F D'acapito

W Blanc

B Dussardier

Different Er 3+ environments in Mg-based nanoparticles-doped optical fibre preforms

Keywords: silica, erbium, nanoparticles, magnesium, EXAFS, fluorescence, optical fiber

Developing new rare-earth ions-doped optical fibres for power amplifiers and lasers requires continuous improvements in fibres spectroscopic properties. To overcome some limitations inherent to silica glass, it is proposed to embed rareearth ions in dielectric nanoparticles. In this article we focus on the modifications of the Er 3+ ions spectroscopy in Mg-silicate nanoparticles doped into optical fiber preforms. Through EXAFS and fluorescence measurements, we demonstrate that different local environments are experienced by Er 3+ ions, attributed to the depolymerization of the phosphate network. These results gain insight into the tayloring of luminescence properties.

Introduction

Developing new rare-earth ions (RE)-doped optical fibres for power amplifiers and lasers requires continuous improvements in the fibres spectroscopic properties besides reduction in device size and economical efficiency. However, some potential applications of REdoped fibres suffer from limitations in terms of spectroscopic properties resulting from RE clustering or inappropriate local environment when they are inserted into silica. An interesting solution consists in using silica as a mechanical host and support of the fibre optical waveguide, and in embedding RE-ions within nanoparticles (NP) of appropriate composition and structure [1,2,3]. The original route proposed by LPMC to obtain NP in fibres is based on the industrial MCVD (Modified Chemical Vapor Deposition) process. As silicate systems have a strong stable immiscibility when they contain divalent metal oxides, we take advantage of thermal treatments inherent to this process to obtain NP through the phase separation mechanism. Through this route, NP are grown in-situ within the material when alkalineearth ions (Mg, Ca and Sr) are incorporated into the fibre core [4,[START_REF] Blanc | Glass Technology -European[END_REF]6]. In particular cases the effect of the divalent metal is also to prevent the clustering of Er 3+ in phosphate phases as shown for similar samples in [7]. In these samples, the spectroscopic behaviour of Er 3+ ions depends on the concentration of the alkaline-earth ions [8]. To explain these modifications, we study, in this article, the local order around Er 3+ by Extended Xray Absorption Spectroscopy (EXAFS) which permits to cast a link between local geometry and optical response. EXAFS at the Er -L III edge has already revealed to be effective in the description of the site of Er 3+ [7] and Yb 3+ [2] in optical fibre preforms with nanoparticles obtained by phase separation. It revealed a marked affinity between the RE and phosphate phases.

Experimental

Sample preparation

Preforms were fabricated by the conventional MVCD technique [9]. The so-called 'solution doping technique' [10] was applied to incorporate magnesium and erbium ions: the core porous layer, doped with GeO 2 and P 2 O 5 , is soaked with an alcoholic solution of ErCl 3 : 6H 2 O and MgCl 2 : 6H 2 O of desired concentrations. After drying of the solvent, the core layer is sintered down to a dense glass layer. Then the tube is collapsed into a solid rod, referred to as preform, at an elevated temperature higher than 1800 • C. Preforms are stretched into 125-µm fibres using a drawing tower at Preprint submitted to J. Non Cryst. Solids September 16, 2013 As previously observed, the mean size of the nanoparticles increases with Mg concentration [8]. It is about 50 nm in Mg-0.1 up to hundreds of nm in Mg-1.5. Optical losses induced by the NP have been discussed in a previous paper [8]. The erbium concentration of 0.01 mol/l was kept constant in the doping solution and the erbium ion concentration in samples was previously estimated to be 200 ppm [8].

Photoluminescence

Emission spectra of Mg-doped core of the preforms were recorded at room temperature. The beam from a continuous 980 nm, 250-mW fiber-coupled laser diode was coupled into the tested preform through a singlemode fiber-coupler. The 1550 nm fluorescence was collected from the second coupler arm and it was directed, through an isolator, to an optical spectrum analyzer (Anritsu MS9701C). The resolution of the OSA was 1 nm. Emission spectra reported in Fig. 2 1 537 eleven points. FWHM and wavelength of the maximum of fluorescence intensity are reported in Fig. 3.

[Mg]solution (mol/l) -0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 -0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6

X-ray Absorption Spectroscopy

XAS measurements have been carried out at the GILDA-CRG beamline BM08 at the European Synchrotron Radiation Facility [11]. The monochromator was equipped with a pair of Si(311) crystals and was run in dynamical (horizontal) focusing mode [12]. The harmonic rejection as well as the vertical focusing was achieved by using a pair of Pd-coated mirrors with a cutoff energy E cuto f f = 18 keV. This optical arrangement permitted to obtain a focal spot of about 200×200 µm on the sample location, small enough to probe the core of the preform. The absorption coefficient was measured in fluorescence mode by using an energy resolving detector (12-elements array High Purity Ge). The spectra are shown in Fig. 4 whereas the related Fourier Transforms are shown in Fig. 5. The structural model used for the quantitative data analysis is the same used in previous literature for this class of systems [13,14,15]: a RE ion linked to a S iO 4 tetradedron with a well defined Er-O distance and Er-O-Si bond angle. Here we note that, due to the similar backscattering amplitude and phase of Si, P and Mg it is not possible to distinguish these three atoms in the second coordination shell. The data analysis was carried out with the ATHENA and ARTEMIS [16] codes and the theoretical XAS paths were generated with the Feff8.10 code [17] starting from the Er 2 S i 2 O 7 crystal structure [18]. The EXAFS signal was modelized by using the main scattering signals originating from the Er-O-Si triangular configuration as thoroughly discussed in [14]. The Debye Waller factor of the second shell was fixed to 0.002 Å 2 whereas the number of oxygen first neighbors was linked to the Er-O bond length by using the data from the Bond Valence Method (BVM, [19]). BVM has been suggested in literature as an effective method in XAS data analysis to reduce the correlation between fit parameters [20]. By using the parameters published in [21] the relationship between the number of O atoms N O and the Er-O bond length R Er-O can be approximated as :

N O ≈ 63.343 -67.499 × R Er-O + 18.695 × R 2 Er-O
The results of the quantitative analysis are shown in Table1.

Results and Discussion

From Fig. 2 and Fig. 3, Er 3+ fluorescence properties can be separated into three Mg-concentration domains. At low concentration (Mg-0 and Mg-0.1), FWHM is about 20-25 nm and the main peak position is at 1536 nm. These values correspond to those measured in a Pdoped silica fiber [START_REF] Desurvire | Erbium-doped fiber amplifiers: Principles and applications[END_REF]. For intermediate Mg concentration (Mg-0.3 and Mg-0.5), FWHM increases up to 45-50 nm, the peak position shifts to shorter wavelengths and the spectrum appears poorly structured. As particle sizes increase compare to previous samples, one may expect that the fluorescence spectrum could be influenced by reabsorption effect. However, this effect would shift peak position to longer wavelengths. For the highest Mg concentration (Mg-1 and Mg-1.5), FWHM and peak position values are 35-40 nm and 1533 nm, respectively, whereas structures appear in the long wavelength side of the line. These two values are close to those reported for samples without P (open symbols in Fig. 3). They can also be compared with values reported for alumino-germano-silicate fibre (type II in [START_REF] Desurvire | Erbium-doped fiber amplifiers: Principles and applications[END_REF]): FWHM and peak position are 40 nm and 1533 nm, respectively. The EXAFS spectra (Fig. 4) show less differences respect to optical fluorescence with the exception of Mg-0 sample which contains more oscillating components than the others as also confirmed by the Fourier Transform (FT) (Fig. 5). As previously reported for this sample [7], the additional signals above the first coordination shells are due to the formation of nuclei with the ErPO 4 structure. This P-cage effect has been already reported [START_REF] Saitoh | [END_REF]. However, the fluorescence spectrum of Mg-0 does not present any features of crystalline phase [24] suggesting that this phase is not spatially extended and consist of only a few coordination shells (around 4-5 Å ). The Mg-doped samples exhibit a dominant oscillation due to a Er-O coordination shell (Fig. 4). The FT confirms this idea as it shows only a dominant main peak at about 1.8 Å (Fig. 5). This situation is typical of rare-earth ions embedded in a glassy matrix. By increasing Mg concentration, R Er-O smoothly decreases from 2.31 down to 2.26 Å and the resulting N 0 decreases from 7 to 6 (Table 1). Both fluorescence and EXAFS measurements demonstrate that Er 3+ ions experience different environements with Mg concentration. In the Mg-0 sample, a well organised solvation shell is formed by P around Er as already observed with this RE [START_REF] Saitoh | [END_REF]24] and Yb [2]. In the Mg-doped samples, the composition of the environment can not be determined exactly from EXAFS measurements as Mg, Si and P can not be distinguished with EXAFS analyses. However, we have reported that, in Mg-doped samples, Er ions are located in the nanoparticles which contain also P and Mg ions [25]. Concentrations of these three elements in the NP are expected to be higher than the mean concentrations reported in 2.1. Then, Mg-0.3 and Mg-0.5 results could be explained by assuming a P-rich and/or well polymerized environment. Indeed, the degree of glass polymerization has been found to be correlated to the RE-O bond length in phosphate glasses [26]. In the case of ultraphosphate glasses, N 0 =7.3 and R Er-O =2.29 Å have been reported by [26], in accordance with Mg-intermediate concentration results. Moreover, FWHM of Er 3+ emission bands in phosphate glasses is usually larger compare to those reported in silicate [27]. When Mg concentration increases, phosphate network tends to depolymerize [28,29]. In the less polymerized metaphosphate glass, N 0 =6.3 and R Er-O =2.23 Å [26], as observed for the highest Mg concentration. FWHM tends also to decrease when Mg is added to a phosphate glass [30]. However, for the highest Mg concentrations, the presence of P in the local environment of Er can not be asserted from these measurements as same characteristics are obtained in P-doped samples (Mg-1 and Mg-1.5) and P-free samples (Mg-0.5noP and Mg-0.7noP).

Sample

N O R Er-O (Å) σ 2 O (Å 2 ) R Er-S i (Å) Mg-0 4

Conclusion

In this article we report on the modification of the erbium ions environment by changing Mg concentration. Different environments are identified by comparing fluorescence and EXAFS measurements. These changes are attributed to the depolymerization of the phosphate network. These results gain insight into the tayloring of luminescence properties of RE-doped optical fibres.

Figure 1 :

 1 Figure 1: SEM pictures of Mg-0.1 (left) and Mg-1.5 (right) preforms.

Figure 2 :FWHM

 2 Figure 2: Emission spectra of Mg-doped preforms (continuous lines) and fibres (dashed lines) recorded at room temperature. Excitation wavelength is 980 nm. Each spectrum is shifted vertically by 0.2.

Figure 3 :

 3 Figure 3: FWHM (triangles) and wavelength of the maximum of fluorescence intensity (squares) vs Mg concentration. Open symbols are related to the samples without P.

