V Doebele

A Benoit-Gonin

F Souris

L Cagnon

P Spathis

P E Wolf

A Grosman

M Bossert

I Trimaille

C Noûs

E Rolley
email: rolley@phys.ens.fr

Direct observation of homogeneous cavitation in nanopores

We report on the evaporation of hexane from porous alumina and silicon membranes. These membranes contain billions of independent nanopores tailored to an ink-bottle shape, where a cavity several tens of nanometers in diameter is separated from the bulk vapor by a constriction.

For alumina membranes with narrow enough constrictions, we demonstrate that cavity evaporation proceeds by cavitation. Measurements of the pressure dependence of the cavitation rate follow the predictions of the bulk, homogeneous, classical nucleation theory, definitively establishing the relevance of homogeneous cavitation as an evaporation mechanism in mesoporous materials. Our results imply that porous alumina membranes are a promising new system to study liquids in a deeply metastable state.

A porous material imbibed with a liquid can dry by two processes: recession of a liquidvapor interface [START_REF] Mason | Determination of the pore-size distributions and pore-space interconnectivity of Vycor porous glass from adsorption-desorption hysteresis capillary condensation isotherms[END_REF] or formation of vapor bubbles within the material by cavitation [START_REF] Ravikovitch | Experimental confirmation of different mechanisms of evaporation from ink-bottle type pores: Equilibrium, pore blocking, and cavitation[END_REF][START_REF] Monson | Understanding adsorption/desorption hysteresis for fluids in mesoporous materials using simple molecular models and classical density functional theory[END_REF].

Fundamental understanding of which of these processes is effective is crucial for many applications, ranging from characterizing porous materials [START_REF] Ravikovitch | Experimental confirmation of different mechanisms of evaporation from ink-bottle type pores: Equilibrium, pore blocking, and cavitation[END_REF][START_REF] Monson | Understanding adsorption/desorption hysteresis for fluids in mesoporous materials using simple molecular models and classical density functional theory[END_REF] to controlling the shrinkage of concrete [START_REF] Maruyama | Role of alcohol-ethylene oxide polymers on the reduction of shrinkage of cement paste[END_REF]. In particular, this requires establishing whether cavitation occurs in the bulk of pores or on their surface, and, in the first case, whether pores are large enough for homogeneous classical nucleation theory [START_REF] Blander | Bubble nucleation in liquids[END_REF] (CNT) to hold or whether confinement has to be considered [START_REF] Vishnyakov | Monte Carlo simulation test of pore blocking effects[END_REF][START_REF] Vishnyakov | Nucleation of liquid bridges and bubbles in nanoscale capillaries[END_REF][START_REF] Rasmussen | Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores[END_REF][START_REF] Bonnet | Thermally activated condensation and evaporation in cylindrical pores[END_REF].

Results from previous studies lead to contradictory conclusions. As illustrated by Fig. 1, evaporation by cavitation is expected for pores presenting an ink-bottle geometry, where a wider cavity is separated from the outside vapor by a constriction narrow enough for capillarity to block the liquid-vapor interface at the cavitation pressure (Fig. 1(c)) [START_REF] Ravikovitch | Experimental confirmation of different mechanisms of evaporation from ink-bottle type pores: Equilibrium, pore blocking, and cavitation[END_REF][START_REF] Morishige | Adsorption hysteresis in ink-bottle pore[END_REF].

Homogeneous-like cavitation has thus been reported in materials with interconnected pores presenting cavities separated by constrictions, realizing such an ink-bottle geometry. These materials are either ordered (SBA-16 mesoporous silica [START_REF] Rasmussen | Cavitation in Metastable Liquid Nitrogen Confined to Nanoscale Pores[END_REF][START_REF] Morishige | Adsorption hysteresis in ink-bottle pore[END_REF][START_REF] Morishige | Change in desorption mechanism from pore blocking to cavitation with temperature for nitrogen in ordered silica with cagelike pores[END_REF], zeolites [START_REF] Maruyama | Cavitation of water in hardened cement paste under short-term desorption measurements[END_REF]) or disordered (cements [START_REF] Maruyama | Cavitation of water in hardened cement paste under short-term desorption measurements[END_REF],Vycor [START_REF] Morishige | Hysteresis Critical Point of Nitrogen in Porous Glass: Occurrence of Sample Spanning Transition in Capillary Condensation[END_REF][START_REF] Bonnet | On condensation and evaporation mechanisms in disordered porous materials[END_REF], controlled porous glasses [START_REF] Machin | Properties of three capillary fluids in critical region[END_REF][START_REF] Reichenbach | Cavitation and pore blocking in nanoporous glasses[END_REF]). In all cases, the evidence for cavitation is only indirect, relying on the interpretation of light or x-ray scattering data or, more often, on the observation of a sharp drop in the evaporation isotherm at a given pressure which is then compared to some model, usually CNT. However, in these materials, the pores have a very small diameter (several nanometers). Attractive interaction with walls should then affect the cavitation threshold, making the identification of cavitation through comparison to CNT ambiguous.

