Antoine Izoulet
email: antoine.izoulet@tridelta.fr

Sophie Guillemet-Fritsch

Claude Estournès
email: estournes@chimie.ups-tlse.fr

Jonathan Morel
email: jonathan.morel@tridelta.fr

Microstructure control to reduce leakage current of medium and high voltage ceramic varistors based on doped ZnO

Keywords: Leakage current, Microstructure, Pyrochlore, Varistor, Zinc oxide

The leakage current (I f ) and the non-linearity coefficient (α) are crucial parameters in varistors. This work, deals with the optimization of the electrical characteristics of medium and high voltage varistors based on the ZnO-Bi 2 O 3 -Sb 2 O 3 system. First, the aim was to allow the formation and the stabilization of the spinel phase by reducing the heating rate. To do so, the right sintering temperature and dwell time have to be chosen to obtain a breakdown field suitable for industrial requirements. During cooling, the spinel phase can return to a pyrochlore phase, which contributes to the increase of leakage current. In a second part, this reaction was prevented by faster cooling in an appropriate temperature range. The fine-tuning of both the heating and cooling phases leads to a significant decrease of the leakage current. Moreover, the value of the non-linearity coefficient was increased by 80%, due to better and more homogeneous wetting of the ZnO grains by the Bi-rich phase.

Introduction

Mainly developed from zinc oxide (ZnO), MOVs (Metal Oxide Varistors) present a strong nonlinear current-voltage characteristic. [START_REF] Matsuoka | Non-ohmic properties of zinc oxide ceramics[END_REF][START_REF] Gupta | Applications of zinc oxide varistors[END_REF] Varistors are characterized by the threshold (or breakdown) voltage U s (=U 1mA ), at which point they switch from a very high conductive state to a strongly insulating state, and the coefficient of non-linearity α. In normal operation (i.e. below its threshold voltage), or when returning to normal conditions following a surge operation, the varistor lets a small current through, the leakage current I f . The leakage current must be as low as possible.

It is well known that the electrical properties of MOVs are directly related to the microstructure and to the formulation of the material. [START_REF] Clarke | Varistor ceramics[END_REF] U s is mainly governed by ZnO grain size. With the composition used in this work, i.e. ZnO-Sb 2 O 3 -Bi 2 O 3 (ZSB), the microstructure consists of a majority of ZnO grains in a continuous and interconnected network of bismuth-rich phases. These Bi-rich phases are located at the grain boundaries and at the triple junctions of ZnO grains. The Bi-rich network also includes a phase with a spinel structure Zn 7 Sb 2 O 12 mainly localized at triple junctions of ZnO grains. [START_REF] Wong | Microstructure and phase transformation in a highly non-ohmic metal oxide varistor ceramic[END_REF][START_REF] Santhanam | Microstructural evolution of multicomponent ZnO ceramics[END_REF][START_REF] Inada | Microstructure of nonohmic zinc oxide ceramics[END_REF][START_REF] Clarke | The microstructural location of intergranular metal-oxide phase in a zinc oxide varistor[END_REF] Olsson et al. showed that this configuration is necessary to obtain the desired varistor effect and to allow efficient oxygen transport during post-sintering annealing. But this continuous Bi-rich skeleton around ZnO grains can also provide an additional current contributing to an increased leakage current. [START_REF] Olsson | Development of interfacial microstructure during cooling of a ZnO varistor material[END_REF] Stucki and Greuter [START_REF] Stucki | Key role of oxygen at zinc oxide varistor grain boundaries[END_REF] correlated the increase of I f to the presence of an excess of oxygen in these areas. The present work focused on the Bi-rich phases to try to minimize their amount in the ceramic. However, due to the multiplicity of different polymorphic phases of Bi 2 O 3 , [START_REF] Metz | Electrical properties of varistors in relation to their true bismuth composition after sintering[END_REF][START_REF] Onreabroy | Electrical properties of a ZnO-CoO-Bi 2 O 3 varistor[END_REF] and their proportions, which varies strongly with the sintering conditions and the formulation, it is difficult to control their formation and the amount produced. Moreover due to the high volume of medium (LDC1) and high (LDC2) voltage varistors (50 and 72 cm 3 respectively), a large thermal gradient occurs between the core and the surface, complicating the control of the distribution of such secondary phases. Nevertheless in the ZSB system, other Bi-rich phases appear. The pyrochlore phase (PY) Bi 3 Zn 2 Sb 3 O 14 , is formed after reaction between Sb 2 O 3 , ZnO and Bi 2 O 3 , in the early stages of sintering. This phase decomposes from 850 • C to yield the spinel phase (SP). [START_REF] Inada | Microstructure of nonohmic zinc oxide ceramics[END_REF] However, this last reaction is reversible upon cooling. It has been described [START_REF] Metz | Electrical properties of varistors in relation to their true bismuth composition after sintering[END_REF][START_REF] Clayton | The electrical and defect properties of Bi 3 Zn 2 Sb 3 O 14 pyrochlore: a grain-boundary phase in ZnO-based varistors[END_REF] that the PY phase has both ionic and electronic conductivities which tend to increase the leakage current. Olsson et al. [START_REF] Olsson | Development of functional microstructure during sintering of a ZnO varistor material[END_REF] argue that the PY phase has a detrimental effect on the efficiency of the ZnO/ZnO interface to act as an electrical barrier. The PY structure is essential to microstructure development because it avoids ZnO grain growth, it retains the Bi 2 O 3 , thereby avoiding its premature volatilization. Moreover, it might reduce grain boundary mobility. [START_REF] Gupta | Applications of zinc oxide varistors[END_REF][START_REF] Mergen | Microstructural relations on BZS Pyrochlore-ZnO mixtures[END_REF] But the PY phase is undesirable after sintering. Greuter et al. [START_REF] Greuter | High Field Strengh Varistor Material[END_REF] showed that the absence of this phase does not affect the nonlinear properties of the varistor since it is not directly involved in the nonlinear behavior. The reactions of formation, decomposition and reformation of the PY phase are described below [START_REF] Inada | Microstructure of nonohmic zinc oxide ceramics[END_REF][START_REF] Peiteado | Thermal evolution of ZnO-Bi 2 O 3 -Sb 2 O 3 system in the region of interest for varistors[END_REF] : Several possibilities can be explored to prevent the formation or the reformation of the Bi 3 Zn 2 Sb 3 O 4 phase. First it is possible to change the composition and especially the ratio of Sb 2 O 3 to Bi 2 O 3 . It has been shown that a formulation with a Sb/Bi ratio 1 enables the formation of PY phase during sintering. [START_REF] Peiteado | Thermal evolution of ZnO-Bi 2 O 3 -Sb 2 O 3 system in the region of interest for varistors[END_REF] Another way to control the formation of the PY is to act on the cooling rate. Rapid cooling (quenching) from high temperature would result in the absence of PY phase. However, no varistor behavior is observed for samples quenched from these temperatures. PY phase is responsible for the varistor effect due to the presence of potential barriers. [START_REF] Wong | Microstructure and phase transformation in a highly non-ohmic metal oxide varistor ceramic[END_REF][START_REF] Mergen | Microstructural relations on BZS Pyrochlore-ZnO mixtures[END_REF] Peiteado et al. [START_REF] Peiteado | Microstructure engineering to drastically reduce the leakage currents of high voltage ZnO varistor ceramics[END_REF] decided to reduce the proportion of PY in the final ceramic by stabilizing the spinel phase and preventing the reformation of Bi 3 Zn 2 Sb 3 O 14 upon cooling. The structure of the spinel phase allows the incorporation of various cations in its empty octahedral and tetrahedral sites. This phase will therefore be able to dissolve cations from other additives. It has been reported that cations such as Co, Ni or Mn were incorporated in order to improve the properties of the varistor and its stability. [START_REF] Inada | Formation mechanism of nonohmic zinc oxide ceramics[END_REF][START_REF] Kato | Effect of Ni and Cr one the stability of leakage current for spinel-reduced ZnO varistor ceramics[END_REF] Peiteado obtained optimized varistors by adding both TiO 2 and SiO 2 , the titanium cations thus can enter in the spinel lattice and stabilize this phase, and SiO 2 reacts with ZnO to form the insulating Zn 2 SiO 4 phase.