  Photoelectron Wavevector k [ °A -1 ]

Figure 4 :

 4 Figure 4: EXAFS spectra of the various Mg-doped samples (0.1-1.5) compared with a sample without divalent codopant (Mg-0). For Mg-1.5 also the fit is shown (+) with the residual (×).

Figure 5 :

 5 Figure 5: Fourier Transforms of the spectra shown in Fig.4. For Mg-1.5 also the fit is shown (+) with the residual (×).

Table 1 :

 1 Results of the quantitative analysis of the EXAFS data. Sample Mg-0 is fitted with a crystalline ErPO 4 model.

			2.30(2)	0.01(5)	-
		4	2.37(2)	0.01(5)	-
	Mg-0.1	7(1)	2.31(6)	0.013(6)	3.7(2)
	Mg-0.3	6.9(7)	2.30(4)	0.013(4)	3.7(1)
	Mg-0.5	6.9(7)	2.29(4)	0.011(4)	3.7(1)
	Mg-0.5noP	6(1)	2.24(3)	0.006(1)	3.6(1)
	Mg-0.7noP	7(1)	2.28(6)	0.014(6)	3.6(2)
	Mg-1	6.3(7)	2.26(4)	0.012(4)	3.6(1)
	Mg-1.5	6.2(7)	2.26(4)	0.012(4)	3.6(1)

Acknowledgements

Authors greatly acknowledge Michèle Ude and Stanislaw Trzesien (LPMC) for the preparation of the preforms, Olivier Tottereau (Univ. Nice Sophia Antipolis, CNRS, CRHEA, UPR 10, Valbonne, France) for SEM/EDX measurements and Ivan Kasik for EPMA measurements realized at the Institute of Photonics and Electronics of the Academy of Sciences of the Czech Republic. GILDA is a project jointly financed by CNR and INFN.


Marc Henry

Koen Jochmans

Bernard Salanié

INFERENCE ON MIXTURES UNDER TAIL RESTRICTIONS

Keywords: AMS 2000 subject classifications: 62G05, 62G20, 62G30, 62G32 mixture model, nonparametric identification and estimation, tail empirical process. 1

Two-component mixtures are nonparametrically identified under tail-dominance conditions on the component distributions if a source of variation is available that affects the mixing proportions but not the component distributions. We motivate these restrictions through several examples. One interesting example is a location model where the location parameter is subject to classical measurement error. The identification analysis suggests very simple closed-form estimators of the component distributions and mixing proportions based on ratios of intermediate quantiles. We derive their asymptotic properties using results on tail empirical processes, and we provide simulation evidence on their finite-sample performance.

1. Introduction. Although finite mixtures are widely used, the dominant mode of inference in such models is parametric; see, e.g., [START_REF] Mclachlan | Finite mixture models[END_REF]. Recent work has established sufficient conditions for (nonparametric) identification of multivariate mixtures; see, notably, [START_REF] Allman | Identifiability of parameters in latent structure models with many observed variables[END_REF], [START_REF] Bonhomme | Nonparametric estimation of finite mixtures[END_REF], [START_REF] Hall | Nonparametric identification of component distributions in a multivariate mixture[END_REF], and [START_REF] Kasahara | Nonparametric identification of finite mixture models of dynamic discrete choices[END_REF]. Unfortunately, fewer identifying restrictions are available when the model of interest is univariate. Several contributions have established identification of mixtures of location models assuming that the error distribution is symmetric (see [START_REF] Bordes | Semiparametric estimation of a two-component mixture model[END_REF], [START_REF] Butucea | Semiparametric mixtures of symmetric distributions[END_REF], [START_REF] Hunter | Inference for mixtures of symmetric distributions[END_REF]). Here, we provide conditions that identify univariate component distributions and associated mixing proportions, and we propose estimators and analyze their properties. As in most of the recent work (see, e.g., [START_REF] Bordes | Semiparametric estimation of a two-component mixture model[END_REF] and [START_REF] Hall | Nonparametric identification of component distributions in a multivariate mixture[END_REF]), our analysis here is confined to mixtures of two components, which is arguably the dominant case of interest.

The models we consider are characterized by an exclusion restriction. That is, we assume the existence of a source of variation that shifts the mixing proportions but leaves the component distributions unchanged. Such an exclusion restriction is natural in several important applications, such as in a measurement-error model or a hidden Markov model. It is also an implication of the fundamental conditional-independence restriction underlying the results of [START_REF] Allman | Identifiability of parameters in latent structure models with many observed variables[END_REF] and [START_REF] Hall | Nonparametric identification of component distributions in a multivariate mixture[END_REF]. In [START_REF] Henry | Partial identification of finite mixtures in econometric models[END_REF] it was shown that, given such an exclusion restriction, both the component distributions and the mixing proportions can be restricted to lie in a non-trivial set. Here we achieve point-identification by complementing the exclusion restriction with a mild restriction on the relative tail behavior of the component distributions. This restriction is quite natural in location model, for example, but can be motivated more generally, as we will discuss in more detail below.

Our identification argument readily suggests plug-in estimators of the mixing proportions and the component distributions. The estimator of the former is based on ratios of intermediate quantiles. The latter is a simple least-squares estimator. Both estimators are available in closed form and are therefore easy to implement. The convergence rate of the estimators is determined by the theory of tail empirical processes (see, e.g., del Barrio, Deheuvels and van de Geer [2007] for a recent account) and is slower than the parametric one, although it can be brought close to it. The intuition for this is the need to rely on the tail behavior of the component densities to infer the mixing proportions. Moreover, if the mixing proportions were known-or could be estimated at the parametric rate-the tail restrictions could be dispensed with and the implied estimator of the component distributions would equally converge at the conventional rate.

Below we first describe the mixture model under study and show identification of its components. These results are next used to motivate the construction of estimators. We provide distribution theory for these estimators and investigate their small-sample performance by means of a small Monte Carlo experiment. All technical details are collected in the Appendix. We omit conditioning variables; the identification analysis extends straightforwardly, and the distribution theory could be extended by using the local empirical-process results in [START_REF] Einmahl | Gaussian approximation of local empirical processes indexed by functions[END_REF].

2. Mixtures with tail restrictions. Let (Y, X) ∈ R×X for X ⊆ R be a pair of random variables for which F (y|x) ≡ P(Y ≤ y|X = x) decomposes as the two-component mixture (2.1)

F (y|x) = G(y) λ(x) + H(y) (1 -λ(x))
for distribution functions G : R → [0, 1] and H : R → [0, 1] and a function λ : X → (0, 1) that maps values x ∈ X into mixing proportions. Everything that follows can be adapted to a situation in which G and H are supported on a subset of the real line with an appropriate change of notation. We complete the mixture model with the following assumption.

Assumption 1. The functions G and H are absolutely continuous on R, and λ is non-constant and bounded away from zero and one on X .

The assumption that the component distributions do not depend on X means that X acts like an instrumental variable. Non-constancy of λ implies instrument relevance. Bounding λ away from zero and one ensures that the mixture is irreducible for all x. This could be relaxed at the cost of more complicated notation. However, bounding the weight function in this manner rules out the possibility of achieving identification of G and H via an identification-at-infinity argument, which is an approach that has been studied in a different context; see, e.g., [START_REF] Andrews | Semiparametric estimation of the intercept of a sample selection model[END_REF].

Our setup accommodates many models. We give two leading examples.

Example 1 (Mismeasured treatment). Let T denote a binary treatment indicator. Suppose that T is subject to classification error. That is, rather than observing T , we observe misclassified treatments, say X. The distribution of the outcome variable Y given X = x is

F (y|x) = P(Y ≤ y|T = 1, X = x) λ(x) + P(Y ≤ y|T = 0, X = x) (1 -λ(x))
for λ(x) = P(T = 1|X = x). The conventional ignorability assumption states that X and Y are independent conditionally on T = t for t ∈ {0, 1}. Then the decomposition of F (y|x) is as in (2.1) with G(y) = P(Y ≤ y|T = 1) and H(y) = P(Y ≤ y|T = 0). Note also that λ is non-constant unless misclassification in T is completely random.

The problem of identification when the treatment indicator is mismeasured has received considerable attention, especially in the context of regression models (see, e.g., [START_REF] Bollinger | Bounding mean regressions when a binary regressor is mismeasured[END_REF], [START_REF] Lewbel | Estimation of average treatment effects with misclassification[END_REF], and [START_REF] Mahajan | Identification and estimation of regression models with misclassification[END_REF]). The conditional ignorability assumption entertained here is an assumption of non-differential misclassification error and has been routinely made elsewhere (see, e.g., [START_REF] Carroll | Measurement error in nonlinear models: a modern perspective[END_REF]).

Example 2 (Hidden Markov model). In a hidden Markov model, outcomes Y are i.i.d. conditional on belonging to a latent state S that follows a Markov process. Suppose that S ∈ {0, 1} has first-order Markov dependence. Then, letting X be the first-order lag of Y , we have that

F (y|x) = P(Y ≤ y|S = 1) P(S = 1|X = x) + P(Y ≤ y|S = 0) P(S = 0|X = x),
which fits (2.1). Further, λ(x) = P(S = 1|X = x) indeed varies with x unless Y and S are independent or S is independently and identically distributed. The hidden Markov model, or regime-switching model, nests switching regressions as well as several versions of stochastic-volatility models.

Our identification approach complements the recent results on hidden Markov models that are based on the availability of multivariate data (see [START_REF] Allman | Identifiability of parameters in latent structure models with many observed variables[END_REF] and [START_REF] Gassiat | Non parametric finite translation mixtures with dependent regime[END_REF]).

To link our model to the literature on multivariate mixtures we also discuss a third example of a mixture that can be cast into (2.1).

Example 3 (Multivariate mixture). Suppose (Y, X) are independent conditional on a latent binary factor T . Then P(X ≤ x, Y ≤ y) factors as a bivariate two-component mixture. Further, because Y is independent of X given T , the conditional distribution of Y given X admits a representation as in (2.1), and so does the distribution of X given Y .

Multivariate mixtures with conditionally-independent components have recently received quite some attention. [START_REF] Hall | Nonparametric identification of component distributions in a multivariate mixture[END_REF] showed that a two-component mixture is point identified with trivariate data but is set identified when only bivariate data are available. These bounds can equally be established using the results in [START_REF] Henry | Partial identification of finite mixtures in econometric models[END_REF].

We will show below that all of G, H, and λ are identified under a dominance condition on the tails of the component distributions. This condition is given in the next assumption.

Assumption 2 (Tail dominance).

(i) The left tail of G is thinner than the left tail of H, i.e., lim y↓-∞ G(y)/H(y) = 0. (ii) The right tail of G is thicker than the right tail of H, i.e., lim y↑+∞ (1 -H(y))/(1 -G(y)) = 0.
Assumption 2 can be motivated by a situation in which large values of the outcome are very likely to have been drawn from G, and small values are very likely to have been generated from H. An important special case where these tail conditions readily follow from the structure of the model is in a location model, as the following example shows.