In contrast, two experiments performed on nanoporous silicon membranes with ink-bottle pores directly evidenced a two-step shape of the evaporation isotherm of nitrogen around 77 K, consistent with a cavitation mechanism, but at a much larger pressure than predicted by the CNT [START_REF] Wallacher | Capillary condensation in linear mesopores of different shape[END_REF][START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF]. This increase is suggestive of heterogeneous cavitation [START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF], in strong contrast with the results reported for the more complex geometries above. Moreover, a similar discrepancy has been found for the cavitation of dibromomethane in Vycor [START_REF] Mitropoulos | On the formation of nanobubbles in Vycor porous glass during the desorption of halogenated hydrocarbons[END_REF].

These results cast doubt on the very principle of using extensions of homogeneous CNT [START_REF] Ravikovitch | Experimental confirmation of different mechanisms of evaporation from ink-bottle type pores: Equilibrium, pore blocking, and cavitation[END_REF][START_REF] Bonnet | Thermally activated condensation and evaporation in cylindrical pores[END_REF] to predict the cavitation threshold in nanoporous materials.

In this paper, we elucidate this paradoxical situation by studying evaporation of hexane from silicon and alumina membranes with pores' transverse size in the range 20-60 nm, large enough to allow a direct comparison to bulk CNT [START_REF] Bonnet | Thermally activated condensation and evaporation in cylindrical pores[END_REF]. For both membranes, we directly evidence that cavity evaporation proceeds at a well-defined pressure, consistent with cavitation. For silicon membranes, however, the evaporation pressure strongly depends on the cavity geometry, inconsistent with homogeneous cavitation. In contrast, for alumina membranes, we accurately check the activated nature of the evaporation process, and show that it is quantitatively described by bulk CNT. This definitively demonstrates the relevance of homogeneous cavitation as an evaporation mechanism in porous materials. Our samples are fabricated using a two step procedure [20]. We first synthesize 1 cm 2 nanoporous alumina (poAl) and silicon (poSi) membranes with parallel pores, by electroetching of highly p-doped silicon [START_REF] Grosman | Capillary condensation in porous materials. hysteresis and interaction mechanism without pore blocking/percolation process[END_REF] or anodization of aluminum wafers [START_REF] Masuda | Ordered metal nanohole arrays made by a two-step replication of honeycomb structures of anodic alumina[END_REF]. In both cases, the pores, about 100 µm long, are closed on one side of the membrane and open on the other.

As illustrated in Fig. 2(a), the poAl pores are well organized with a narrow distribution in diameter around an average value of several tens of nanometers, tunable through the anodization conditions [START_REF] Lee | Structural engineering of nanoporous anodic aluminium oxide by pulse anodization of aluminium[END_REF][START_REF] Park | Porous anodic aluminum oxide: Anodization and templated synthesis of functional nanostructures[END_REF]. We study two alumina membranes with pore lengths l = 57 and 76 µm determined by scanning eectron microscopy (SEM). Their respective average pore diameters are d = 60 nm and d = 25 nm determined by combining the interpore distance measured by SEM with the membrane porous volume deduced from adsorption isotherms. In contrast, the poSi pores have a polygonal cross section (Fig. 3(c)), with a wider distribution of transverse sizes d around a mean value increasing with the sample porosity [START_REF] Grosman | Capillary condensation in porous materials. hysteresis and interaction mechanism without pore blocking/percolation process[END_REF] . Most poSi samples have a 70% porosity corresponding to d in the range 13-40 nm ( d = 26 nm) and l between 5 and 60 µm. In a second step, we deposit alumina at the pore mouth to obtain the desired ink-bottle geometry. For the 60 nm poAl and poSi samples, we use successive evaporations of 2 nm of aluminum followed by oxidation.