3Sb 2 O 4 + 4ZnO + 3Bi 2 O 3 + 3O 2 → 2Bi 3 Zn 2 Sb 3 O 14 (> 500 • C) 2Bi 3 Zn 2 Sb 3 O 4 + 17ZnO → 9Zn
Given the large number of additives necessary to obtain optimized nonlinear properties, the incorporation of further substances may modify these characteristics. Indeed, interactions between components are not excluded. Our aim is to limit drift of the electrical characteristics. Finally changing the Sb/Bi ratio will result in a modification of the threshold voltage. Indeed we want to obtain a breakdown field between 180 and 200 V mm -1 at 1 mA DC.

In this study, it was decided to work on another way to control the microstructure, i.e. the sintering cycle, trying to adapt the cycle to minimize the presence of PY phase in the varistor. Firstly, we defined a new sintering cycle which gives the right good decomposition of PY phase and better stabilization of the SP phase. In order to succeed, the heating rate up to the sintering temperature was studied. The sintering temperature and the dwell time were explored to obtain the best non-linear properties and an appropriate breakdown field. To complete this first step, we focused on how to limit SP phase decomposition during cooling by studying the influence of cooling rate in a selected range of temperatures.

Experimental procedure

Sample preparation

The samples, based on the ZSB system, were produced following the conventional way of preparing ceramics. The raw materials were industrial grade chemicals. The formulation of the samples includes: ZnO, Sb 2 O 3 (Ms), Bi 2 O 3 (Mb), Co 3 O 4 , Mn 2 O 3 , NiO, B 2 O 3 , Al(NO 3 ) 3 •9H 2 O, with Ms/Mb = 2.30. The oxide mixture was ground energetically in distilled water for 5 h. Binders were added and the solution spray dried. The green ceramics were pressed at 100 MPa (LDC1) or at 95 MPa (LDC2) in cylindrical pellets 58 mm in height and 48.5 mm (LDC1) or 57.8 mm (LDC2) in diameter. A first heat treatment (400 • C -10 • C h -1 ) was performed to remove the binders.

In the present study, the influence of several parameters on the final properties of the varistors LDC1 was evaluated using a Taguchi method. [START_REF] Taguchi | Introduction to quality engineering[END_REF] In order to limit the number of tests we focused on three factors, presenting no interactions between them, at three levels. The testing of all combinations between each level theoretically requires 3 [START_REF] Clarke | Varistor ceramics[END_REF] (27) tests. To reduce this number, a full two-factor map was constructed (Table 1-a), including 9 tests, which was added to a Latin square 3 × 3 (Table 1-b). The final plan is shown in Table 1-c.

These selected parameters were the sintering temperature, the heating rate from 950 • C to the final temperature, and the sintering time. The temperature is controlled by rings (Ferro). The shrinkage dependence on temperature of these rings is known and listed in abacus. The description of all sintering tests on LDC1 varistors are shown in Table 2.

In the first part of this paper, electrical tests on LDC1 varistors were carried out in order to assess the effects of the parameters altered during the sintering study. Then, tests on varistors with higher energy capabilities (LDC2) were conducted (Table 2).

A F1 F2 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 B 1 2 3 2 3 1 3 1 2 C F1 F2 F3 1 1 1 1 2 2 1 3 3 2 1 2 2 2 3 2 3 1 3 1 3 3 2 1 3 3 2
One varistor was sintered at 1095 • C for 2.5 h (MOVA), another at 1115 • C for 5 h (MOVB) and a last one at 1115 • C for 5 h with accelerated cooling (MOVC). Usual tests consist of measuring tension at low current (from 1 mA to 10 A) and at high current (from 500 A to 100 kA).

Characterization

Prior to the electrical characterization, the varistors were embedded within a dielectric glass layer. Then two opposite surfaces of the coated varistor were polished and an aluminum electrode deposited by sputtering on each side. The sintered varistors were 40 mm in diameter (LDC1) and 48 mm for superior energy capability (LDC2) and the height was 40 mm for both. The current-voltage of the varistors was measured at low current (pre-breakdown and breakdown zone) using a DC generator Spellman Electronics. For the determination of the U(I) curve at high currents, a current impulse 8/20 s generator was used. The residual voltage at 10 kA was determined using a Yokogawa 120 oscilloscope via a 10-fold voltage divider.

The current leakage I f was determined at room temperature under a voltage of 0.5 × U 1mA (DC).

The coefficient of nonlinearity α given by (1) was determined from 1 mA to 10 mA using the following relation:

α = log(I 1 /I 2 ) log(U 1 /U 2 ) (1)
The clamping voltage R given by Eq. ( 2) allows the evaluation of the extent of the nonlinear zone and the performance of varistor. should as low as possible.

R = U 10 kA (V) U 1 mA (V) (2)
Electrical tests were carried out according to ICE 60099-4 standard.

The microstructure of samples and the nature of the phases were examined using a scanning electron microscope (SEM -JEOL JSM-6400) coupled to an energy dispersive X-ray spectrometry (EDX) system from Oxford Instruments.

Results and discussion

• LDC1 varistors Fig. 1 shows the influence of the sintering parameters on breakdown field (E s ) at I = 1 mA. Each point corresponds to the average value of tests performed on 5 different samples.

An increase in heating rate led to an increase of the breakdown field while the opposite effect was noted for the dwell time. The influence of the sintering temperature is not straightforward. It is well known that the switching voltage is related to the ZnO grain size. The higher the sintering temperature and/or the longer the dwell time, the larger the ZnO grain size is. With large ZnO grains, the threshold voltage (or threshold field) should be low. Nevertheless we note that the increase of the sintering temperature does not lead to the desired effect. The minimum was reached at 1110 • C, and at 1115 • C the threshold field value was higher (Fig. 1).

The coefficient of non-linearity α is only slightly influenced by the heating rate and the dwell time (Fig. 2). As noted for the breakdown field, the variation of α with the sintering temperature was not monotonous. Nevertheless, the temperature of 1115 • C seemed to have a beneficial effect on non-linearity compared to the other temperatures. The variation of α usually follows the variation of the threshold voltage: α was inversely proportional to the threshold voltage.

Fig. 3 shows the effect of the sintering parameters on the leakage current measured 0.5 × U 1 mA (V). The sintering temperature had a major effect on I f , the minimum value 0.36 A was obtained at 1115 • C. The value of I f was also very sensitive to the other parameters. The value of I f was minimal for a slow heating rate (10 • C h -1 ) and a sintering time of 1 h.

Among the different sintering parameters, the sintering temperature has a major effect on the electrical characteristics of varistors. The threshold voltage is very sensitive to the changes in each sintering cycle. A fast heating rate leads to a smaller grain size and therefore a significant increase of the threshold voltage. We determined the optimum dwell time at 1 h. The grain size, and thus, the voltage are mainly dependent on the sintering temperature. Nevertheless obtaining a higher threshold field shows that the grain size of ZnO is not the only parameter governing E s .

• LDC2 varistors The optimized sintering parameters were adapted for varistors of higher energy capacity, i.e. a larger diameter (48 mm). Because of industrial specifications, the value of the threshold field E s has to be between 170 and 190 V mm -1 . Due the higher volume, the heating rates and the sintering times have to be optimized and are different from those of the previous experimental plan. The results obtained were compared with data obtained Fig. 2. Graph of average effect: influence of the sintering parameters on the nonlinearity coefficient α (calculated between 10 -3 and 10 -2 A cm -2 ). from a standard commercial varistor. The details are given in Table 3.

Concerning samples A and B, increases of both the sintering time and temperature had a positive effect on the leakage current, the coefficient of non-linearity and the ratio (Table 3) but the sintering time had a negligible effect on E s . Indeed, a slight decrease in E s was observed. The cooling at 150 • C h -1 performed on MOV C seems to improve the varistors' electrical properties.