Example 4 (Location models). Suppose that Y = µ(T ) + U , where T is a binary treatment indicator, U ∼ F independent of T , and µ(t

) ≡ E[Y |T = t]. Then (2.1) yields F (y|x) = F (y -µ(1)) P(T = 1|X = x) + F (y -µ(0)) P(T = 0|X = x).
Suppose that µ(0) < µ(1), and that the derivatives of -ln 1 -F (u) and -ln F (u) grow without bound as u ↑ +∞ and u ↓ -∞, respectively. Then Assumption 2 is satisfied for the distributions G(y) = F (y -µ(1)) and H(y) = F (y -µ(0)).

Theorem 1 formally states our identification result.

Theorem 1 (Identification). Under Assumptions 1 and 2, G, H, and λ are identified.

The proof is constructive. Fix x ∈ X and choose x ∈ X so that λ(x ) = λ(x ). Then

F (y|x ) F (y|x ) = 1 + λ(x ) (G(y)/H(y) -1) 1 + λ(x ) (G(y)/H(y) -1) , 1 -F (y|x ) 1 -F (y|x ) = 1 + λ(x ) ((1 -G(y))/(1 -H(y)) -1) 1 + λ(x ) ((1 -G(y))/(1 -H(y)) -1) ,
follow from re-arranging (2.1). Taking limits, Assumption 2 further implies that

ζ -(x , x ) ≡ lim y↓-∞ F (y|x ) F (y|x ) = 1 -λ(x ) 1 -λ(x ) , ζ + (x , x ) ≡ lim y↑+∞ 1 -F (y|x ) 1 -F (y|x ) = λ(x ) λ(x ) , (2.2)
which are well-defined. These two equations can be solved for the mixing proportion at x . This yields

(2.3) λ(x ) = 1 -ζ -(x , x ) ζ + (x , x ) -ζ -(x , x ) .
Since λ is non-constant, for any x ∈ X there exists a x ∈ X for which such a system of equations can be constructed. The function λ is therefore identified on its entire support. To establish identification of G and H, first note that, using (2.1),

D(y) ≡ G(y) -H(y) = F (y|x ) -F (y|x ) λ(x ) -λ(x ) ,
which is identified. Further, for any x ∈ X ,

H(y) = F (y|x) -λ(x) D(y), G(y) = H(y) + D(y),
so that both component distributions are identified. This concludes the proof of Theorem 1.

If we only assume one-sided tail dominance, then one of the component distributions remains identified.

Corollary 1 (One-sided dominance). Under Assumption 1, G is identified if Assumption 2(i) holds and H is identified if Assumption 2(ii) holds.

As an illustration, consider a two-regime stochastic volatility model. Go back to Example 2 and assume that the outcome has conditional mean zero and conditional variance Sσ

2 G + (1 -S)σ 2 H for positive constants σ 2 G , σ 2 H . Suppose that σ 2 G > σ 2 H .
Then G is the distribution associated with a regime that is characterized by higher volatility. In this case, both tails of G dominate those of H and so Assumption 2(i) fails, but the distribution of the lower-volatility regime, H, remains identified.

3. Estimation and inference. The proof of Theorem 1 suggests simple estimators for both the distributions G and H and the mixing function λ. We will characterize their asymptotic distribution under the following sampling conditions. Assumption 3. X = {x 1 , x 2 , . . . , x q } for some finite integer q ≥ 2 and {Y i , X i } n i=1 constitutes a random sample on (Y, X).

The results below can be extended to weakly-dependent sequences using convergence results for tail empirical processes in [START_REF] Rootzén | Weak convergence of the tail empirical process for dependent sequences[END_REF].

Our procedure builds on estimators of the limit quantities in (2.2). These estimators are constructed based on order statistics of the data. For each x ∈ X , let n x ≡ n i=1 1{X i = x}, and let

F n (y|x) ≡ n -1 x n i=1 1{Y i ≤ y, X i = x}.
For integers ι and κ, let ιn x and r κn x be the (ι + 1)th and (n -κ)th order statistics of the subsample in which the instrument takes on value x. For each (x , x ) ∈ X we estimate ζ -(x , x ) and ζ + (x , x ) by

(3.1) ζ - ιn (x , x ) ≡ F n ( ιn x |x ) F n ( ιn x |x ) , ζ + κn (x , x ) ≡ 1 -F n (r κn x |x ) 1 -F n (r κn x |x )
.

We will choose the order statistics so that ιn x ↓ -∞ and r κn x ↑ +∞ as n ↑ +∞ at an appropriate rate.

Assumption 4 (Order statistics). For all x ∈ X , ι/n x ↓ 0 and ι/ √ n x ln ln n x ↑ +∞, and κ/n x ↓ 0 and κ/ √ n x ln ln n x ↑ +∞ as n ↑ +∞.

Note that F n ( ιn x |x ) > 0 and F n (r κn x |x ) < 1 must hold for ζ - ιn (x , x ) and ζ + κn (x , x ) to be well defined, and so it is recommended to inspect the empirical distribution functions when selecting the order statistics.

With the estimators in (3.1) at hand we can turn to the estimator of λ. Observe that, because λ need not be strictly monotonic, estimating λ(x) by an average of plug-in estimates of (2.3) can be problematic because the denominator of some of the contributions can be zero or be arbitrarily close to it. To avoid such difficulties, we proceed in a different manner. Write • for the Euclidean norm. Let

ζ + (x j ) ≡ ζ + (x 1 , x j ), ζ + (x 2 , x j ), . . . , ζ + (x j-1 , x j ), ζ + (x j+1 , x j ), . . . , ζ + (x q , x j ) ,
and define ζ -(x j ) in an analogous fashion. Because λ is non-negative, it follows from (2.3) that it can equally be expressed as λ

(x) = 1 -ζ -(x) / ζ + (x) -ζ -(x)
, where 1 is a vector of (q -1) ones.

λ ικ n (x) = 1 -ζ - ιn (x) ζ + κn (x) -ζ - ιn (x)
.

This estimator is immune to the problem just mentioned.

Given an estimator of the mixing proportions, estimators of the distribution functions are readily constructed via minimum distance. Recall that

F (y|x) = H(y) + λ(x) D(y),
for each (y, x). Letting F (y) ≡ (F (y|x 1 ), F (y|x 2 ), . . . , F (y|x q )) and λ ≡ (λ(x 1 ), λ(x 2 ), . . . , λ(x q )) , we thus have that, for each fixed y, (H(y), D(y)) = arg min α,β F (y) -α -λβ 2 . As our estimators of H(y) and D(y), we therefore use

(H ικ n (y), D ικ n (y)) ≡ arg min α,β F n (y) -α -λ ικ n β 2 ,
where F n (y) and λ ικ n are defined in an obvious manner. Finally, an estimator of G(y) is then easily constructed as G ικ n (y) ≡ H ικ n (y) + D ικ n (y). These estimators are convenient because they are available in closed form. Indeed, H ικ n (y) and D ικ n (y) are obtained from the ordinary least-squares regression of F n (y|x 1 ), F n (y|x 2 ), . . . , F n (y|x q ) on the q estimated mixing proportions λ ικ n (x 1 ), λ ικ n (x 2 ), . . . , λ ικ n (x q ) and a q-vector of ones.

To ensure that the limit distributions of our estimators are free of asymptotic bias, we need an additional condition on the relative tails of the component distributions.

Assumption 5 (Tail rates). For each x ∈ X ,

G( ιn x )/H( ιn x ) = o p (1/ √ ι) and (1 -H(r κn x ))/(1 -G(r κn x )) = o p (1/ √ κ)
hold.

The following two examples specialize Assumption 5 to densities with log-concave tails and Pareto tails, respectively.

Example 5 (Log-concave tails). Suppose that G and H have log-concave tails, i.e.,

-ln (1 -G(y)) ∼ y σ + G α + G , -ln (1 -H(y)) ∼ y σ + H α + H , as y ↑ +∞, for real numbers α + G , α + H > 1 and σ + G , σ + H > 0, and
-ln G(y) ∼ y σ - G α - G , -ln H(y) ∼ y σ - H α - H , as y ↓ -∞, for real numbers α - G , α - H > 1 and σ - G , σ - H > 0. Then Assumption 4 implies Assumption 5 if both (i) α + G < α + H , or α + G = α + H and σ + G > σ + H ; and (ii) α - G > α - H , or α - G = α - H and σ - G < σ - H hold.
Example 5 does not cover location models with log-concave distributions, for which the α and σ parameters of H equal those of G. In such models, the tails of the error density converge to zero too fast to ensure zero asymptotic bias. To assess the sensitivity of our methods, the simulation experiment below explicitly deals with a mixture of normals, where Assumption 5 fails.

Example 6 (Pareto tails). Let C denote a generic constant. Suppose that G and H have Pareto tails, i.e.,

(

1 -G(y)) ∼ C y -α + G , (1 -H(y)) ∼ C y -α + H , as y ↑ +∞, for positive real numbers α + H > α + G and G(y) ∼ C (-y) -α - G , H(y) ∼ C (-y) -α - H , as y ↓ -∞,
for positive real numbers α - G < α - H . Then Assumption 4 implies Assumption 5.

The main step in our asymptotic analysis is summarized in the following lemma.

Lemma 1 (Intermediate quantiles). Let Assumptions 1-5 hold. Then, as n ↑ +∞, for each

x , x ∈ X , √ κ ζ + κn (x , x ) -ζ + (x , x ) A ∼ ζ + (x , x ) Z + (x ) -ζ + (x , x ) Z + (x ), √ ι ζ - ιn (x , x ) -ζ -(x , x ) A ∼ ζ -(x , x ) Z -(x ) -ζ -(x , x ) Z -(x ),
where the Z + (x) and Z -(x) are standard-normal variates that are both mutually independent and independent across x.

The asymptotic behavior of λ ικ n , G ικ n , and H ικ n can readily be deduced from Lemma 1 by an application of the delta method.

Start with λ ικ n . Lemma 1 implies that

√ ι ζ - ιn (x) -ζ -(x) d → N 0, Σ(x) , Σ(x) ≡ ζ -(x)ζ -(x) + diag{ζ -(x)}.
Write the vector of partial derivatives of 1 -ζ -(x) with respect to the ζ -(x i , x) as d(x). Then we can state the following theorem.

Theorem 2 (Weights). Let Assumptions 1-5 hold. Then, for each x ∈ X ,

√ ι λ ικ n (x) -λ(x) d → N (0, σ 2 (x)), σ 2 (x) ≡ d(x) Σ(x)d(x) ζ + (x) -ζ -(x) 2 ,
as n ↑ +∞.

Consistency of a plug-in estimator of the asymptotic variance in this theorem is immediate.