For the 25 nm poAl sample, we used continuous atomic layer deposition (ALD) based on the chemical reaction between trimethylaluminum and water. SEM images show that both methods yield alumina constrictions smaller than 10 nm in diameter [25]. This upper bound is consistent with the maximal constriction diameter for observing cavitation, estimated to be 6 nm using Refs. 28-30 [20].

Condensation and evaporation of hexane in these samples were studied in optical cells, regulated at a temperature slightly below the ambient temperature, with a stability of about 1 mK. A capillary line connecting the cell to a tank of hexane at saturated vapor pressure immersed in a temperature-controlled bath is used to fill or empty the membrane through a precision microvalve at a very small flow rate. The vapor pressure P V in the cell is measured by a pressure gauge. The amount of fluid in the pores is determined through the change ∆n of the membrane optical index measured by white light interferometry (WLI) [START_REF] Casanova | Direct observation of cooperative effects in capillary condensation: The hysteretic origin[END_REF][START_REF] Pacholski | Biosensing using porous silicon double-layer interferometers: Reflective interferometric Fourier transform spectroscopy[END_REF].

Figure 2 shows the successive sorption isotherms measured for the 60 nm poAl membrane as the pore aperture is progressively reduced. Here, P V is converted to P L = (RT /v L ) ln(P V /P sat ), the pressure of the liquid in equilibrium with the vapor under the assumptions of ideal gas and incompressible liquid (R is the gas constant, T the temperature and v L the liquid molar volume). Condensation takes place at a well-defined pressure, independent of the pore aperture as expected for pores closed at one end. For the native membrane, evaporation occurs at a slightly lower pressure, probably due to some pore corrugation [START_REF] Bruschi | Adsorption in alumina pores open at one and at both ends[END_REF][START_REF] Puibasset | Adsorption/desorption hysteresis of simple fluids confined in realistic heterogeneous silica mesopores of micrometric length: A new analysis exploiting a multiscale Monte Carlo approach[END_REF][START_REF] Morishige | Nature of adsorption hysteresis in cylindrical pores: Effect of pore corrugation[END_REF][START_REF] Doebele | Condensation et évaporation de l'hexane dans les membranes d'alumine poreuse[END_REF]. Progressively reducing the pore aperture shifts the evaporation to much lower pressures, in agreement with the expectation that evaporation is controlled by meniscus recession in constrictions (Fig. 1 either due to the initial pore diameter distribution and/or uneven deposition. The salient observation is that, for all coated samples, a sharp drop of the liquid content is observed at the same pressure, P cav -20 MPa, irrespective of the aperture reduction. The fraction of pores emptying at this pressure increases at each deposition step, and reaches nearly 100% for the last step. Since, at this stage, the constrictions necessarily remain distributed in diameter, the fact that the evaporation pressure is sharply defined demonstrates that evap-oration takes place within the cavities, with the constrictions remaining filled, in agreement with the cavitation mechanism depicted in Fig. 1(c). We stress that, in contrast to most experiments with porous materials, this evidence is direct, and does not rely on comparing the evaporation pressure value to any model.

We observe a similar behavior in poSi ink bottles prepared in the same way as poAl (Fig. 3(a)). However, in this case, the limiting evaporation pressure P evap is much larger, around -10 MPa. In order to confirm this difference, we perform complementary experiments on duplex-layer poSi membranes, similar to those previously used for studying nitrogen evaporation [START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF]. A bottom layer with large pores -the cavities -is connected to the vapor reservoir through a top layer with narrow pores -the constrictions -(Fig. 3(c)). These constrictions are much longer than those obtained by alumina deposition, allowing WLI to simultaneously monitor the fluid content in the constrictions and the cavities [START_REF] Casanova | Direct observation of cooperative effects in capillary condensation: The hysteretic origin[END_REF]. As shown in Fig. 3b, and similarly to Refs.17, 18, cavities empty before the constrictions. When the length l and diameter d of the cavities are identical to those for the alumina-coated poSi sample ( d = 26 nm, l = 20 µm), we find P evap -9.5 MPa, close to the -10 MPa obtained in the latter case, showing that this value is intrinsic of the cavity layer.