These changes in the sintering cycle led to a reduction by more than 80% of the value of the leakage current and at the same time increase the coefficient α by 111%. There was no major change in the value of the threshold field. Finally, the decrease of R was mainly due to a strong decrease in the residual tension U res10 kA (-1290 V).

The current-voltage characteristics of the four varistors are plotted in Fig. 4.

In order to check the influence of the sintering parameters on the microstructure, SEM observations were performed on a standard varistor and on varistor C (Figs. 5 and6). The two ceramics clearly show different microstructures. Both present the characteristic elements of a varistor microstructure: ZnO Table 3 Electrical measurements of LDC2 samples T f : sintering temperature -E s = U 1mA /h -U res10 kA : residual tension at 10 kA -I f : leakage current at 0.75 × U 1 mA -α: non-linearity coefficient calculated between J = 10 -3 and 10 -2 A cm -2 -R: clamping voltage R = U res10 kA /U 1 mA .

Sample ID

Sintering conditions (heating rate -T f -dwell time) grains appear in dark gray, spinel phase in light gray and the bright elements are Bi-rich phases. The size of the ZnO grains ranges from 10 to 15 m for each sample. Nevertheless, a lot of bright phases, in the form of clusters, are observed at the triple junctions. The MOV C shows many Bi-rich intergranular phases at the homo-and triple junctions (Fig. 6). Moreover ZnO grain wetting seems to be improved in varistor C. SEM observations on polished samples (Fig. 7) confirm the difference in microstructure between standard varistor A and sample C. Unlike the standard varistor, which exhibited some Bi-rich clusters, the MOV C contained only Bi-rich phases located at the grain boundaries. X-ray cartography (Figs. 8 and9) confirmed the absence of the PY phase. It also showed that Sb and Bi were not located in the same phase.

E s (V mm -1 ) U res10 kA (V) I f (A) α R Stand. 70 • C h -1 -1095 • C -2.

Discussion

The heating rate and sintering temperature appeared to be the most crucial parameters to control in order to improve the electrical properties of the ceramics. The decrease of the heating rate, before the sintering temperature, provides both a high nonlinear coefficient α and a low leakage current. However there is also a slight decrease in the threshold field.

The threshold field decrease accompanying the decrease of heating rate is due to an increase of the grain size and therefore a smaller number of grain boundaries. The decrease of the heating rate also allows more time for PY decomposition into the spinel phase. This PY decrease thus explains the decrease in leakage current as observed by Metz and Clayton. [START_REF] Metz | Electrical properties of varistors in relation to their true bismuth composition after sintering[END_REF][START_REF] Clayton | The electrical and defect properties of Bi 3 Zn 2 Sb 3 O 14 pyrochlore: a grain-boundary phase in ZnO-based varistors[END_REF] The value of threshold field observed at 1115 • C which was higher than the one observed at 1110 • C could be explained by a change in the grain boundary voltage. Olsson and Dunlop 21 demonstrated that according to the nature of the interfaces, the breakdown voltage can vary by a few volts from one grain boundary to another one. The sharp decrease of leakage current at 1115 • C could be explained by a higher ratio of Bi 2 O 3 evaporation.

The transposition of the optimized conditions to higher energy capability varistors confirms the trend that the electrical properties can be significantly improved when optimizing the sintering parameters. Without modifying E s , the R parameter is strongly reduced. It is well known that it is the conductivity of ZnO grains that controls the high current region behavior. [START_REF] Gupta | Applications of zinc oxide varistors[END_REF][START_REF] Houabes | The effect of aluminium oxide on the residual voltage of ZnO varistors[END_REF] The low current region, which is governed by the grain boundaries, seems to be more resistive. This is confirmed by the decrease of I f . So, the sintering modifications seem to improve the non-linearity at high currents and the high resistivity at low currents. SEM observations agree with this pattern. MOVC, which is sintered with an optimized cycle, presents a homogeneous microstructure, devoid of PY. The higher cooling rate, associated with a slow heating rate, prevented the reappearance of PY. This faster cooling also avoids dewetting of ZnO grains by Bi 2 O 3 . Indeed, compared to a standard varistor, which had a majority of clusters of Bi 2 O 3 -rich phases at the triple junctions, MOV C exhibited an interconnected network composed of the Bi-rich phase that totally wets all the ZnO grains. This homogeneity and the continuity of the coating ZnO grains provide better non-linear properties.

Conclusion

The improvement of electrical performance, including the reduction of leakage current, by the control of microstructure in ZnO based varistors was investigated. The sintering cycle was optimized to reduce the final amount of pyrochlore phase (in medium and high voltage varistors). The reduction of the heating rate associated with appropriate sintering temperature and dwell time lead to a decrease of the leakage current and an increase of the non-linearity coefficient α. The new parameters we suggest allow a better spinel phase formation from the pyrochlore phase and its stabilization. An accelerated cooling rate in a selected range of temperatures avoids the reverse reaction SP to PY. This new sintering cycle provides varistors with high non-linearity and a strongly reduced leakage current. The microstructure of the optimized varistors showed no pyrochlore phase and a better ZnO grain wetting by the intergranular Bi-rich phase. These results show that the varistor behavior at low and high current can be improved without changing the formulation. It will be interesting to study the varistor behavior and the leakage current behavior at high temperature. Adding some minor dopants, to modify the resistivity of grain boundaries and the conductivity of ZnO grains, could lead to highly nonlinear varistors showing high degradation resistance.

Fig. 1 .

 1 Fig. 1. Graph of average effect: influence of the sintering parameters on the breakdown field E s (at 1 mA DC).

Fig. 3 .

 3 Fig. 3. Graph of average effect: influence of the sintering parameters on the leakage current I f at 0.5 × U 1 mA (V) DC.

Fig. 4 .

 4 Fig. 4. I-U characteristics of different varistors ( : Standard, ᭹: MOV A, MOV B, : MOV C).

Fig. 5 .

 5 Fig. 5. SEM images of standard MOV (a) and of MOV C (b) (×1000). ZnO grains appear in dark gray, spinel phase in light gray and the bright white regions are Bi-rich phases.

Fig. 6 .

 6 Fig. 6. SEM images of standard MOV (a) and of MOV C (b) (×2000). ZnO grains appear in dark gray, spinel phase in light gray and the bright white regions are Bi-rich phases.

Fig. 7 .

 7 Fig. 7. SEM images of polished standard MOV (a) and of polished MOV C (b). ZnO grains appear in dark gray, spinel phase in light gray and the bright elements are Bi-rich phases. Arrows indicate the differences in morphology of Bi-rich phases between standard MOV (clusters) and MOV C (thin layers).

Fig. 8 .

 8 Fig. 8. SEM image of MOV C (×2000) (ZnO grains: dark gray, spinel phases: light gray, Bi-rich phases: bright).

Fig. 9 .

 9 Fig. 9. X-ray mapping of MOV C.

  2,33 Sb 0,67 O 14 + 3Bi 2 O 3 (> 850 • C)

	3Zn 2,33 Sb 0,67 O 4 → Zn 7 Sb 2 O 12 (> 950 • C)
	3Zn 7 Sb 2 O 12 + 3Bi 2 O 3
	→ 2Bi 3 Zn 2 Sb 3 O 14 + 17ZnO (slowcooling)

Table 1

 1 Work methodology (F = factor).

Table 2

 2 Sintering conditions for the study of LDC1 (φ = 40 mm, h = 40 mm) and LDC2 (φ = 48 mm, h = 40 mm) varistors.

	Type of varistor	Sample	Sintering parameters
			Heating rate ( • C h -1 )	Sintering temperature ( • C)	Sintering time (h)
		1	10	1105	0
		2	10	1105	0.5
		3	10	1105	1
		4	15	1110	0.5
	LDC1 ( 40 mm)	5	15	1110	1
		6	15	1110	0
		7	20	1115	1
		8	20	1115	0
		9	20	1115	0.5
		MOV A	20	1095	2.5
	LDC2 ( 48 mm)	MOV B	20	1115	5
		MOV C	20	1115	5 a

a Fast cooling.