From Lemma 1 we can deduce the asymptotic covariance between √ ιζ - ιn (x l , x i ) and √ ιζ - ιn (x k , x j ) for i = j, l = i, and k = j. Writing this covariance as γ i,j (l, k), we have

γ i,j (l, k) =            ζ -(x l , x i ) ζ -(x l , x j ) if l = k -ζ -(x k , x j ) ζ -(x j , x i ) if l = j, k = i -ζ -(x l , x i ) ζ -(x i , x j ) if l = j, k = i -ζ -(x j , x i ) ζ -(x i , x j ) -ζ -(x i , x j ) ζ -(x j , x i ) if l = j, k = i 0 otherwise
. This gives the asymptotic covariance between λ ικ n (x i ) and λ ικ n (x j ) as

υ(x i , x j ) ≡ d(x i ) Ω(x i , x j )d(x j ) ζ + (x i ) -ζ -(x i ) ζ + (x j ) -ζ -(x j ) ,
where Ω(x i , x j ) is the asymptotic covariance between √ ιζ - n (x i ) and √ ιζ - n (x j ), constructed using γ i,j (l, k) when i = j and using Σ(x i ) when i = j. Collect the various υ(x i , x j ) in a q × q matrix and call this matrix Υ. Then

√ ι(λ ικ n -λ) d → N (0, Υ) follows. For any function A of x, let A(x) ≡ q -1 q i=1 A(x i ).
To be able to state the asymptotic variance of the estimators of H(y) and G(y) for given y we need to introduce the q × 1 vectors

ξ H ≡ - λ(x) λ -λ(x) λ -λ(x) F (y) -F (y|x) , ξ G ≡ 1 -λ(x) λ -λ(x)] λ -λ(x) F (y) -F (y|x) ,
where we leave the dependence on y implicit. Theorem 3 provides the limit distributions of the estimators of the component distributions.

Theorem 3 (Distributions). Let Assumptions 1-5 hold. Then, for each y ∈ R,

√ ι H ικ n (y) -H(y) d → N 0, σ 2 (y; H) , √ ι G ικ n (y) -G(y) d → N 0, σ 2 (y; G) ,
where σ 2 (y;

H) ≡ ξ H Υξ H and σ 2 (y; G) ≡ ξ G Υξ G , as n ↑ +∞.
As a corollary to Theorem 3 we obtain uniform confidence bounds for G and H.

Corollary 2 (Uniform inference). Let Assumptions 1-5 hold. Then

lim n↑+∞ P sup y∈R √ ι (G ικ n (y) -G(y)) σ n (y; G) > z τ /2 = τ, lim n↑+∞ P sup y∈R √ ι (H ικ n (y) -H(y)) σ n (y; H) > z τ /2 = τ,
where z τ denotes the 1 -τ quantile of the standard normal distribution, and σ n (y; G) and σ n (y; H) denote plug-in estimators of σ(y; G) and σ(y; H), respectively. .320, .679) [.116, .118] Left: G and (λ(0), λ(1)). Right: H and (1 -λ(0), 1 -λ(1)). Upper: .80 quantile. Middle: .90 quantile. Lower: .95 quantile. 10, 000 replications and n = 1, 000. Left: G and (λ(0), λ(1)). Right: H and (1 -λ(0), 1 -λ(1)). Upper: .80 quantile. Middle: .90 quantile. Lower: .95 quantile. 10, 000 replications and n = 1, 000. Left: G and (λ(0), λ(1)). Right: H and (1 -λ(0), 1 -λ(1)). Upper: .80 quantile. Middle: .90 quantile. Lower: .95 quantile. 10, 000 replications and n = 1, 000.

for some constant C. With G and H having log-concave tails, it then follows from this expression that, asymptotically, r behaves like

α + G √ ln n x . So, because 1 -H(r) 1 -G(r) ∼ exp r σ + G α + G - r σ + H α + H , we have that 1 -H(r) 1 -G(r) = O p exp(-(ln n x ) α + H /α + G ) if α + H > α + G O p (1/n x ) if α + H = α + G and σ + H < σ + G ,
from which the conclusion follows.

Technical details for Example 6. The argument is very similar to the one that was used to verify Example 5. We treat the right tail only. We have

κ n x = (1 -G(r)) (1 + o p (1)) = C r -α + G (1 + o p (1)) . Assumption 5 requires that (1 -H(r))/(1 -G(r)) = o(1/ √ κ), that is, that r α + G -α + H = o p (1/ √ κ). This rate condition is satisfied when n κ α + G -α + H α + G = o p (1/ √ κ),
which can be achieved by setting κ = o(n γ + ) and ι = o(n γ -) for

γ + ≡ α + H -α + G α + H -α + G /2 , γ -≡ α - H -α - G α - H -α - G /2 .
Both conditions are weaker than Assumption 4 and therefore implied by it.

Proof of Lemma 1 . Throughout the proof we fix (x , x ) and omit them as arguments unless doing so would cause confusion. We focus on the limit behavior of

√ κ(ζ + ιn -ζ + ); the corresponding result for √ ι(ζ - ιn -ζ -) follows in an analogous fashion. First note that √ κ(ζ + ιn -ζ + ) = √ κ(ζ + ιn -ζ κ ) + √ κ(ζ κ -ζ + ), for ζ κ ≡ (1 -F (r|x ))/(1 -F (r|x )).
Using Assumption 5 together with a small calculation shows that

√ κ(ζ κ -ζ + ) = √ κ   λ(x ) + 1-H(r) 1-G(r) (1 -λ(x )) λ(x ) + 1-H(r) 1-G(r) (1 -λ(x )) - λ(x ) λ(x )   = √ κ O p 1 -H(r) 1 -G(r) = o p (1),
and so the second term is asymptotically negligible. Moving on to the first term, note that

√ κ(ζ + ιn -ζ κ ) = n x κ ζ κ G n x (r) -G n x (r) where G nx (y) ≡ √ n x F n (y|x) -F (y|x) . Let α n (u) ≡ √ n U n (u)
-u where U n denotes the empirical measure of an i.i.d. sample of size n from U[0, 1]. Then, because F (•|x) is a continuous function for all x ∈ X by Assumption 1, G n x (y) = α nx 1-F (y|x ) and G n x (y) = α nx 1-F (y|x ) by an application of the probability integral transform. Thus we may write

(A.1) √ κ(ζ + ιn -ζ κ ) = n x κ ζ κ α n x 1 -F (r|x ) - n x κ α n x 1 -F (r|x ) .
Now, for the first right-hand side term in this expression, by definition of the order statistic r and by adding and subtracting F n (r|x ) and re-arranging,

α n x 1 -F (r|x ) = α n x κ n x + 1 √ n x G n x (r) = α n x κ n x 1 + √ n x κ G n x (r)
.

By the law of the iterated logarithm,

G n x (r) = O a.s ( √ ln ln x ), so ε ≡ - √ n x /κ G n x (r) = o a.s (1).
Hence, the first term of

√ κ(ζ + ιn -ζ κ ) equals (A.2) n x κ ζ κ α n x 1 -F (r|x ) = n x κ ζ κ α n x κ n x {1 -ε} = ζ κ B(1 -ε) + o p (1) A ∼ ζ + Z + (x ),
where B(1 -ε) is a sequence of standard Brownian motions and Z + (x ) is a standard-normal random variable. Here, the second transition follows from the central limit theorem for tail empirical processes (see Theorem 2.1 in [START_REF] Einmahl | Limit theorems for tail processes with application to intermediate quantile estimation[END_REF]), which states that Proof of Theorem 3. Since ι/n ↓ 0 as n ↑ +∞, and the empirical distribution function F n (y|x) is uniformly consistent, the estimators H n (y) and D n (y) are asymptotically equivalent to the intercept and slope coefficients from the infeasible regression of F (y|x i ) on λ ικ n (x i ) (i = 1, 2, . . . , q). Furthermore, √ ι (D ικ n (y) -D(y)) = q i=1 √ ι (λ ικ n (x i ) -λ(x i )) F (y|x i ) -F (y|x) q i=1 λ(x i ) -λ(x) The result then follows.

Proof of Corollary 2. Follows directly from Theorem 3 as its proof is uniform in y.

  Letting ζ - ιn (x) and ζ + κn (x) denote the estimated counterparts of ζ -(x) and ζ + (x), we estimate λ(x) by

Fig

  Fig 1. α = .5

Fig

  Fig 2. α = 1

Fig

  Fig 3. α = 2

  (t) = o p (1), and the fact that Brownian motions are almost surely continuous. For the second right-hand side term in (A.1), on observing that 1 -F (r|x ) = 1 -F (r|x ) ζ κ we can proceed in the same fashion to arrive at(A.3) n x κ α n x 1 -F (r|x ) = ζ + n x κ ρ ζ + α n x κ n x ζ + ρ {1 -ε} + o p (1) A ∼ ζ + Z + (x ),where ρ ≡ lim n↑+∞ n x /n x and Z + (x ) is a standard-normal random variable that by Assumption 3 is independent of Z + (x ). Combining (A.1) with (A.2) and (A.3) then gives√ κ(ζ + ιn -ζ κ ) A ∼ ζ + Z + (x ) -ζ + Z + (x )as claimed.Proof of Theorem 2. By Lemma 1√ ι(ζ - ιn (x , x) -ζ -(x , x)) and √ ι(ζ - ιn (x , x) -ζ -(x , x)) have an asymptotic covariance that is equal to ζ -(x , x)ζ -(x , x) when x = x and to ζ -(x , x)(1+ζ -(x , x))otherwise. The result then follows from an application of the delta method.

  n (y) -D(y))+o p (1), since|λ ικ n (x)-λ(x)| = o p (1) for all x ∈ X . Therefore, √ ι (D n (y) -D(y)) = √ ι F (y) -F (y|x)] λ -λ(x) λ -λ(x) (λ n -λ) + o p (1),and so√ ι (H ικ n (y) -H(y)) = ξ H √ ι (λ ικ n -λ) + o p (1), √ ι (G ικ n (y) -G(y)) = ξ G √ ι (λ ικ n -λ) + o p (1).

Acknowledgements. Some of the results presented here previously circulated as part of Henry, Kitamura and Salanié [2010], whose published version (Henry, Kitamura and Salanié [2013]) only contains results on partial identification. We are grateful to Victor Chernozhukov and Yuichi Kitamura for fruitful discussions. Parts of this paper were written while Henry was visiting the University of Tokyo Graduate School of Economics and while Salanié was visiting the Toulouse School of Economics. The hospitality of both institutions is gratefully acknowledged. Henry's research has received funding from SSHRC Grants 410-2010-242 and 435-2013-0292, and NSERC Grant 356491-2013. Jochmans' research has received funding from the SAB grant "Nonparametric estimation of finite mixtures".