On the face of these results, we could be tempted to conclude, as in Ref. [START_REF] Grosman | Cavitation in Metastable Fluids Confined to Linear Mesopores[END_REF], that evaporation in poSi samples proceeds by heterogeneous cavitation on the cavity surface [35].

However, this is surprising as hexane is believed to perfectly wet most types of surfaces and should nucleate homogeneously. Moreover, as detailed in the Supplemental Material, P evap in poSi can be changed from -6 MPa to -12 MPa by varying the cavity diameter and length. This large change seems inconsistent with a heterogeneous cavitation mechanism where the cavitation pressure would only weakly depend on the pore surface. An alternative scenario, suggested by recent NMR studies [START_REF] Puibasset | Structure and permeability of porous silicon investigated by self-diffusion nmr measurements of ethanol and heptane[END_REF][START_REF] Kondrashova | Scale-dependent diffusion anisotropy in nanoporous silicon[END_REF], could be that the cavities of the bottom layer communicate with one another and are connected to the vapor by a small number of wide channels through the top layer (the total volume of these channels being too small to be detected by WLI). In this case, the bottom layer would empty by a percolation mechanism [START_REF] Mason | Determination of the pore-size distributions and pore-space interconnectivity of Vycor porous glass from adsorption-desorption hysteresis capillary condensation isotherms[END_REF], consistent with the observation of a sharply defined evaporation pressure. Testing such a scenario would require further studies.

In contrast, for poAl, repeating the experiment of Fig. 2 -20. C. Here, Γ is measured from the exponential decay of the number of filled pores following a quench of the pressure reservoir from 60 mb down to a lower pressure ranging between 52 and 55 mbar, as illustrated in the inset for the 60-nm membrane. The fraction of filled pores φ is measured through the logarithm of the optical transmission and normalized to its value at time t=0, corresponding to a 10% transmission(see Supplemental Information). The cavitation rate per unit volume Γ is deduced by dividing the decay rate by the pore volume, which is computed using d = 56 nm and d = 27 nm. These values lie within the error bars of the measured values above, and are such that Γ(P L ) is identical for the two membranes.

Lines corresponding to the CNT predictions for different values of the attempt rate or the surface tension (see text).

In order to test whether cavitation in poAl is quantitatively described by CNT, we measure its nucleation rate Γ(P L ). A specific feature of our experimental system is that the mem-branes contain a very large number of independent pores, of the order of several 10 10 /cm 2 . This feature allows us to determine the cavitation rate in a single-shot experiment, in contrast to acoustically driven [START_REF] Pettersen | Experimental investigation of cavitation in superfluid 4 He[END_REF] or thermally controlled [START_REF] Azouzi | A coherent picture of water at extreme negative pressure[END_REF] bulk cavitation experiments, where the cavitation statistics is determined over thousands of cycles. To this aim, starting from a slightly larger pressure, we quench the reservoir pressure to a stable value corresponding to P L around -20 MPa, and monitor the temporal decay of the number of filled pores by measuring the light transmission through the membrane. The totally empty or filled membrane, being nearly homogeneous on the scale of the light wavelength, only weakly scatters.

In contrast, when stochastic cavitation takes place, the pores randomly empty, giving rise to local fluctuations of the refractive index and strong light scattering, resulting in a reduced transmission. This effect can be directly evidenced by illuminating the membrane with a wide collimated light beam and precisely quantified by measuring the transmission of a laser beam [20].

We have performed such experiments for the two alumina membranes. As expected for a stochastic process, the number of filled pores decreases exponentially with time with a time constant τ (inset of Fig. 4 for the 60-nm membrane). Repeating this quench at different depths yields τ (P L ), from which we deduce the cavitation rate per unit time and unit volume Γ(P L ) = (V p τ (P L )) -1 .

Figure 4 shows that Γ(P L ) increases by nearly 3 orders of magnitude when changing P V , hence P L , by only 5%. This large increase exemplifies the exponential dependence of the relaxation time on the energy barrier.