Raphaël Lachièze-Rey

Matthias Reitzner

U-statistics in stochastic geometry

come

U( f ; η) = ∑ x k ∈η k = f (x k )
otherwise, where η k = is the class of k-tuples x k = (x 1 , . . . , x k ) of distinct points from η. Remark that since the sum is over all such k-tuples, f can be assumed to be symmetric without loss of generality.

An abundant literature deals with the asymptotic study of U( f ; ηp ) as p → ∞ when ηp is a binomial process, i.e. a set of p iid variables over X. We are concerned here with Poisson input, i.e. η is a Poisson measure over X which intensity is a non-atomic locally finite measure µ on X. So that the definition makes any sense, the basic assumption is that

f ∈ L 1 s (X k ) = L 1 s (X k ; µ k ).
In the sequel of this section, let µ be a non-atomic locally finite measure on (X, X ), η a Poisson measure with intensity µ, and k ≥ 1.

chaotic decomposition and multiple integrals

Theorem 1. Let f ∈ L 1 s (X k ) such that U( f ; η) ∈ L 2 (P) . We have the L 2 decomposition

U( f ; η) = k ∑ n=0 I n (h n ).
(1)

Here I n is the n-th order stochastic integral over η defined in Chapter [17]. The functions h n have been explicitely computed in [25, Lemma 3.3],

h n (x n ) = n k X n f (x n , x k-n )dµ k-n (x k-n ) (2)
for x n ∈ X n , and h n is a function of L 1 s (X n ) ∩ L 2 s (X n ).

Remark 1. Somewhat counterintuitively, f ∈ L 1 s (X k ) ∩ L 2 s (X k ) does not imply that EU( f ; η) 2 < ∞ (see [25]), but in most examples f is bounded and has a bounded support, which makes the latter condition automatically satisfied.

As is apparent in Theorem 1, each U-statistic of order k is a finite sum of multiple integrals of order n ≤ k, and it is not difficult to prove that conversely any multiple integral of order n ≥ 1 can be written as a finite sum of U-statistics which orders are smaller or equal to n. From a formal point of view, it is therefore equivalent to study the asymptotics of finite sums of U-statistics or of finite sums of multiple integrals. U-statistics are more likely to appear in applications, but the homogeneity of multiple integrals make them easier to deal with, and some of the Malliavin operators of U-statistics have a particularly intuitive form. Consider for instance the case where F = I k ( f ) is a multiple integral of order k. The Malliavin derivative, the Orsntein-Uhlenbeck, and inverse Orstein-Uhlenbeck operators, take the following form

D x F = kI k-1 ( f (x, •)), x ∈ X, LF = -kI k ( f ), L -1 F = -k -1 I k ( f ).
(3)

For a U-statistic F, one can still derive D x F, LF, L -1 F using the linearity of those operators and the decomposition (1). The object of this section is really the study of sums of multiple integrals which order is bounded by some k ≥ 1. The chaotic decomposition also yields that any L 2 variable can be approximated by such a sum, allowing us in some cases to pass on limit theorems stated here to infinite sums. The following result gives the first two moments of U-statistics.

Proposition 1. Let f ∈ L 1 s (X k ). Then E|U( f ; η)| < ∞ and EU( f ; η) = X k f (x k ) dµ k (x k ).
If furthermore U( f ; η) ∈ L 2 (P),

Var(U( f ; η)) = k ∑ n=1 n! h n 2 (4)
where h n is given in Theorem 1 and h n is the usual L 2 (X n )-norm.

Proof. The first statement is a direct consequence of the Slyvniack-Mecke formula, while the second stems from the orthogonality between multiple integrals

I n (h n ), 0 ≤ n ≤ k.

Hoeffding decomposition

Let N ≥ 1, ηp = {X 1 , . . . , X p } be a family of i.i.d. variables with common distribution µ on X. Given a measurable kernel h over X k , k ≥ 1, the traditional Hoeffding decomposition (see e.g. Vitale [31]) is written

U(h, ηp ) = k! p k σ N k (h) = k! p k k ∑ m=0 k m σ p m (H m ),
where

σ p m (H m ) = 1 N m ∑ 1≤i 1 <i 2 <•••<i m ≤p H m (X i 1 , . . . , X i m ), 0 ≤ m ≤ k,
and each kernel H m is symmetric and completely degenerated, i.e.

EH m (x 1 , . . . , x m-1 , X m ) = X H m (x 1 , . . . , x m-1 , y)dµ(y) = 0
for µ (m-1) -a.e. x 1 , . . . , x m-1 . This property implies in particular the orthogonality of the σ p m (H m ), 1 ≤ n ≤ k. If µ is a probability measure, the H m are uniquely defined and can be expressed explicitely via an inclusion-exclusion formula

H m (x 1 , . . . , x m ) = m ∑ n=0 (-1) m-n ∑ 1≤i 1 <•••<i n ≤m k n -1 h n (x i 1 , . . . , x i n ) (5)
where h n is defined in (2). As is clear in this last formula, this decomposition is different from (1) because in the latter multiple integration is performed with respect to the compensated measure ηµ, while in σ p m (H m ) the compensation occurs in the kernel H m .

The Hoeffding rank m 1 is defined as the smallest index m such that H m = 0, and we can see through (5) that it is equal to the smallest index n such that h n = 0.

We furthermore have

H m 1 = k m 1 -1 h m 1 .
As proved in [7] for binomial processes or [16] for Poisson processes, the stochastic integral of order m 1 dominates the sum, and limit theorems for geometric U-statistics can then be derived by studying this term, see Section 2.1.2.

Contraction operators

Let f ∈ L 1 s (X q ), g ∈ L 1 s (X k ).
If f and g satisfy the technical conditions defined in Chapter [17], one can define for 1 ≤ r ≤ l ≤ min(q, k) their contraction function of index (r, l), denoted f ⋆ r l g. It has k + qrl variables as arguments, decomposed in (x r-l , y q-r , z k-r ), where x r-l ∈ X r-l , y q-r ∈ X q-r , and z k-r ∈ X k-r . We have f ⋆ r l g(x r-l , y q-r , z k-r ) := f (x l , x r-l , y q-r )g(x l , x r-l , z k-r ) dµ l (x l ).

Remember that each function appearing here is symmetric, whence the order of the arguments does not matter. Contraction operators are used below to assess the distance between a stochastic integral and the normal law. See [4] for more information on contraction operators.

Rates of convergence

Let F be a L 2 variable of the form

F = k ∑ n=0 I n (h n ) (6)
for some kernels h n ∈ L 2 s (X n ), n ≥ 1. We assume that those kernels satisfy the technical conditions mentioned in Chapter [4] so that their mutual contraction kernels are well defined. This model englobes U-statistics, as outlined by Theorem 1, as well as finite sums of U-statistics and multiple integrals.

In applied situations, the set-up consists of a fixed integer k ≥ 1, and, for t > 0, a family of measures µ t on X, and a family of kernels h n,t ∈ L 2 s (X n ; µ n t ), 1 ≤ n ≤ k. We study the random variables

F t := ∑ n=1 I n (h n,t ), (7)
and more precisely the existence of numbers a t , b t > 0 and of a random variable V such that

Ft := F t -a t √ b t → V
in the weak topology. In all the applications, µ t is either of the form

• µ t = tµ for some reference measure µ on the space X, or • µ t = 1 X t µ where X t ⊂ X depends on t.

The following two settings occur in the most important applications. If η = η t is a Poisson point process on X = R d the measure µ will often be the Lebesgue measure ℓ d , or for X = R d × M a product measure µ = ℓ d ⊗ ν with a probability measure ν on a topological marks space (M, M ).

If η = η t is a Poisson 'flat' process on the Grassmannian X = A d i of affine idimensional subspaces (flats) of R d , the intensity measure µ(•) will be a translation invariant measure on A d i . The Poisson flat process is only observed in a compact convex window W ⊂ R d with interior points. Thus, we can view η t as a Poisson process on the set [W ] defined by

[W ] = h ∈ A d i : h ∩W = / 0 .

Central Limit theorem

Let F be of the form (6). Let N ∼ N (0, 1), σ 2 = Var(F). The next result, which Wasserstein bound has been established in [15], and Kolmogorov bound in [8], gives a bound on the distance between F and N in terms of the contractions between the kernels of F.