4. Simulation experiment. As a numerical illustration, we simulate data from a location version of the misclassification problem in Example 1. The observed outcomes were generated from the model Y = αT + σU, U ∼ N (0, 1), where the treatment indicator, T , takes on the values -1 and 1, and X = {0, 1}. Treatment was assigned according to λ(1) = P(T = 1|X = x) = 1 4 = 1 -λ(0), and X was generated so that P(X = 1) = 1 2 . Under such a data generating process, the R 2 of the regression of Y on T is

We set σ = 1 and present results for α ∈ {.5, 1, 2}, corresponding to R 2 ∈ {.20, .50, .80}, and, therefore,

Note that larger values of α make the relative thickness of the tails of G and H more pronounced.

The sample size was fixed to n = 1, 000, and 10, 000 Monte Carlo replications were performed at each design point. We estimated the distributions G, H and the weights λ(0), λ(1) using the .80, .90, and .95 quantile of the subsample with X = 0 for r κn 0 and the .20, .10, and .05 quantile of the empirical distribution function of Y given X = 0 for ιn 0 . To ensure our estimates of G and H to be monotonic, we first computed G ικ n and H ικ n via our least squares procedure described above and then applied a re-arrangement algorithm [START_REF] Chernozhukov | Improving point and interval estimators of monotone functions by rearrangement[END_REF] on the estimates.

Figures 123present the results for the three values of α. Each figure provides the mean (solid line) over the point estimates of G and H, the 95% and 5% envelopes of the empirical distribution of the point estimates (dashed lines), as well as the true curves (solid red line). In each figure, the left plots concern G, the right plots concern H. The upper (middle; lower) panels corresponds to the estimates obtained by using the .20th (.10th; .05th) quantile of the empirical conditional distribution of Y given X = 0 for ιn 0 and the .80th (.90th; .95th) quantile of the same distribution for r κn 0 . Below each plot we provide the mean of the estimates of λ(0) and λ(1) (left) and of 1 -λ(0) and 1 -λ(1) (right), together with their standard deviation in square brackets.

For all design points considered the estimates component distributions reflect the shape of the true distributions well. The bias tends to decrease as the cut-off points ιn 0 and r κn 0 are chosen further in the tails of the relevant empirical distribution. This decrease in bias is most pronounced for the design with α = .5. For the other designs the effect is much smaller because the tail dominance is much stronger in these cases due to the larger values of α. Choosing ιn 0 and r κn 0 too far in the tails of the relevant empirical distribution results in larger dispersion of the estimates, however, as is apparent from the plots. This bias-variance trade-off is intuitive. The variance inflation is further concentrated around the tails of G and H, which is again in line with expectations.

Similar conclusions can be drawn for the estimator of the mixing proportions. Indeed, the mixing proportions, too, give roughly correct estimates of the unobserved proportions; recall that λ(0) = 3/4 and λ(1) = 1/4. The bias decreases as α increases. A bias-variance trade-off is again visible. Indeed, the standard deviation of the point estimates increases roughly be a factor of two as we go down the plots within a given figure.

APPENDIX

Technical details for Example 4. Let ϕ(u) ≡ -ln(1 -F (u)) and let ϕ be its first derivative. A mean-value expansion gives

where * is some value that lies between y -µ(1) and y -µ(0). As ϕ increases without bound as y ↑ +∞ and µ(1) > µ(0), the expression on the right-hand side tends to zero as y increases. Assumption 2(i) is verified in the same way.

Proof of Theorem 1. The proof is given in the main text.

Proof of Corollary 1. For brevity we only consider identification of H. Let x , x be as in the proof of Theorem 1. Under Assumption 2(ii) we can still determine ζ + (x , x ) = λ(x )/λ(x ), from which we can learn the ratio

Recalling that, for any x ∈ X ,

the result follows on evaluating this expression at x = x .

Technical details for Example 5 . We verify the second rate; the first follows in an analogous fashion. Throughout, fix x ∈ X and omit it as argument whenever possible. Assumptions 2(ii) and 4 imply that

Further, because κ/n x = 1 -F n (r|x), adding and subtracting F (r|x) gives

Put together, we find κ n x = C (1 -G(r)) (1 + o p (1))


Christopher Trapani

José Echeveste
email: echeveste@ircam.fr

Real Time Tempo Canons with Antescofo

With recent advances in score-following capabilities, it has become possible to envision new timing strategies, to realize previously impractical methods of coordination between a live performer and electronics. Our work centers on the challenge of synchronizing at the end of a musical phrase, where events and processes are timed not from an initial trigger, but occur according to a relative distance towards a predicted future attack.

The key software component is Antescofo, a score-following tool which relies on a strong coupling, through a dynamic programming language, between a machine listening module and a reactive engine. This language allows hierarchical, concurrent, and heterogeneous computer processes to be organized over time, and for external events to be anticipated, with a runtime system that triggers electronic actions in response to variations in performance.

Various programming strategies were implemented, honed and tested with a live performer. The musical material of these sketches focused on the idea of flexible canons, on interactions between a live instrument and a second voice generated from a buffered real-time recording. The different canonic strategies make use of Antescofo's live tempo calculations to create dynamic tempo shifts, to force voices to converge, and to control precise canon effects.

INTRODUCTION

Electroacoustic performance has long relied on the method of front-end synchronization, allowing live players to interact with electronics by triggering events in real time, with processes timed to unfold from these interspersed cues. With a reliable score-following tool coupled with a domain specific language, different strategies of synchronization have become feasible, notably the possibility of aligning at the end of a musical phrase.

Using the tempo canon principles of Conlon Nancarrow ( [START_REF] Gann | The Music of Conlon Nancarrow[END_REF], [START_REF] Thomas | Nancarrow's canons: Projections of temporal and formal structures[END_REF]) as a model, voices at related but independent speeds catch up at a predetermined convergence point (Fig 1). Unlike Nancarrow's rigid grids however, a degree of rhythmic flexibility and vitality becomes possible with the dynamic score-following tool Antescofo, whose programming structure allows high-level macro controls that adapt to realtime tracking data, adapting the playback speed of the canons accordingly to accompany the inevitable fluctuations of a live performance. Electronic events are timed no longer from a given starting point, but in relation to an approaching moment whose distance is continually predicted and recalculated.

ANTESCOFO

Synchronizing electronic actions with a musician performance

Research around the Antescofo system focuses on how to achieve a high-level musical interaction between live musicians and a computer, specifically in the context of mixed music, where the temporal development of musical processes depends on active listening and complex synchronization strategies [START_REF] Cont | On the creative use of score following and its impact on research[END_REF]. We have proposed in [START_REF]Antescofo: Anticipatory synchronization and control of interactive parameters in computer music[END_REF] a novel architecture (Fig 2) that relies on a strong connection between artificial machine listening and a domain-specific real-time programming language for compositional and performative purposes. To instruct the artificial listening engine, the user creates an augmented score whose language integrates both programming tools and musical terms, allowing a unique and flexible temporal organization. The augmented score outlines both the instrumental and the electronic parts and the instructions for their real-time coordination during performance, the specific events that should be recognized in real time, and the actions corresponding to these triggers. During a performance, the language runtime evaluates the augmented score and controls processes synchronously with the musical environment, with the aid of artificial machine listening. This approach has been implemented in the Antescofo system and validated in many live performances.

Some elements of the language

An Antescofo score contains both an instrumental part and the accompaniment actions. As explained above, details of both notated music and electronics are combined in a single augmented score. During live performance, Antescofo reacts to data from the listening machine and from the external environment. This reactive system dynamically takes account of tempo fluctuations and the values of external variables, in order to synchronize accompaniment actions to the real-time interpretation. The possibility of timing events and the actions relative to the tempo, as in a classical score, is one of the main strengths of Antescofo. Within the augmented score language, the composer can decide to associate actions to certain events with delays (either in absolute time or relative to the tempo), to group actions together, to define timing behaviors, to structure groups hierarchically, and to allow these groups to act in parallel. An augmented score is a sequence of instrumental events and actions. The syntax for the instrumental part allows the description (pitches + duration) of events such as notes, chords, trills, glissandi and improvisation boxes [START_REF]Antescofo: Anticipatory synchronization and control of interactive parameters in computer music[END_REF]. Actions are divided into atomic actions which perform an elementary computation and compound actions. Compound actions group multiple actions; these are triggered more or less directly by an event.

NOTE C4 1.0 1/2 receive_action $v1 $v2 CHORD (D4 F#4) 1/2 receive_action $v3 $v4
This excerpt from an Antescofo score shows two musical events with two electronic actions. The first corresponds to a single note (middle C) with a duration of one beat. An electronic action is triggered by Antescofo one eighth-note after the C is detected, where the message "$v1 $v2" is sent to the object receive action.

REALIZATION

Sketches written by Christopher Trapani and played by the clarinetist Jérôme Comte from the Ensemble intercontemporain formed the basis of a six-month study, culminating with a 3-minute final sketch viewable here [5]. In this section we will detail the various challenges addressed during this collaboration. These sketches were realized in the Max/MSP environment. The live performer is recorded into an audio buffer, while a Phase Vocoder object (SuperVP [START_REF] Roebel | Transient detection and preservation in the phase vocoder[END_REF], [START_REF] Roebel | Efficient spectral envelope estimation and its application to pitch shifting and envelope preservation[END_REF]) reads this buffer at specific positions and speeds. The Antescofo program calculates the correct position and speed values according to incoming real-time data, communicating with the Phase Vocoder.

Set-up

The challenge of creating a dynamic canon lies in building, estimating, and updating in real time the relation between the temporality of the score (in beats) and absolute time. All parameters that control the unfolding of the canon depend on this relationship. The Antescofo language allows the composer to manage this kind of control in a precise fashion.

Description of Real-Time Canons

A canon can be defined as the superimposition of a phrase upon a version of itself. In our work, the live musician provides the first voice of the canon, whereas the electronics provide a manipulated second voice in reply. A two-voice canon can thus be characterized as a symbiotic phrase with two components:

• the canonic reply played back from a buffer (the line segment [start,end] in the figures 4),

• the phrase performed live by the musician onto which the canonic reply is superimposed (the line segment [start',sync] in the figures 4).

Keeping in mind that the playback position can never overtake that of the live recording for obvious causal reasons, different types of real-time canons can be defined (Fig 4) The temporal data pertaining to each canon is specified in the score. These specifications are virtual and relative to the tempo so that during performance, these values can be translated into a real and absolute scale.

The canon (4a) starts at the same time as the musician but unfolds at a slower speed. The canon (4b) starts after the musician has played the end of a phrase. This corresponds to a superposition of two independent voices. The canon (4c) begins after the live voice at a slower tempo that will gradually increase until catching up the musician at a predefined convergence point.