To our knowledge, these results are the first evidence for the stochastic and activated nature of cavitation in nanoporous materials. The relaxation rate can be compared to the CNT prediction [START_REF] Blander | Bubble nucleation in liquids[END_REF] Γ = Γ 0 exp(-E b /k B T ), where Γ 0 is an attempt rate and the energy barrier E b is given by:

E b = 16πσ 3 3(P V -P L ) 2 (1)
where σ is the surface tension. Different expressions for Γ 0 [START_REF] Blander | Bubble nucleation in liquids[END_REF][START_REF] Maris | Nucleation of bubbles in liquid helium at negative pressure[END_REF] lead to Γ 0 = 2.10 38 m -3 s -1 within a factor of 10. As shown by Fig. 4, using this value and the bulk surface tension (0.185 J/m 2 at 19 • C [START_REF] Lemmon | Thermophysical Properties of Fluid Systems[END_REF]) leads to predicted rates that are about 10 7 too small over the full pressure range, corresponding to a predicted cavitation pressure 15% larger than observed. Keeping Γ 0 = 2.10 38 m -3 s -1 , an approximate agreement with experiments requires us to reduce the surface tension σ by about 9% with respect to its bulk value. A similar difference has been measured in the case of bulk cavitation of heptane and ascribed to a reduction of the surface tension due to the large curvature of the critical germ [START_REF] Bruot | Curvature dependence of the liquid-vapor surface tension beyond the Tolman approximation[END_REF]. However, adjusting only Γ 0 or σ does not allow us to match the experimental Γ vs P L data over the full pressure range. Such a match requires that we combine a 5% reduction of σ with an increase of Γ 0 by a factor of order 500. This requirement might point to the fact that the above expressions of the CNT attempt rate are underestimated.

To summarize, we show that cavitation of hexane in porous alumina ink bottles with cavity diameters of several tens of nanometers closely follows the predictions of the bulk, homogeneous, classical nucleation theory. This result confirms that homogeneous cavitation is a relevant mechanism in nanoporous systems, as assumed by previous studies on materials with interconnected pores. Our approach opens the way to performing similar experiments with pores of smaller diameter (below 10 nm) in order to quantitatively test extensions of CNT in the presence of confinement. In contrast, we confirm earlier experiments, finding that porous silicon ink bottles empty more easily than expected from CNT . More studies will be required to understand whether this is due to heterogeneous cavitation or to a nonideal structure of poSi.

Finally, our work opens new prospects for fundamental studies of cavitation. In contrast to cavitation acoustically driven at MHz frequencies, our experiments are essentially static, allowing us to precisely measure the relaxation rate at a given pressure. Also, in contrast to the so-called artificial-tree technique [START_REF] Wheeler | The transpiration of water at negative pressures in a synthetic tree[END_REF], where cavitation is probed in macroscopic cavities closed by a single porous layer, nanoporous membranes with independent nanopores tolerate the existence of a small density of leaky constrictions. PoAl membranes are thus a promising system to address points of current debate, such as the influence of superfluidity of liquid helium on its cavitation [START_REF] Qu | Cavitation density of superfluid helium-4 around 1 K[END_REF], or the origin of the much larger cavitation pressure observed for water in the artificial-tree geometry [START_REF] Caupin | Exploring water and other liquids at negative pressure[END_REF] as compared to that observed in quartz inclusions [START_REF] Zheng | Liquids at large negative pressures: Water at the homogeneous nucleation limit[END_REF]. Beyond these examples, by allowing us to decrease the liquid pressure down to its tensile limit, these membranes open a new route to study liquids in deeply metastable states.

This paper is dedicated to the memory of our late co-author and colleague A. Grosman.

We thank K. Davitt for her critical reading. We acknowledge the financial support of Agence Nationale de la Recherche through the project CavConf, ANR-

FIG. 1 .

 1 FIG. 1. Cavitation in an ink-bottle geometry. (a) Straight cylindrical pore opened to a vapor reservoir, which empties through recession of the liquid-vapor meniscus. The smaller the pore diameter, the smaller the evaporation pressure. (b,c) Two possible evaporation mechanisms for a cavity ended by a cylindrical constriction. In panel(b), the constriction empties at its equilibrium pressure, triggering further evaporation in the wider cavity through meniscus recession; in panel (c), if the constriction is narrow enough for its evaporation pressure to lie below the cavitation threshold, the cavity empties by cavitation, while the constriction remains filled with liquid.

FIG. 2 .