Theorem 2. Put B(F) = max max 1 h n ⋆ l r h n , max 2 h n ⋆ l r h m , max n=1,...,k h n 2 B ′ (F) = max(|1 -σ 2 |, B(F), B(F) 3/2 )
where max 1 is over

1 ≤ r ≤ n ≤ k, 1 ≤ l ≤ r ∧ (n -1), max 2 is over 1 ≤ l ≤ r ≤ n < m ≤ k.
There exists a constant C k > 0 not depending on the kernels of F such that

d W (F, N) ≤ σ -1 C k B(F) (8) d K (F, N) ≤ C k B ′ (F). (9)
We reproduce here the important steps of the proof for the Wasserstein bound. The main result, due to Peccati, Sole, Taqqu, Utzet [20], is a general inequality on the Wasserstein distance between a Poisson functional F with variance σ 2 > 0 having a finite Wiener-Ito expansion and the normal law. We have

d W (F, N) ≤ 1 σ E[(σ 2 -D x F, -D x L -1 F L 2 (X) ) 2 ] (10) + 1 σ 2 X E[(D x F) 2 |D x L -1 F|]µ(dx).
To translate those inequalities into bounds on the contraction norms, we use the multiplication formula from [20], which yields that the multiplication of mutiple integrals is a linear combinations of multiple integrals. For

k, q ≥ 1, f ∈ L 2 s (X q ), g ∈ L 2 s (X k ), I q ( f )I k (g) = q∨k ∑ r=0 r! q r k r r ∑ l=0 r l I q+p-r-l ( f ⋆l r g), (11)
where the symmetrized contraction kernels f ⋆l r g are the average of kernels f ⋆ l r g over all possible permutations of the variables.

If for instance F = I k ( f ) is a single multiple integral, (3) gives the value of the Malliavin operators, and a computation then yields the bound (8) with f k = f ; f i = 0 for i = k, see [22,Prop. 5.5]. If F is a general functional with a finite decomposition, such as a U-statistic (see (1)), Malliavin operators are computed using linearity and yield the bound (8), see the proof of Th. 3.5 in [15].

Concerning Kolmogorov distance, Schulte [28,29] has derived a Stein bound similar to (10), but with more terms on the right-hand side (Theorem 1.1), reflecting the effect that test functions are indicator functions, more irregular than the Lipschitz functions involved in Wasserstein distance. This bound was later improved by Eichelsbacher and Thäle [8,Th. 3.1], reducing the number of additional terms. With similar computations as in the Wasserstein case, one can then prove [8,Th. 4.1] that those additional terms only add contraction norms f i ⋆ r l f j 3/2 at the power 3/2, up to a constant, yielding the bound B ′ (F).

Remark 2. The terms in B ′ (F) bounding the Kolmogorov distance are smaller than the original terms present in B(F) if the bound goes to 0, and don't change the bound magnitude or its eventual convergence to 0.

Remark 3. The constant C k explodes as k → ∞. In other papers [25], [18], similar bounds are derived in more specific cases, with a different method. The constants are more tractable and allow for instance to approximate accurately the distance from a Gaussian to an infinite series of multiple integrals by that of its truncation at some order (see for instance [28]).

Theorem 3 (4th moment theorem). Assume furthermore that kernels h k are nonnegative. Then for some C ′ k > 0

B(F) ≤ C ′ k EF 4 -3σ 4 .
• In view of ( 8), the convergence of the 4-th moment to that of a Gaussian therefore implies central limit, with a bound for Wasserstein distance. In this case, as noted in [8], using (9) yields a similar bound for Kolmogorov distance. The positiveness of the kernels is adapted to U-statistics with a non-negative kernel. • It is highly remarkable that the convergence of the 4-th moment to that of the Gaussian variable is therefore sufficient for such variables to converge to the normal law. The only technical requirement is that the variables F 4 t are uniformly integrable.

Example 1 (De Jong's theorem). Let f 2 be a non-zero degenerate symmetric kernel from L 1 s (X 2 ), i.e. such that

X f 2 (x, y)µ(dx) = 0 for µ-a.e. y ∈ X.
This degeneracy property implies that U( f 2 , η) = I( f 2 ; η), we also assume that f 2 ∈ L 4 s (X 2 ). De Jong [11] derived a 4-th moment central limit theorem for binomial Ustatistics of the form U( f 2 ; ηp ), where p ∈ N goes to infinity and ηp is a sequence of p iid variables with law µ. In the Poisson framework, (9) yields Berry-Essen bounds between F = U( f 2 ; η) = I 2 ( f 2 ; η) and N:

d W ( F, N) ≤ C 2 1 f 2 2 b( f 2 ) d K ( F, N) ≤ C 2 1 f 2 2 max(b( f 2 ), b( f 2 ) 3/2 ) where b( f 2 ) = max f 2 ⋆ 0 2 f 2 , f 2 ⋆ 1 1 f 2 , f 2 ⋆ 1 2 f 2 .
See Eichelsbacher and Thäle [8, Th. 4.5] for details. In [21], Peccati and Thäle derive bounds on the Wasserstein distance between such a U-statistic and a target Gamma variable, also in terms of contraction operators.

Local marked U-statistics

For many applications, it is useful to assume that the state space X is of the form S × M where S is a subset of R d containing the points t i of η, and (M, M ) is a mark space, i.e. a locally compact space endowed with some probability measure ν. The space M contains marks m i that will be randomly assigned to each point of the process. In this setting, assume that η t has intensity measure µ t = 1 X t ℓ d ⊗ ν and let F t ∈ L 2 (P) be a U-statistic F t = U( f ; η t ). Let the kernel f of the U-statistic be locally square integrable on

X t = [-t 1/d ,t 1/d ] d × M and stationary, i.e. for µ k - almost all (t k , m k ) ∈ X k t , z ∈ R d , f (t k + z, m k ) = f (t k , m k ). ( 12
)
The tail behavior of the function f is fundamental regarding the limit of variables F t as t → ∞.

Definition 1. A measurable function f : (R d × M) k → R is rapidly decreasing if it
is locally square integrable, stationary, and if it satisfies the following integrability condition: There exists a non-vanishing probability density κ on (R d ) k-1 such that for p = 2, 4,

A p ( f ) = (R d ) k-1 ×M k f (0, t k-1 , m k ) p κ(t k-1 ) 1-p dℓ k-1 d (t k-1 )dν k (m k ) < ∞.
The slight abuse of notation f (0, t k-1 , m k-1 ) means that t k = (0, t k-1 ) = (0,t 2 , . . . ,t k-1 ), and m k = (m 1 , . . . , m k ).

We have in this case the following result, which is a consequence of Theorem 6.2 and Example 2.12-(ii) in [16] : Theorem 4. Let F t = U( f ; η t ) where f is a rapidly decreasing function, and µ t = [25] first established this result in the case where f is the indicator function of a ball of R d (any non-vanishing continuous density κ can be chosen in this case because f (0, •) has a compact support).

1 X t ℓ d ⊗ ν with X t = [-t 1/d ,t 1/d ] d × M. Then, with a t = EF t , b t = Var(F t ), we have for some C 1 ,C 2 ,C 3 > 0 not depending on t, C 1 t ≤ b t ≤ C 2 t d W ( Ft , N 1 ) ≤ C 3 t -1/2 . Remark 4. Reitzner & Schulte
Remark 5. A similar result holds if F is simply assumed to be a finite sum of stochastic integrals which kernels are rapidly decreasing functions, the U-statistics being a particular case.

Geometric U-statistics

Coming back to the general framework, assume F t = U( f ; µ t ) where f ∈ L 2 s (X k ) is fixed and µ t = tµ for some measure µ on X. Then F t admits the decomposition ( 7) where

h n,t (x n ) = t k-n k n X k-n f (x n , x k-n )dµ k-n (x k-n ), x n ∈ X n .
One can then see that the term h 1,t dominates the other terms in the variance expression (4), provided this term does not vanish. In any case the important feature is the Hoeffding rank of the U-statistic

n 1 := inf{n : h n = 0},
because it turns out that I n 1 (h n 1 ,t ) is the predominant term in (7), in the sense that

F t -I n 1 (h n 1 ,t ) = o(F t )
for the L 2 norm as t → ∞. It yields the following result (Theorem 7.3 in [16]).