Canon speed calculation

In order to determine the precise speed for each canon in the score, the Antescofo program uses real-time tempo calculations as well as temporal estimations of upcoming events. The tempo of the musician as estimated by the listening machine of Antescofo is the key to calculating these estimates of future events. Thus, the speed values are reestimated throughout the entire duration of the canon as soon as new information concerning the position or the tempo of the musician arrives.

Figure 5 represents the different relationships which can be found in the three types of canons mentioned above.

Let t the current position of the musician in the absolute scale; the computation of the speed follow this equation:

canonSpeed = end -t ′ sync -t (1)
where synct is the estimation of the time before the synchronization point and endt ′ is the estimation of the time that remains to be played by the canon until the synchronization point. This is the main piece of code in the score that computes the speed of the canon each time the tempo is updated: $NOW, $RNOW, and $RT_TEMPO are internal variables that represent, respectively, the current time in seconds, the current position of the musician in beats, and the current value of the tempo.

The whenever statement allows actions to be launched when a given condition is verified. Each time the variables corresponding to this condition are updated, the predicate is re-evaluated. Here the body of the whenever is executed each time the variable $RT_TEMPO changes. This process continues until the live musician has reached the end of the canon ($RNOW $\geq$ $end_pos).

$t' represents the audio time-chunk in the buffer already played; @date(t') is the date of the last update of $t' variable. The tempo estimation is used to predict the dates of $end and $sync. $speed represents the speed of the playback and is used as the control parameter of the phase vocoder.

Real-Time Canons with speed curve constraints

Traditionally, canons are defined by a constant speed ratio. Phase vocoder playback however offers the possibility of dynamic tempo flexibility following a predefined pattern. These shifts in speed can be easily stored as a function graph, with care taken to avoid increased playback speeds that would cause the playback buffer to overtake the recording. Equally applicable to prolongation and converging canons, these tempo maps provide a function trans- This strategy is also used to create a specific style of canon (type e in Fig. 1) where two voices cover the same length, both beginning and ending together, but with dynamic changes of speed within this frame. Nancarrow makes frequent use of this effect, switching the speeds between a faster and a slower voice at the exact midpoint of the phrase for a flawless alignment. In our example, the tempo shifts dynamically. From an initial shared starting point, the playback voice unfolds at a slower speed than the live player's, giving the impression of a prolongation canon with a malleable tempo. Later in the canon, phrases in playback are repeated faster than their live renditions, with the tempo again continually calibrated against the approaching convergence point. The figure 7 shows a possible curve to constraint the speed of the canon. If t is the relative position of the musician in the score, the sent speed is:

canonSpeed = f (t) * sync -t ′ sync t f (x) dx (2)
where f(t) is the current value of the curve, sync-t ′ is the time in beats that remains to be played by the canon until the synchronization point and sync t f (x) dx the integral of the curve from the current instant to the synchronization point.

In practice, the speed calculation is updated every 0.05 seconds.

Metadata interactions

Attack recognition data gathered in real time by Antescofo also serves as a tool for storing and recalling the positions of various events. As the live voice is followed, the timing in ms of each detected event is stored as a marker in the buffer. Though playback speed is continually in flux, the positions of these markers remain constant and function as signposts for various meta-processes.

As one basic example, these markers can be read as multiple entry points, so that canonic replies can begin not only at the start of the recorded buffer, but at any recognized event within the musical phrase. They may also serve as signposts for dynamic transposition, sending predetermined transposition values to the phase vocoder when the playback voice reaches the matching phrase.

Markers are also valuable for managing data outside of the signal realm. Using the onset and pitch data for each note in an Antescofo score, it is possible to keep track of the current pitch of each canonic voice, even as the playback rate changes. In our example, a process of adding and subtracting the frequencies of active pitches is established, with the results displayed in real time using the Bach interface [START_REF] Agostini | Bach: An environment for computer-aided composition in max[END_REF]. The resulting pitches are used to pilot the transposition of concatenative synthesis in CataRT [START_REF] Schwarz | Real-time corpus-based concatenative synthesis with catart[END_REF], producing a constantly shifting halo of samples whose harmonic content conforms to the sum and difference tones of the current pitches in the two canonic voices.

Control at the Convergence Point

The most significant challenge of creating converging canons is how to treat the arrival point, since perfect synchrony between a live voice and a buffer is impossible. One shortterm solution involves a last second crossfade in playback, from the buffer to the live voice. While this allows the final attack to be reliably aligned, the drawback is a loss of precision in the passage leading up to the convergence point. The playback speed is also systematically reset to 1. (parity) at the convergence point to prevent playback from overtaking the buffer's playhead. A watchdog mechanism is executed to prevent playback from overtaking the buffer's playhead each time the speed value of the canon is updated: The watchdog mechanism is managed in the same body as the speed computation. The time needed for the position of the playhead ($t') to catch up to the live player at the speed sent just before ($speed) is computed in the variable $delay. The message ph_voc speed 1.0 is scheduled to be launched $delay seconds later. If the tempo is again updated, the abort watchdog command will cancel the message launch and re-schedule a new one.

The final moments of each converging canon involve increasingly perilous calculations. As the predicted arrival point approaches, tempo calculations have more immediate consequences, and small changes in the live player's timing can create drastic alterations in playback speed. With less time left to correct course, the danger of veering ahead of the live buffer's recording point increases. A provisional solution has been the introduction of a braking mechanism which performs a secondary prediction. If the next event anticipated by Antescofo has not yet arrived after a given percentage of the allotted window (80% by default), the speed automatically slows along an exponential curve, so that playback gradually slows until the next predicted point of recalibration arrives. A prototype braking patch was created and tested during the work period, and should be integrated into the patch after further experimentation.

CONCLUSION

The success of this project demonstrates Antescofo's capacity for sophisticated time-related processes, handling both live performance and real-time computing through a single streamlined engine. This could open the door to novel and imaginative reinventions in the realm of rhythm, extending the existing paradigm and encouraging new experiments with ancient musical devices in a wholly contemporary context. The musical examples explored thus far have been brief and intuitive forays that only scratch the surface of the new possibilities offered by this approach.

If tempo canons can be realized in Antescofo with just a few lines of code, one can imagine a larger-scale work with several local-level tempo loops, converging and diverging in fractal-like patterns. The main barrier to composing such music today is the notation and execution of such a highly intricate level of rhythmic interplay, but a tool that generates canonic responses with real-time adaptive capabilities could lead to exciting new forays in the realm of labyrinthine rhythmic layering and detailed canonic writing.

The most important breakthrough for Nancarrow -the primary factor that paved the way for his unprecedented experimentation with multiple layers of tempo and timing -was being allowed to work out his ideas in visual and mathematical terms, directly onto the piano roll. The outcome of this project could signal a similar breakthrough in the electroacoustic realm: a new tool for intricatelytimed canonic writing that could prove to be useful for many composers, enabling them to succinctly and reliably describe complex numerical relationships without recourse to notation.

Figure 1 :

 1 Figure 1: (1a) conventional canon. (1b) converging canon. (1c) diverging canon. (1d) converging diverging canon. (1e) diverging converging canon.

Figure 2 :

 2 Figure 2: General architecture of Antescofo. The score is an input of both the listening machine and the reactive engine. The events recognized by the listening machine are signaled to the reactive engine which schedule the actions at the right time.

Figure 3 :

 3 Figure 3: Antescofo follows the clarinetist's performance which is recorded in parallel into a buffer. Phase Vocoder objects read this buffer at specific positions and speeds controlled by Antescofo as outlined in the score.

  .

Figure 4 :

 4 Figure 4: Different types of real-time canons

Figure 5 :

 5 Figure 5: The diagram represents the different values to be considered for the speed calculation. At the current time t, the dates end (the canon end) and sync (the final synchronization point) are estimated according to the tempo value. The t ′ value corresponds to the time at which the live musician played the current position of the buffer.

  whenever($RT_TEMPO) { $t':=$t'+((@date($t')-$NOW) * $speed) $end:=$NOW+($end_pos-$RNOW) * 60/$RT_TEMPO $sync:=$NOW+($sync_pos-$RNOW) * 60/$RT_TEMPO $speed := ($end-$t')/($sync-$NOW) ph_voc speed $speed }until ($RNOW ≥ $end_pos)

Figure 6 :

 6 Figure 6: Real-Time Canons with speed curve constraint

Figure 7 :

 7 Figure 7: Example of a curve specified by the composer to constraint the speed of the canon.

  canon speed computation ... group watchdog { $delay:=($NOW-$t')/($speed-1) $delay ph_voc speed 1.0 } }until ($RNOW ≥ $end_pos)

Acknowledgments

This work is made possible by a grant from the French National Research Agency (ANR) INEDIT Project (ANR-12-CORD-0009).


Holger Paul Keeler

Bartłomiej Błaszczyszyn

SINR in wireless networks and the two-parameter Poisson-Dirichlet process

Keywords: SINR process, Poisson-Dirichlet process, factorial moment measures

Stochastic geometry models of wireless networks based on Poisson point processes are increasingly being developed with a focus on studying various signal-to-interference-plus-noise ratio (SINR) values. We show that the SINR values experienced by a typical user with respect to different base stations of a Poissonian cellular network are related to a specific instance of the so-called two-parameter Poisson-Dirichlet process. This process has many interesting properties as well as applications in various fields. We give examples of several results proved for this process that are of immediate or potential interest in the development of analytic tools for cellular networks. Some of them simplify or are akin to certain results that are being developed in the network literature. By doing this we hope to motivate further research and use of Poisson-Dirichlet processes in this new setting.

I. INTRODUCTION

To derive accurate analytic tools of cellular networks, stochastic geometry models have been developed with the almost standard assumption that the network base stations are positioned according to a Poisson point process. The aim of these models is often to derive distributional characteristics of various signalto-interference-plus-noise ratio (SINR) values, which are, due to information theoretic arguments, related to network performance characteristics and user quality of service metrics. Besides tractability and 'worst-case' arguments, the Poisson assumption is justified by a recent convergence result [START_REF] Błaszczyszyn | Using Poisson processes to model lattice cellular networks[END_REF] showing that a large class of stationary network configurations give results for functions of incoming signal strengths, such as the SINR, as though the placement of the base stations is a Poisson process when sufficiently large log-normal shadowing is incorporated into the model 1 .

A stochastic process known as the two-parameter Poisson-Dirichlet process has been thoroughly studied over the years owing to the discovery of its many interesting properties and relations to other random structures and applications in various fields such as population genetics, number theory, Bayesian statistics and economics [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF][START_REF] Feng | The Poisson-Dirichlet Distribution and Related Topics: Models and Asymptotic Behaviors[END_REF]. In this letter we detail how a specific case of this well-studied process is equivalent to (what we call) the process of signal-to-total-interference-ratio (STIR) values experienced by a typical user with respect to different base stations of a Poissonian cellular network. The STIR process is trivially related to the signal-to-interference-ratio (SIR) process and further to the SINR one. We then list and apply results that have been derived in different settings and suggest results that may be useful in the future.