 2 FIG. 2. Cavitation in an alumina membrane. Starting from a native alumina membrane with 60 nm-diameter pores (a), successive 2-nm-thick alumina layers are deposited at the mouth of the pore, reducing the pore aperture (b: 8 layers). Sorption isotherms of hexane are measured at 18 • C (c) as described in the text. The membrane fluid content is deduced from ∆n, the change of the membrane optical index with respect to the empty state. The (superimposed) dashed curves are the condensation isotherms, and the continuous lines are the evaporation isotherms, for increasing deposits of alumina at the pore mouth (black is for native, green for 8 layers, blue for 10 layers, and red for 12 layers). For intermediate coatings, the noise is due to a loss of contrast of the interference pattern (resulting from a strong light scattering [20]). The sharp drop at -20 MPa is the signature of cavitation.

FIG. 3 .

 3 FIG. 3. Evaporation in porous silicon ink bottles: (a) Isotherms for an alumina-coated poSi sample (left part of panel c). Superimposed dashed lines correspond to condensation and solid lines to evaporation. Black: lines are poSi as prepared ( d = 26 nm, pore length l = 20 µm); green/blue/red lines are poSi coated by 2/6/8 alumina layers. For as-prepared poSi, the condensation isotherm is much less steep than in the case of poAl, reflecting the wider distribution of pore diameters. (b) Isotherms obtained on a duplex-layer sample formed by successively electro-etching a top layer with small pores d = 12 nm and a bottom layer with large pores d = 26 nm (right part of panel c). The contributions of the two layers are plotted separately in black for the top layer and in red for the bottom layer. (c) binarized transmission electron microscopy (TEM) images of the cross section of cavities ( d = 26 nm, red) and of the constrictions of the duplex sample ( d = 12 nm, black.

  FIG. 4. Cavitation rate Γ as a function of liquid pressure P L for poAl membranes with average pore diameters d = 60 nm and d = 25 nm at 19

					1			54.90
	-3 s -1 )	10 18					53.65 mbar 54.24
	Cavitataion rate (m	10 10	14 16	5	-20 60 nm pores 0 10 7 , bulk 0 , bulk 0 450 , bulk 0.950 0.914 25 nm pores 0 10 -2	52.5	-19.5 53.08 1	t (h)	-19 2
					P L (MPa)

  17-CE30-0002, including the funding of F. S. and M. B., and of Université Grenoble Alpes, which funded V.D. through an AGIR PhD grant. A. G., M. B., I. T., E. R., V. D., A. B.-G., F. S., P. S., and P. E. W.participated in the measurements and analyses; P. E. W. and E. R. wrote the paper with contributions from L. C., and P. S. C. N. represents a consortium who has contributed to the conceptualization and methodology of this work.


Patrice Guillotreau

Kevin Bistoquet

How Eurostat can assist CO2 assessment in small island developing states: a post-Covid estimation of the Seychelles carbon footprint

Keywords: Environmentally-Extended Input-Output (EE-IO) model, air emission accounts, carbon footprint, Covid-19, Seychelles economy

Small Island Developing States (SIDS) are particularly vulnerable to climate change and ought to pay attention to their own contribution through the CO2 emissions resulting from the domestic production and consumption levels. Although poorly responsible of the worldwide carbon emissions by the modest level of their domestic demand, they can nonetheless contribute to the problem because of the global demand for their exported commodities. However, the carbon footprint is hardly assessed by SIDS because of a lack of data about greenhouse gas emissions or national account statistics. Taking the opportunity of the Covid-19 pandemic and the resulting economic shock, an environmentally-extended input-output model based on Eurostat data on air emissions is used to disentangle the CO2 emissions embodied in the domestic production and international trade, respectively, and to clearly identify the origin of emissions by industry. Not surprisingly, the consumption-based carbon footprint of Seychelles is deemed lower (6.79 tCO2 per capita) than the production-based inventory (9.55 tCO2 p.c.) for this small open economy relying to a large extent on exports of canned tuna and tourism services, hence a decrease of carbon dioxide emissions (-16%) in 2020 because of the Covid-19 pandemic. Could it be the right time to re-frame the international specialization of Seychelles?