Theorem 5. For some C 1 ,C 2 ,C 3 > 0 not depending on t,

C 1 t 2k-n 1 ≤ b t ≤ C 2 t 2k-n 1 . (i) If n 1 = 1, U( f ; µ t ) follows a central limit theorem and d W ( Ft , N) ≤ C 3 t -1/2 , d K ( Ft , N) ≤ C 3 t -1/2 . (ii) If n 1 > 1, U( f ; µ t )
does not follow a CLT and Ft converges to a Gaussian chaos of order n 1 (see [16,).

For a deeper understanding we refer to the proof of Theorem 6. Remark 8. The speed of convergence to the Gaussian chaos in (ii) is studied by Peccati and Thäle [21] in case the limit is a Gamma distributed random variable.

Regimes classification

The crucial difference in Theorems 4 and 5 is the area of influence of a given point x ∈ η t . In the case of a local U-statistic, a typical point x ∈ η t interacts with a stochas- tically bounded number of neighbors, that are more likely near in view of Assumption 12. The situation is different for a geometric U-statistic, where a point potentially interacts with any other point, regardless of the distance. Both these regimes can be seen as two particular cases of a continuum.

Let α t > 0 be a scaling factor, X t = [-t 1/d ,t 1/d ] d × M, µ t = 1 X t ℓ d ⊗ ν, and F t = U( f t ; η t ), where f t is obtained by rescaling a rapidly decreasing function f :

f t (x k ) = f (α t x k ), x k ∈ X k t . (13)
Say that f has non-degenerate projections if the functions

f n (x n ) = X k-n t f (x n , x k-n )dµ k-n t , x n ∈ X n ,
well defined in virtue of (12), are not µ-a.e. equal to 0. It is trivially the case if for instance f = 0 and f ≥ 0 µ-a.e.. Concerning notations, every spatial transformation of a point x = (t, m) ∈ R d × M, such as translation, rotation, or multiplication by a scalar number, is only applied to the spatial component t. Subsequently, any spatial transformation applied to a k-tuple of points x k = (x 1 , . . . , x k ) is applied to the spatial components of the x i 's. The quantity v t = α -d t is relevant because it gives the magnitude of the number of points interacting with a typical point x. The case v t = α t = 1 is that of local U-statistic. If v t = t is roughly the volume of X t , it corresponds to geometric U-statistics. In this case it is useless to assume that f is rapidly decreasing, as only the behavior over X 1 is relevant for the problem. Theorem 6. Assume that f t is of the form (13), where f is a rapidly decreasing function with non-degenerate projections. With the notations above, there are

C 1 ,C 2 ,C 3 > 0 such that C 1 ≤ b t tv 2k-2 t max(1, v -k+1 t ) ≤ C 2 , and d W ( Ft , N) ≤ C 3 t -1/2 max(1, v -k+1 t ) 1/2 d K ( Ft , N) ≤ C 3 t -1/2 max(1, v -k+1 t ) 1/2 .
Concerning the bound for Kolmogorov distance, it is not formally present in the literature. It relies on the fact that in Theorem 2, B ′ (F) ≤ CB(F) for some C > 0 in the case where σ → 1 and B(F) → 0. Then one can simply reproduce the proof of [16], entirely based on an upper bound for B(F). Remark 9. Theorems 4 and 5-(i) can be retrieved from this theorem by setting respectively v t = 1 or v t = t. Remark 10. If some projections do vanish, the convergence rate can be modified, and the limit might not even be gaussian, as it is the the case for the degenerate geometric U-statistics of Th. 5-(ii).

Remark 11. Depending on the asymptotic behavior of v t , we can identify four different regimes:

1. Long interactions: v t → ∞, CLT at speed t -1/2 , the first chaos I 1,t (h t,1 ) dominates (geometric U-statistics). 2. Constant size interactions: v t = 1, CLT at speed t -1/2 , all chaoses have the same order of magnitude (local U-statistics). 3. Small interactions: v t → 0,tv -k+1 t → ∞, CLT at speed (tv -k+1 t ) -1/2 , higher order chaoses dominate. In the case of random graphs (k = 2), the corresponding bound in (tv t ) -1/2 has been obtained in [15]. 4. Rare interactions: tv -k+1 t → c < ∞, the bound does not converge to 0. In the case k = 2, it has been shown in [15] that there is no CLT but a Poisson limit in the case c > 0 (see Chapter [4] for more on Poisson limits).

Other limits and multi-dimensional convergence

Besides the Gaussian chaoses appearing in Theorem 5-(ii), some characterizations of non-central limits have also been derived for Poisson U-statistics.

Multidimensional convergence

We consider in this section the conjoint behavior of random variables

F t = (F 1,t , . . . , F k,t ) where F m,t = I q m (h m,t ) for 1 ≤ m ≤ k, with h m,t ∈ L 2 s (X q m ) for some q m ≥ 1, for t > 0; 1 ≤ m ≤ k.
Call σ 2 t = ∑ k m=1 Var(F m,t ). Any L 2 candidate for the limit of σ -1 t F t should have as covariance matrix

C m,n = lim t σ -2 t EF m,t F n,t , 1 ≤ m, n ≤ k
if those limits exist. In this case there is indeed asymptotic normality if all contraction norms h m,t ⋆ l r h n,t go to 0 for r = 1, . . . , q k , and every l = 1, . . . , r ∧ (q k -1), under technical conditions on the kernels related to technical condition of chapter [4], see [22, Th. 5.8],[3, Th. 2.4] for details. These articles contain explicit bounds on the speed of convergence with a specific distance related to thrice differentiable functions on (R d ) k , and the convergence is stable, in the sense of [3].

If now F t = (F 1,t , . . . , F k,t ) where each F m,t is a U-statistic, one can consider the random vector G t constituted by all multiple integrals with respect to kernels from the decompositions of the F m,t , as defined in (2). One can then infer conditions for asymptotic normality of F t by applying the previous considerations to G t .

As noted in Remark 11, some U-statistics behave asymptotically as Poisson variables. Asymptotic joint laws of U-statistics can also converge to random vectors with marginal Poisson laws, and it can also happen that they converge to an hybrid random vector which has both Gaussian and Poisson marginals, here again the reader is referred to Chapter [4].

Gamma

Similar results to those of Section 2.1 with Gamma limits have been derived by Peccati and Thäle [21] for Poisson chaoses of even order. The distance used there is

d 3 (U,V ) = sup h∈H 3 |Eh(U) -h(V)|
where H 3 is the class of functions of class C 3 with all first 3 derivatives uniformly bounded by 1. We again denote by f ⋆l r g the symmetrized contraction kernels . For ν > 0, let F(ν/2) be a Gamma distribution with mean and variance both equal to ν/2. We introduce the centered unit variance variable G(ν) := 2F(ν/2)ν.

Theorem 7. Let F = I k (h k ) for some even integer k ≥ 2.

We have

d 3 (I k (h k ), G(ν)) ≤ D k max{k! h k -2ν; h k ⋆ p p h k ; h k ⋆ l r h k 1/2 ; h k ⋆q/2 q/2 h k -c k h k }
where the maximum is taken over all p = 1, . . . , k -1 such that p = k/2 and all (r, l) such that r = l and l = 0, or r ∈ {1, . . . , k} and l ∈ {1, . . . , min(r, k -1)}. Also

c k = 4 (q/2)! q q/2 2 .
Remark 12. In the case of double integrals (k = 2), the authors of [21] provide a 4th moment theorem, in the sense that under some technical conditions, a sequence of double stochastic integrals converge to a Gamma variable if their first moments converge to those of a Gamma variable.

Remark 13. This result enables to give an upper bound on the speed of convergence to the second Gaussian chaos in Theorem 5 in the case n 1 = 2, if this limit is indeed a Gamma variable.