INRIA-ENS, 23 Avenue d'Italie, 75214 Paris, France Bartek.Blaszczyszyn@ens.fr, keeler@ens.fr 1 This result is actually more general and holds under log-normal-type distributions, e.g. the Suzuki model [START_REF]Wireless networks appear Poissonian due to strong shadowing[END_REF].

For related work, there is a number of Poisson-based models with a focus on calculating the distribution of the SINR; see [START_REF] Mukherjee | Analytical Modeling of Heterogeneous Cellular Networks[END_REF][START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF] and references therein. Błaszczyszyn and Keeler [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF] characterized the SINR process by obtaining its factorial moment measures. The densities of these measure lead to the joint probability density of the order statistics of the SINR process, which can be used to calculate the coverage probability under some signal combination and interference cancellation models [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF]. Invariance properties of Poisson models have been investigated in connection to general random marks [START_REF] Błaszczyszyn | Equivalence and comparison of heterogeneous cellular networks[END_REF] and the special case of log-normal shadowing marks [START_REF]Wireless networks appear Poissonian due to strong shadowing[END_REF]. Equivalent results in relation to the Sherrington-Kirkpatrick spin glass model have been derived independently in physics, as detailed by Panchenko [START_REF] Panchenko | The Sherrington-Kirkpatrick model[END_REF].

The two-parameter Poisson-Dirichlet process is examined by Pitman and Yor [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF], hence it is also called the Pitman-Yor process, though it was introduced earlier by Pearman, Pitman, and Yor [START_REF] Perman | Size-biased sampling of poisson point processes and excursions[END_REF]. Handa [START_REF] Handa | The two-parameter Poisson-Dirichlet point process[END_REF] derived the factorial moment density (or correlation function) of the process and other useful results. Kingman [START_REF] Kingman | Poisson Processes[END_REF] covers the Poisson point process and its relationships to subordinators and the original (i.e. one-parameter or Kingman's) Poisson-Dirichlet process. In physics a related but different one-parameter process is sometimes also called the Poisson-Dirichlet process [START_REF] Panchenko | The Sherrington-Kirkpatrick model[END_REF] 2 , which is exactly our STIR process. This process and Kingman's one are both special cases of the two-parameter Poisson-Dirichlet process.

We believe that we are the first to illustrate these connections and in doing so it is our hope that certain results on the twoparameter Poisson-Dirichlet process will be adopted and used to develop analytic tools for studying SINR-based characteristics in communication networks.

II. NETWORK MODEL AND QUANTITIES OF INTEREST

We consider the "typical user" approach where one assumes a typical user is located at the origin. On R 2 , we model the base stations with a homogeneous or stationary Poisson point process Φ = {X} with density λ. Define the path-loss function as ℓ(|x|) = (K|x|) β , with constants K > 0 and 2 < β < ∞ assumed henceforth. Given Φ, let {S X } X∈Φ be a collection of independent and identically and arbitrarily distributed random variables representing the random propagation effects (i.e fading and/or shadowing) from the origin to X. Let S be equal in distribution to S X . In this paper we will always (tacitly) require the moment condition E(S point process on the positive R + , as

Θ = {Y } := ℓ(|X|) S X : X ∈ Φ . (1)
Lemma 1. The propagation process {Y } is an inhomogeneous Poisson point process with intensity measure Λ Θ ([0, t)) = at 2 β , where

a := λπE[S 2 β ] K 2 . ( 2
)
This invariance result 4 has been observed a number of times; see, e.g. [START_REF] Błaszczyszyn | Using Poisson processes to model lattice cellular networks[END_REF] for a proof, [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF] for related work, and [START_REF]Wireless networks appear Poissonian due to strong shadowing[END_REF][START_REF] Błaszczyszyn | Equivalence and comparison of heterogeneous cellular networks[END_REF][START_REF] Panchenko | The Sherrington-Kirkpatrick model[END_REF] for generalizations with random marks .

We define the SINR process on the positive half-line R + as

Ψ = {Z} := Y -1 W + (I -Y -1 ) : Y ∈ Θ , (3)
where the constant W ≥ 0 is the additive noise power, and

I = Y ∈Θ Y -1 , (4)
is the power received from the entire network (so that I -Y -1 is the interference).

To represent the signal-to-total-received-power-and-noise ratio, we define the STINR process on (0, 1] as

Ψ ′ = {Z ′ } := Y -1 W + I : Y ∈ Θ . (5)
Working with Ψ ′ is algebraically simpler and information on it gives information on Ψ by the relation

Z = Z ′ /(1 -Z ′ ) and Z ′ = Z/(1 + Z).
For n ≥ 1, we define the factorial moment measure

M ′(n) (t ′ 1 , . . . , t ′ n ) = M ′(n) ((t ′ 1 , 1] × • • • × (t ′ n , 1]) of the STINR process {Z ′ } as M ′(n) (t ′ 1 , . . . , t ′ n ) = E    (Z ′ 1 ,...,Z ′ n )∈(Ψ ′ ) ×n distinct n j=1 1(Z ′ j > t ′ j )    , (6)
where 1 is an indicator function. The equivalent measure of Ψ = {Z} is defined by analogy but in relation to the rectangle

(t 1 , ∞] × • • • × (t n , ∞]. Both measures require two integrals. For x ≥ 0 define I n,β (x) = 2 n ∞ 0 u 2n-1 e -u 2 -u β xΓ(1-2/β) -β/2 du β n-1 (C ′ (β)) n (n -1)! (7)
where

C ′ (β) := 2π/(β sin(2π/β)) = Γ(1 + 2/β)Γ(1 -2/β), ( 8
)
and Γ is the gamma function. Note that

I n,β (0) = 2 n-1 /(β n-1 [C ′ (β)] n ). For all x i ≥ 0 define J n,β (x 1 , . . . , x n ) = (1 + n j=1 x j ) n × [0,1] n-1 n-1 i=1 v i(2/β+1)-1 i (1 -v i ) 2/β n i=1 (x i + η i ) dv 1 . . . dv n-1 , (9)
4 One can assume arbitrary propagation effects by setting, e.g. S = 1, and replacing λ with λ ′ = λE(S where

η 1 = v 1 v 2 . . . v n-1 , η 2 = (1 -v 1 )v 2 . . . v n-1 , η 3 = (1 -v 2 )v 3 . . . v n-1 , • • • , η n = 1 -v n-1 .
(10) For reasonably low n (i.e. n ≤ 20), both these integrals are numerically tractable [START_REF] Keeler | Studying the SINR process in Poisson networks by using its factorial moment measures[END_REF].

Let ti = ti (t ′ 1 , . . . , t ′ n ) := t ′ i /(1 - n j=1 t ′ j )
and the define the n-dimensional unit simple

∆ n = {(t ′ 1 , . . . t ′ n ) : t ′ 1 , . . . t ′ n ≥ 0, t ′ 1 + • • • + t ′ n ≤ 1}
, and 1 ∆n denotes the corresponding indicator function. We now present the factorial moment measures [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF].

Proposition 2. For t ′ i ∈ (0, 1], the factorial moment measure of order n ≥ 1 of the STINR process [START_REF] Mukherjee | Analytical Modeling of Heterogeneous Cellular Networks[END_REF] satisfies

M ′(n) (t ′ 1 , . . . t ′ n ) = n! n i=1 t-2/β i × I n,β (W a -β/2 )J n,β ( t1 , . . . , tn )1 ∆n (t ′ 1 , . . . , t ′ n ). (11
) Furthermore, for t i ∈ (0, ∞) the SINR process (3) has the moment measure

M (n) (t 1 , . . . t n , ) = M ′(n) (t ′ 1 , . . . , t ′ n ) , ( 12
)
where t i = t ′ i /(1 -t ′ i ) and t ′ i = t i /(1 + t i ). Let M ′(n) 0 and M (n) 0
respectively denote the factorial moment measures of the STIR and SIR processes, i.e., M ′(n) and M (n) with W = 0. Hence

M ′(n) (•) = Īn,β (W a -β/2 )M ′(n) 0 (•), (13)
M (n) (•) = Īn,β (W a -β/2 )M (n) 0 (•), (14) where
Īn,β (W

a -β/2 ) = I n,β (W a -β/2 ) I n,β (0) . ( 15
)
This ability to separate the noise term in the factorial moment measures (and densities) is convenient and is reminiscent of factoring out the noise term in the distribution of the SINR under Rayleigh fading, an assumption that is not required, however, in our present setting.

III. TWO-PARAMETER POISSON-DIRICHLET PROCESS

One way to define the two-parameter Poisson-Dirichlet process [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF] for two given parameters 0 ≤ α < 1 and θ > -α is to first introduce a sequence of random variables { Ṽi } by

Ṽ1 = U 1 , Ṽi = (1 -U 1 ) . . . (1 -U i-1 )U i , i ≥ 2, ( 16
)
where U 1 , U 2 , . . . are independent beta variables such that each

U i has B(1 -α, θ + iα) distribution. Note that ∞ i=1 Ṽi = 1 with probability one. Denote the decreasing order statistics { Ṽ(i) } of { Ṽi } by {V i } (V 1 ≥ V 2 ≥ . . . )
, then define the twoparameter Poisson-Dirichlet distribution with parameters α and θ, abbreviated as PD(α, θ), to be the distribution of {V i }. By considering {V i } (or equivalenently { Ṽi }) as atoms of a point process, we see PD(α, θ) as a distribution of a point process. The above approach of defining the PD(α, θ) distribution is related to problems on so-called size-biased sampling and stick-breaking or the residual allocation model, where PD(α, θ) plays a central role. In fact, the distribution of { Ṽi } coincides with that of the size-biased permutation of {V i } [START_REF] Pitman | Random discrete distributions invariant under sizebiased permutation[END_REF].

Another way to define a Poisson-Dirichlet process [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF], more aligned with our setting, is to use the concept of a subordinator having almost surely increasing trajectories. For s ≥ 0, let σ s be a subordinator, and, assuming it has zero drift, its Laplace transform is

E[exp(-zσ s )] = exp -s ∞ 0 (1 -e -zr )Λ(dr) , ( 17
)
where Λ is a measure on (0, ∞), called the Lévy measure, characterizing the subordinator without drift. When multiplied by s, this measure (sΛ(dr)) can be identified with the intensity measure of the Poisson point process of jumps the subordinator makes in the interval (0, s). Let us order and denote these jumps by

V 1 (σ s ) ≥ V 2 (σ s ) ≥ . . . . Clearly σ s = ∞ i=1 V i (σ s ). Let 0 < α < 1,
and then σ s is called an α-stable subordinator if Λ(dr) = Dr -α-1 dr for some constant D > 0, which implies Proposition 6] says that for any s > 0 the sequence {V 1 (σ s )/σ s , V 2 (σ s )/σ s , . . . } has PD(α, 0) distribution 5 .