(www.sustainabledevelopment.un.org). Agriculture lands, populations and infrastructures are often concentrated in the coastal zones, making SIDS highly vulnerable to climatic extreme events. Although poorly responsible of the worldwide carbon emissions by the modest level of their domestic demand, they can nonetheless contribute to the problem because of the global demand for their exported commodities. Assessing the carbon dioxide emissions of SIDS would allow to identify the main sources of emission and to implement targeted abatement policies ("charity begins at home"). The UN international Mauritius programme of action for the sustainable development of SIDS (2005) and the Bali Action Plan (2007) have first defined some Nationally Appropriate Mitigation Actions (NAMAs) to reduce the impact of SIDS and other developing countries on climate change through "common but differentiated responsibilities and respective capabilities" (UNDP 2008). The NAMAs aim at inventing pathways and concrete steps for developing countries to reduce their emission trajectory. Several conditions are required to follow the adopted action plans: the initiatives must be measurable, verifiable and reportable (Ibid.). Developing such actions can be seen as a means of meeting sustainable development aspirations. It can also shed new light on particular industries which are vital although overlooked for SIDS economies, such as natural resource-based industries and tourism [START_REF] Burns | The changing landscape of climate change: NAMAs, SIDS and tourism[END_REF]Vishan 2010, Pratt 2015).

Depending to a great extent on foreign trade, the economic benefits of these activities for SIDS in terms of income can be outweighed by the negative environmental consequences of trade liberalization [START_REF] Sannassee | Trade openness and CO2 emission: from a SIDS. In Handbook of environmental and sustainable finance[END_REF]. While developed countries tax more heavily polluting activities, the pollution haven hypothesis would tend to concentrate the most polluting industries in developing countries [START_REF] Zheng | Multiple environmental policies and pollution haven hypothesis: evidence from China's polluting industries[END_REF]. Separating the production-based from the consumption-based CO2 emissions would show the respective responsibilities of developed trade partners and of small developing nations in global emissions [START_REF] Nath | Climate Change, Sea Level Dynamics, and Mitigation[END_REF]. By reducing the weight of their most emission intensive activities, the SIDS economies can also become less dependent to fossil fuel energy, providing the double dividend of a "no-regret" strategy, i.e. a net gain even if the climate change does not result in too severe consequences (Hallegate 2009). They can do so by shifting to "green" or "blue" energy sources and deploy defensive strategies to enhance the environmental productivity [START_REF] Nath | Climate Change, Sea Level Dynamics, and Mitigation[END_REF].

Today, the emission levels of carbon dioxide are calculated by both the International Energy Agency (IEA) and the Carbon Dioxide Information Analysis Center (CDIAC) 1 . However there is no information regarding industry level breakdowns in the data, nor up-to date input-output tables as developed in an environmentally-extended input-output (EE-IO) [START_REF] Hertwich | Carbon footprint of nations: A global, trade-linked analysis[END_REF][START_REF] Wiedmann | Quo Vadis MRIO? Methodological, data and institutional requirements for multi-region input-output analysis[END_REF][START_REF] Moran | Convergence between the Eora, WIOD, EXIOBASE, and Open EU's consumption-based carbon accounts[END_REF][START_REF] Wiebe | Estimating CO2 Emissions Embodied in Final Demand and Trade Using the OECD ICIO 2015: Methodology and Results[END_REF][START_REF] Wu | Carbon emissions embodied in the global supply chain: Intermediate and final trade imbalances[END_REF][START_REF] Brown | Producing environmental accounts with environmentally extended input output analysis[END_REF]). Therefore we propose in this research to develop EE-IO model for the first time in Seychelles on the basis of Eurostat data on air emissions [START_REF] Gajos | National Accounting Matrix with Environmentally Extended IO analysis[END_REF], and to estimate both CO2 footprint and national inventory before and after the COVID-19 pandemic.

