice that -Hermite polynomials are orthogonal with respect to the measure with density . Thus we have
Using (2.1) know that
Observing that function is symmetric and -Hermite polynomials of odd order are odd functions, we deduce that above mentioned identities are trivial for odd Thus, let us concentrate on even Introducing new variable and multiplying both sides of this identity by we get
where Polynomials are called continuous -Hermite polynomials. It can be easily verified (following (1.6) that they satisfy the following three-term recurrence equation:
with Moreover, it is also known that (see, e.g., [13]),
Let us change once more variables in (4.21) and put Then, for all
or for all
Keeping in mind that we see that
Now keeping in mind that for we see that
On the other hand taking into account (4.23) we see that
for Hence we have (2.7).
Proof of Corollary 2.4. Keeping in mind that is symmetric with respect to we deduce that the point of change of modality of must be characterized by the condition Calculating second derivative of the right hand side of (2.1) and remembering that we end up with an equation Now since we get equation in Corollary 2.4 defining
To prove Lemma 2.5 we need the following lemma.
Lemma 4.1. Suppose and Then
Proof. Recall that for we have: and that Thus we have Now notice that for Hence
Proof of Lemma 2.5. (i) We have From this fact we deduce that Now using Lemma (4.1) we have since Now to get (ii) we use routine transformations and sum two geometric series.
Proof of Lemma 3.1. Notice that comparing definitions of and we have Now if we have If then since then
Appendix
Program in Mathematica that generates i.i.d. sequences from .
;
;
However it requires definition of function which is in fact function of this paper. It is quite lengthy. newMM denotes function of this paper. Further function denotes the ratio Parameter denotes number that we insert instead of in the above mentioned formulae. The above procedure produces observation from
Now ; produces table of observation from and plots it in color Then and produce plots for in color. (green) and in color (red).
Acknowledgment
The author would like to thank all three referees for many sugesttions that helped to improve the paper.