Large deviations

There are only few investigations concerning concentration inequalities for Poisson U-statistics. Most results require an nice bound on sup η∈N(X), z∈X D z (F) < ∞. For U-statistics of order ≥ 2 this condition is not satisfied, even if f is bounded. For U-statistics of order 1, this holds if f ∞ < ∞. Therefore we split our investigations into a section on U-statistics of order one and on higher order local U-statistics. We start with a general result. Throughout this section we assume that f ≥ 0 and f = 0.

A general LDI

In this section we sketch an approach developed in [26] leading to a general concentration inequality. For two counting measures η and ν we define the difference η\ν by

η\ν = ∑ x∈X (η(x) -ν(x)) + δ x . ( 14
)
For x ∈ η and f ∈ L 1 s (X k ), we recall that

U( f ; η) = ∑ x∈η F(x; η) with F(x; η) = ∑ x k-1 ∈(η\{x}) k-1 = f (x, x k-1 ).
Assume that in addition to η a second point set ζ ∈ N(X) is chosen. The nonnegativity of f yields

U( f ; η) ≤ U( f ; ζ ) + k ∑ x∈η F(x; η)1(x / ∈ ζ ) = U( f ; ζ ) + k F(x; η) d(η\ζ ) .
The convex distance of a finite point set η ∈ N(X) to some A ⊂ N(X) was introduced in [24], and is given by

d π T (η, A) = max u 2,η ≤1 min ζ ∈A u d(η\ζ )
where u : X → R + is a non-negative measurable function and u 2 2,η = u 2 dη. To link the convex distance to the U-statistic, we insert for u the normalized function F(x; η) -1 2,η F(x; η) and rewrite U(η) in terms of the convex distance as follows:

d π T (η, A) ≥ min ζ ∈A 1 F(x; η) 2,η F(x; η)d(η\ζ ) ≥ 1 k F(x; η) 2,η min ζ ∈A U( f ; η) -U( f ; ζ ) . If we assume F(x; η) ≤ B, then F(x; η) 2 2,η ≤ B ∑ x∈η F(x; η) = BU( f ; η), which implies d π T (η, A) ≥ 1 k √ B min ζ ∈A U( f ; η) -U( f ; ζ ) U( f ; η) 1(∀x ∈ η : F(x; η) ≤ B) . (15)
In [24], a LDI for the convex distance was proved. For η a Poisson point process, and for A ⊂ N(X), we have

P(A)P (d π T (η, A) ≥ s) ≤ exp - s 2 4 .
Precisely as in [26], this concentration inequality combined with the estimate (15) yields the following theorem.

Theorem 8. Assume that ε(•) and B ∈ R satisfy P(∃x ∈ η : F(x; η) > B) ≤ ε(B).

Let m be the median of U( f ; η). Then

P(|U( f ; η) -m| ≥ u) ≤ 4 exp - u 2 4k 2 B(u + m) + 3ε(B) . (16)
In the next sections we apply this to U-statistics of order one and to local U-statistics.

In the applications, the crucial ingredient is a good estimate for ε(B).

LDI for first order U-statistics

There are several concentration inequalities for integrals over Poisson point processes, i.e. U-statistics of order one,

U( f ; η) = ∑ x∈η f (x) = f dη, f ≥ 0 in which case D z U = f (z). Assuming that f ∞ = B < ∞ we have D z U ∞ ≤ B.
A result by Houdre and Privault [10] shows that

P(U -f 1 ≥ u) ≤ exp - f 1 f ∞ g u f 1 ( 17
)
where g(u) = (1 + u) ln(1 + u)u, u ≥ 0 and because f ≥ 0 the 1-norm equals the expectation EU( f ; η). A similar result is due to Ane and Ledoux [1]. proves an estimate involving the 2-norm f 2 instead of the 1-norm. A slightly more general estimate is given by Breton et al. [5].

We could also make use of Theorem 8 and choose B = f ∞ . This yields

P(|U( f ; η) -m| ≥ u) ≤ 4 exp - u 2 4 f ∞ (u + m) , (18)
which is a slightly weaker estimate than (17).

LDI for local U-statistics

In this paragraph we assume that X is equipped with a distance and B(x, r) denotes the ball of radius r around x ∈ X. If U is a local U-statistic which is concentrated on a ball of radius δ t , we have

F(x, η) ≤ f ∞ η(B(x, δ t )) k-1 P(∃x : F(x; η) > B) ≤ E ∑ x∈η 1(F(x; η) > B) ≤ X P(F(x; η) > B)dµ t
and it remains to estimate

P η(B(x, δ t )) > B f ∞ 1 k-1 .
We use the Chernoff bound for the Poisson distribution, namely

P(η t (B d (x, δ t )) > r) ≤ inf s≥0 e E(e s -1)-sr , (19)
because η(B(x, δ t )) is a Poisson distributed random variable with mean

E(x) := Eη t (B d (x, δ t )) = µ t (B d (x, δ t )) ≤ sup x∈X µ t (B d (x, δ t )) =: E . (20)
Because inf s≥0 E(e s -1)sr = r(1ln(r/E)) -E we estimate the right hand side of ( 19) by exp -1 2 r for Ee 2 ≤ r. This leads to

P(∃x : F(x; η) > B) ≤ µ t (X) exp - 1 2 B f ∞ 1 k-1 := ε(B) for B ≥ E k-1 e 2(k-1) f ∞ . We set B = f 1 k ∞ ( u 2 (u+m) ) k-1
k and combine this with the general Theorem 8. Theorem 9. Set E := sup x∈X µ t (B d (x, δ t )). Then for u 2 (u+m) ≥ E k e 2k f ∞ ,

P(|U( f ; η) -m| ≥ u) ≤ 4µ t (X) exp - 1 4k 2 f -1 k ∞ u 2 u + m 1 k .
Clearly, in particular situations more careful choices of ε(B) and B lead to more precise bounds.

Applications

In this section we investigate some applications of the previous theorems in stochastic geometry. In all these cases X is either a subset of R d or a subset of the affine Grassmannian A d i , the space of all i-dimensional spaces in R d . We state some normal approximation and concentration results which follow from the previous theorems. In many cases multi-dimensional convergence and convergence to other limit distributions can be proved in various regimes. We restrict our presentation to certain 'simple' cases without making any attemp for completeness. Our aim is just to indicate recent trends, we refer to further results and investigations in the literature.

Intersection process

Let η t be a Poisson process on the space A d i with an intensity measure of the form µ(•) = tθ (•) with t ∈ R + and a σ -finite non-atomic measure θ . The Poisson flat process is only observed in a compact convex window W ⊂ R d with interior points. Thus, we can view η t as a Poisson process on the set X = [W ] defined by

[W ] = h ∈ A d i : h ∩W = / 0 .
Given the hyperplane process η t , we investigate the (d -k(d -i))-flats in W which occur as the intersection of k planes of η t . Hence we assume k ≤ d/(d -i).

In particular, we are interested in the sum of their j-th intrinsic volumes given by

Φ t = Φ t (W, i, k, j) = 1 k! ∑ (h 1 ,...,h k )∈η k = V j (h 1 ∩ . . . ∩ h k ∩W )
for j = 0, . . . , dk(di), i = 0, . . . , d -1 and k = 1, . . . , ⌊d/(di)⌋. For the definition of the j-th intrinsic volume V j (•) we refer to the Chapter 2 of the current book. We remark that V 0 (K) is the Euler characteristic of the set K, and that V n (K) of an ndimensional convex set K is the Lebesgue measure ℓ n (K). Thus Φ t (W, i, 1, 0) is the number of flats in W and Φ t (W, i, k, d -k(d -i)) is the (d -k(d -i))-volume of their intersection process. To ensure that the expectations of these random variables are neither 0 nor infinite, we assume that 0 < θ ([W ]) < ∞, and that 2 ≤ k ≤ ⌊d/(di)⌋ independent random hyperplanes on [W ] with probability measure θ (•)/θ ([W ]) intersect in a (dk(di))-flat almost surely and their intersection flat hits the interior of W with positive probability. For example, these conditions are satisfied if the hyperplane process is stationary and the directional distribution is not concentrated on a great subsphere.