E[exp(-zσ s )] = exp[-sDΓ(1 -α)z α ]. A crucial observa- tion [3,
Set s = 1 and the constants α = 2/β and D = a, where a is given by ( 2). Then we see that the jumps of the subordinator in the interval (0, s) can be identified with the power values of the signals from all the base stations or, equivalently, the inverse values of the propagation process Θ which, in view of Lemma 1, is an inhomogeneous Poisson process with intensity measure (2a/β)t -1-2/β dt. Consequently, σ 1 represents the interference in our Poisson network model, and its Laplace transform is

E[exp(-zσ 1 )] = E[exp(-zI)] = exp[-aΓ(1 -2/β)z 2/β ].
In other words, the subordinator representation of the Poisson-Dirichlet process PD(α, 0) ([3, Proposition 6]) relates this process to our STIR process. More precisely, denote the increasing orders statistics of {Y } by {Y (i) }, such that Y (1) ≤ Y (2) ≤ . . . , and the decreasing order statistics of {Z ′ } by {Z ′ (i) }. Then we have the following relation, which is a key observation of this letter.

Proposition 3. Assume W = 0. Then the sequence {Z ′ (i) } is equal in distribution to {V i } for α = 2/β and θ = 0. In other words, the STIR process Ψ ′ is a PD(2/β, 0) point process.

The fact that { Ṽi }, defined in [START_REF] Dhillon | Modeling and analysis of k-tier downlink heterogeneous cellular networks[END_REF], form a size-biased permutation of {V i } can be interpreted as follows regarding our STIR process.

Remark 4. Assume when the typical user is choosing its serving base station that, instead of looking for the strongest received signal Y -1

(1) , it makes a randomized decision, picking a base station i with a bias proportional to Y -1 i (hence stronger stations have more chance to be selected). Then its STIR, with respect to the chosen station, has the distribution of Ṽ1 , i.e. B(1-2/β, 2/β). Suppose now that another user positioned with the typical one and subject to the same propagation effects makes its choice of the serving base station by applying the same randomized procedure but excluding the station already selected by the first user. Then the join distribution of the STIR's experienced by these two users is equal to that of the random pair ( Ṽ1 , Ṽ2 ), which can be easily derived from [START_REF] Dhillon | Modeling and analysis of k-tier downlink heterogeneous cellular networks[END_REF]. The above randomized access policy, and the corresponding evaluation of the STIR values, which can be extended to an arbitrary number of users, is of potential interest for managing user hotspots.

IV. SOME USEFUL RESULTS

Appropriately adapted for this setting, we list some interesting results of the PD(2/β, 0) distribution. The first result [3, Proposition 8] applied here shows that the ratio of successive STINR values have beta distributions. Proposition 5. For the STINR process Ψ ′ (W ≥ 0), the random variables

R i := Z ′ (i+1) Z ′ (i) = Y (i) Y (i+1) (18)
have, respectively, B(2i/β, 1) distributions such that P(R i ≤ r) = r i2/β (for 0 ≤ r ≤ 1). Moreover, {R i } are mutually independent.

The fact that each R i is a ratio of Y (i) values indicates that this result (proved in [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF] under assumption W = 0) is invariant of the noise term W . This applies also to the next result, which involves the following variables

A i : = Z ′ (1) + • • • + Z ′ (i) Z ′ (i+1) = Y -1 (1) + • • • + Y -1 (i) Y -1 (i+1)
.

Σ i : = Z ′ (i+1) + Z ′ i+2 + . . . Z ′ i = Y -1 (i+1) + Y -1 (i+2) + • • • Y -1 (i) (20) defined for i = 1, 2, . . . . For γ ≥ 0 let φ β (γ) := 2 β ∞ 1 e -γx x -2/β-1 dx, (19)
ψ β (γ) := Γ(1 -2/β)γ 2/β + φ β (γ). (21)
The next proposition follows [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF]Proposition 11]. Proposition 6. Consider the STINR process Ψ ′ (W ≥ 0). Then 1/Z ′ (i) = 1 + A i-1 + Σ i where A i-1 is distributed as the sum of i -1 independent copies of A 1 , with the characteristic function E[e -γAi-1 ] = (φ β (γ)) i-1 ; Σ i is distributed as the sum of i independent copies of Σ 1 , with the characteristic function E[e -γΣi ] = (ψ β (γ)) -i ; and A i-1 and Σ i are independent.

Remark 7. By observing that Σ

-1 i = Y -1 (i) /(Y -1 (i+1) + Y -1 (i+2)
+ . . .), the above result in the setting of successive-interference cancellation (with no noise, W = 0) can be compared to a result [12, Theorem 1] and its generalization [6, Proposition 21] on the ratio of the k th strongest propagation process and a successively reduced interference term. Moreover, the ratio of independent random variables (1

+ A i-1 )/Σ i = (Y -1 (1) + • • • + Y -1 (i) )/(Y -1 (i+1) + Y -1 (i+2) + • • • )
relates the above result to a recent signal combination model in the STIR (W = 0) scenario [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF]. The difference between the STINR and STIR results suggests that the noise term W can add a significant layer of complexity to the models. Proposition 6 leads to a Laplace transform result (cf [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF]Corollary 12]), which can be compared to a previous observation [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF]Remark 18].

Corollary 8. The inverse of the i th strongest STIR (W = 0) value, 1/Z ′ (i) , has the Laplace transform

E[e -γ/Z ′ (i) ] = e -γ (φ β (γ)) i-1 (ψ β (γ)) -i . (23)
Furthermore, a previous result [START_REF] Keeler | SINR-based k-coverage probability in cellular networks with arbitrary shadowing[END_REF]Corollary 7] gives an expression for the tail of the distribution function of the i th strongest STINR (W ≥ 0) value.

Remarkably, the interference I, as by ( 4), and noise W can be recovered from the STINR processes. Indeed, the first statement of the following result is trivial, while the second one can be proved using the same arguments as [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF]Proposition 10].

Proposition 9. For the STINR process (W ≥ 0), W/I = ∞ i=1 Z ′ (i) -1
-1, and W + I = (L/a) -β/2 , where the limit L := lim i→∞ i(Z ′ (i) ) 2/β , both exists almost surely and for all p-means with p ≥ 1.

The densities of the factorial moment measures M ′(n) of the STINR process can be used to find an expression for the joint probability density of the order statistics of the STINR process [START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF]Proposition 20]. But, the expression on the right-hand-side of [START_REF] Kingman | Poisson Processes[END_REF] appears too unwieldy to differentiate with respect to more than a couple variables. However, using the representation ( 16), the factorial moment density of the PD(α, θ) process was derived in closed-form [10, Theorem 2.1], which implies the following new result for our STINR process. For n ≥ 0 denote c n,α,θ =

n i=1 Γ(θ + 1 + (i -1)α)/(Γ((1 -α)Γ(θ + iα)); in particular c n,2/β,0 = (2/β) n-1 Γ(n)/(Γ(2n/β)Γ(1 -2/β) n ).
Proposition 10. For the STINR process Ψ ′ (W ≥ 0), the n th factorial moment density is given by

µ (n) (t ′ 1 , . . . t ′ n ) := (-1) n ∂ n M ′(n) (t ′ 1 , . . . t ′ n ) ∂t ′ 1 . . . ∂t ′ n (24) = c n,2/β,0 Īn,β (W a -β/2 ) n i=1 t ′-(2/β+1) i 1 - n j=1 t ′ j 2/βn-1
for (t ′ 1 , . . . , t ′ n ) in ∆ n and 0 otherwise. This result follows from including the noise term W , via (13), and using [10, Theorem 2.1]. We are unaware of anybody showing the equivalence of Propositions 2 and 10, either by differentiating the measure [START_REF] Kingman | Poisson Processes[END_REF] or integrating the density (24).

Another way for calculating the joint density of the order statistics of the STIR process is offered by the result [START_REF] Handa | The two-parameter Poisson-Dirichlet point process[END_REF]Theorem 5.4], where the two-parameter Dickman function was introduced as ρ α,θ (s) :

= P(V 1 < 1/s) (25)
where V 1 is the largest value of the PD(α, θ) process, which can be computed as follows:

ρ α,θ (s) := ∞ n=0 (-1) n c n,α,θ n! I n,α,θ (s), (26)
where

I n,α,θ (s) = ∆n n i=1 1 [1,∞) (st ′ i ) t ′α+1
m j=1 t ′ j t ′ m 1 ∆m (t ′ 1 , . . . , t ′ m )1 {t ′ 1 >•••>t ′ n } . (28)
The two non-zero parameters of the Dickman function explains why PD(α, θ) is needed in (25), and not just PD(α, 0).

V. CONCLUSION

We showed the relationship between the SINR process, which is an important object in the study of the performance of cellular networks, and the two-parameter Poisson-Dirichlet process. We presented some results recently proved for the former process, which have interesting interpretation in terms of the STINR process (easily related to the SINR one). Our goal is to encourage further research aimed at building bridges between these two, until now, separate research areas.

1 ×

 1 and for n = 1, 2, . . . , with I n,α,θ (s) = 0 whenever n > s, which makes that the right-hand-side of (26) is actually a finite sum; cf[START_REF] Handa | The two-parameter Poisson-Dirichlet point process[END_REF] Section 4].Remark 11. Recall that when α = 2/β and θ = 0, V 1 is equal in distribution to the strongest STIR value Z ′(1) and thus (26) should be compared to[15, Corollary 7, with k = 1, valid for W ≥ 0] or [16, Theorem 1, taken for single-tier network, valid for SINR values greater than one]. Proposition 12. For the STIR process Ψ ′ (W = 0) and for each m = 1, 2, . . . , the joint probability density of {Z ′ (1) , . . . , Z ′ (m) } is given by f m,β (t ′ 1 , . . . t ′ m ) := c m,2ρ 2/β,2m/β 1 -

β ) < ∞. We define the propagation (loss) process

, considered as a[START_REF]Wireless networks appear Poissonian due to strong shadowing[END_REF] It appears as the thermodynamic (large system) limit in the low temperature regime of Derrida's random energy model and a key component of the so called Ruelle probability cascades, which are used to represent the thermodynamic limit of the Sherrington-Kirkpatrick model for spin glasses.[START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF] We introduce the propagation process for historical reasons, in particular to be consistent with[START_REF] Blaszczyszyn | Studying the SINR process of the typical user in Poisson networks by using its factorial moment measures[END_REF] and papers cited therein. Otherwise the process of received powers can be considered.

To define the original Kingman's Poisson-Dirichlet process with PD(0, θ) distribution, a subordinator known as the Moran[START_REF] Kingman | Poisson Processes[END_REF] or Gamma subordinator[START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF] is used. A subtle combination of this and the α-stable subordinator is used to define the PD(α, θ) process, but we omit the details for brevity.

Acknowledgements: We thank M.K. Karray for carefully proofreading the manuscript.