The fact that the summands in the definition of Φ k i are bounded and have a bounded support makes sure that all moment conditions are satisfied and we can apply Theorem 5: Theorem 10. Let N be a standard Gaussian random variable. Then constants c = c(W, i, k, j) exist such that

d W ( Φt , N) ≤ ct -1/2 , d K ( Φt , N) ≤ ct -1/2 , for t ≥ 1.
Furthermore, it can be shown [25] that the asymptotic variances satisfies VarΦ

t = C Φ t 2k-1 (1 + o(1)) as t → ∞ with a constant C Φ = C Φ (W, i, k, j).
The order of magnitude already follows from the first part of Theorem 5.

For more information we refer to [9] and [18]. In the second paper the Wiener-Itô chaos expansion is used to derive even multivariate central limit theorems in an increasing window for much more general functionals Φ.

Flat processes

For i < d 2 two i-dimensional planes in general position will not intersect. Thus the intersection process described in the previous section will be empty with probability one. A natural way to investigate the geometric situation in this setting is to ask for the distances between this i-dimensional planes, or more general for the socalled proximity functional. The central limit theorems described in the following fits precisely into the setting of this contribution, we refer to [30] for further results.

Let η t be a Poisson process on the space A(d, i) with an intensity measure of the form µ t (•) = tθ (•) with t ∈ R + and a σ -finite non-atomic measure θ . The Poisson flat process is observed in a compact convex window W ⊂ R d . To two i-dimensional planes in general position there is a unique segment

[x 1 , x 2 ] with d(h 1 , h 2 ) = x 2 -x 1 = min y∈h 1 ,z∈h 2 z -y .
The midpoints m(h 1 , h 2 ) = 1 2 (x 1 + x 2 ) form a point process of infinite intensity, hence we restrict this to the point process

{m(h 1 , h 2 ) : d(h 1 , h 2 ) ≤ δ , h 1 , h 2 ∈ η 2 = }
and are interested in the number of midpoints in W .

Π t = Π t (W, δ ) = 1 2 ∑ h 1 ,h 2 ∈η 2 = 1(d(h 1 , h 2 ) ≤ δ , m(h 1 , h 2 ) ∈ W )
It is not difficult to show that EΠ t is of order t 2 δ d-2i . The U-statistic Π t is local on the space A d i . Thus the following theorem due to Schulte and Thaele [30] is in spirit similar to Theorem 4.

Theorem 11. Let N be a standard Gaussian random variable. Then constants c(W, i) exist such that

d K ( Πt , N) ≤ c(W, i)t -d-i 2 .
for t ≥ 1.

Moreover, Schulte and Thäle proved that the ordered distances form after suitable rescaling asymptotically an inhomogeneous Poisson point process on the positive real axis. We add to this a concentration inequality which follows immediately from Theorem 9. Observe that µ t (X) = tθ ([W ]).

Theorem 12. Denote by m the median of Π t . Then

P(|Π t -m| ≥ u) ≤ 4tθ ([W ]) exp - 1 16 u √ u + m for u √ u+m ≥ e 2 t sup h∈[W ] θ (B d (h, δ )).

Gilbert graph

Let η t be a Poisson point process on R d with an intensity-measure of the form µ t (•) = tℓ d (• ∩ W ), where ℓ d is Lebesgue measure and W ⊂ R d a compact convex set with ℓ d (W ) = 1. Let (δ t : t > 0) be a sequence of positive real numbers such that δ t → 0, as t → ∞. The random geometric graph is defined by taking the points of η t as vertices and by connecting two distinct points x, y ∈ η t by an edge if and only if xy ≤ δ t . The resulting graph is called Gilbert graph.

There is a vast literature on the Gilbert graph and one should have a look at Penrose's seminal book [23]. More recent developments are due to Bourguin and Peccati [3], Lachièze-Rey and Peccati [15,16] and Reitzner, Schulte and Thäle [26].

In a first step one is interested in the number of edges

N t = N t (W, δ t ) = 1 2 ∑ (x,y)∈η 2 t, = 1( x -y ≤ δ t )
of this random geometric graph. It is natural to consider instead of the norm functions 1( f (yx) ≤ δ t ) and instead of counting more general functions g(y -x):

∑ (x,y)∈η 2 t, = 1( f (yx) ≤ δ t ) g(y -x).

For simplicity we restrict our investigations in this survey to the number of edges N t in the thermodynamic setting where tδ d t tends to a constant as t → ∞. Further results for other regimes, multivariate limit theorems and sharper concentration inequalities can be found in Penrose's book and the papers mentioned above.

Because of the local definiton of the Gilbert graph, N t is a local U-statistic. Theorem 6 with ν t = tδ d t can be applied. Theorem 13. Let N be a standard Gaussian random variable. Then constants c(W ) exist such that d W ( Ñt , N) ≤ c(W )t -1/2 , d K ( Ñt , N) ≤ c(W )t -1/2 . for t ≥ 1.

A concentration inequality follows immediately from Theorem 9. Observe that µ t (X) = tℓ d (W ).

Theorem 14. Denote by m the median of N t . Then there is a constant c d such that In [26] a concentration inequality for all u ≥ 0 is given using a similar but more detailed approach.

P(|Π t -m| ≥ u) ≤ 4tℓ d (W ) exp - 1 16 u √ u + m for u √ u+m ≥ c d .

Random simplicial complexes

Given the Gilbert graph of a Poisson point process η t we construct the Vietoris-Rips complex R(δ t ) by calling F = {x i 1 , . . . , x i k+1 } a k-face of R(δ t ) if all pairs of points in F are connected by an edge in the Gilbert graph. Observe that e.g. counting the number N (k) t of k-faces is equivalent to a particular subgraph counting. By definition this is a local U-statistics given by

N (k) t = N (k) t (W, δ t ) = 1 (k + 1)! ∑
x 1 ,...,x k+1 ∈η k+1 t, = 1( x ix j ≤ δ t , ∀1 ≤ i, j ≤ k + 1).

Central limit theorems and a concentration inequality follow immediately from the results for local U-statistics. We restrict our statements again to the thermodynamic case where tδ d t tends to a constant as t → ∞. Results for other regimes can be found e.g. in Penrose's book. Because of the local definiton of the Gilbert graph, N (k) t is a local U-statistic. Theorem 6 with ν t = tδ d t can be applied.

velopment by Bárány [2]. In our setting, the function t -k N t is an estimator for the probability p(W, k) and we are interested in its distributional properties. The asymptotic behaviour of Var(N t ) is of order t 2k-1 . Together with Theorem 5, we immediately get the following result showing that the estimator H is asymptotically Gaussian:

Theorem 17. Let N be a standard Gaussian random variable. Then there exists a constant c(W, k) such that d W Ñt , N ≤ c(W, k)t -1 2 .

References

Remark 6 .

 6 Point (i) first appears in [25]. Remark 7. Point (ii) crucially uses the results of Dynkin & Mandelbaum [7].

Theorem 15. Let N be a standard Gaussian random variable. Then constants c(W ) exist such that

for t ≥ 1.

A concentration inequality follows immediately from Theorem 9. Observe that µ t (X) = tθ ([W ]).

Theorem 16. Denote by m the median of N t . Then

Much deeper results concerning the topology of random simplicial complexes are contained in [6, 12] and [14]. We refer the interested reader to the recent survey article by Kahle [13]

Sylvester's constant

Again we assume that the Poisson point process η has an intensity-measure of the form µ t (•) = tℓ d (• ∩ W ), where ℓ d is Lebesgue measure and W ⊂ R d a compact convex set with ℓ d (W ) = 1.

As a last example of a U-statistic we consider the following functional related to Sylvester's problem. Originally raised with k = 4 in 1864, Sylvester's original problem asks for the distribution of the number of vertices of the convex hull of four random points. Put

which counts the number of k-tuples of the process such that every point is a vertex of the convex hull, i.e., the number of k-tuples in convex position.

The expected value of U is then given by

where X 1 , . . . , X k are independent random points chosen according to the uniform distribution on W . The question to determine the probability p(W, k) that k random points in a convex set W are in convex position has a long history, see e.g. the more recent de-