The concept of carbon footprint is defined as a synthetic indicator of ecological sustainability [START_REF] Boutaud | L'empreinte écologique[END_REF]. The carbon footprint of a resident population represents the set of Greenhouse Gas (GHG) emissions, of which CO2 is a major component, associated with the domestic consumption, whatever the originating country where the goods and services are produced. The national inventory is a domestic production-based assessment, including the demand for exports but excluding the emission content of imports. Since the protocol of Kyoto signed in 1997 and the various Conferences of Parties afterwards, many countries are committed to reduce their emissions of carbon dioxide. Several studies have shown a transfer of emissions between developed countries and emerging or developing countries because of emissions embodied in international trade [START_REF] Ahmad | Carbon Dioxide Emissions Embodied in International Trade of Goods, STI Working paper 2003-15[END_REF][START_REF] Aichele | Kyoto and the carbon footprint of nations[END_REF][START_REF] Wiebe | Estimating CO2 Emissions Embodied in Final Demand and Trade Using the OECD ICIO 2015: Methodology and Results[END_REF][START_REF] Wu | Carbon emissions embodied in the global supply chain: Intermediate and final trade imbalances[END_REF]. As largely open economies, the SIDS are particularly concerned by the gap between consumption-based and production-based emissions [START_REF] Robinson | Climate change adaptation in SIDS: A systematic review of the literature pre and post the IPCC Fifth Assessment Report[END_REF]. In particular, in many SIDS, the economic pillars are tourism and primary sectors (mining, agriculture and fishing), which may release a great deal of greenhouse gas in the atmosphere [START_REF] Archer | The economic impact of tourism in the Seychelles[END_REF]Fletcher 1996, Chassot et al. 2021). This is why it is of major importance to assess the Seychelles CO2 footprint and inventory, attempting to disentangle the emissions embodied in trade and domestic production.

Such an assessment is a timely necessity as new challenges are coming along for SIDS like the Republic of Seychelles since the recent pandemic shock. On March 11 th 2020, the World Health Organization declared Covid-19 a worldwide pandemic resulting in a wave of lockdowns for many countries. The International Monetary Fund considered that the global economic growth of 2020 would be reduced by 3.5% in their revised estimation of January 2021, before bouncing back to +5.5% in 2021 and +4.2% in 2022 according to projections (IMF 2021). The African Bank of Development (AfdB), in its outlook of March 2021, predicted the worst growth rate (-12% in 2020) of all East-African countries (AfDB 2021). The World Trade Organization predicted that global trade would contract between 13 and 32% in 2020 (WTO 2020). On the basis of autoregressive models and impulse response functions, the UN Department of Economic and Social Affaires predicted a situation for SIDS that should be even worse than the financial crisis of 2008-09 since the length of recovery depends on the magnitude of the initial shock: SIDS are projected to take about four years to return to the baseline forecast path [START_REF] Kim | How long will it take for LDCs and SIDS to recover from the impacts of COVID-19? United Nations[END_REF]. Consequently, the level of CO2 emissions associated with production and consumption should be somehow affected by the trade restrictions imposed by the pandemic. SIDS rely heavily on foreign trade and supply of commodities for the day-to-day life. In particular, the energy required by the Seychellois households and industries is fossil fuelbased, machinery and transport equipment represent 30% of imports, even the fish canning plant, main private employer of the archipelago, needs a large volume of raw materials like frozen tuna and tin metal cans coming from overseas. A sudden disruption of foreign trade affects dramatically the whole economy [START_REF] Kontovas | Maritime Cargo Prioritisation during a Prolonged Pandemic Lockdown Using an Integrated TOPSIS-Knapsack Technique: A Case Study on Small Island Developing States-The Rodrigues Island[END_REF]. In Seychelles, a comprehensive report assessing the economic consequences of the COVID-19 pandemic for the country was published by the Government and the United Nations in December 2020 [START_REF] Rassool | An assessment of the socio-economic impact of Covid-19 in Seychelles, UNDP Seychelles and Government of Seychelles report[END_REF]. It clearly showed the extent of the social and economic shock caused by the pandemic, with a GDP contraction of 11.5% in 2020 (the IMF 2020 forecast was even higher at -13.8% for 2020). The budget balance as a share of GDP was one of the worst among SIDS at -15.5% in 2020, with a projection of -6% in 2021 [START_REF] Rassool | An assessment of the socio-economic impact of Covid-19 in Seychelles, UNDP Seychelles and Government of Seychelles report[END_REF]. Exports decreased by 38.9% in 2020 (IMF 2021), and the actual GDP contraction in 2020 estimated by the National Bureau of Statistics finally amounted to -11% relatively to previous year [START_REF] Nbs | Supply and Use Table[END_REF].

The Covid-19 pandemic has deeply affected the whole economy but some industries have suffered more than others, like the tourism sectors bearing a 70% reduction in the number of visitors in 2020. Tourism is considered in this country as the main pillar of the economy and would normally represent one fourth of the Seychelles GDP directly, and two-thirds after including indirect and induced effects [START_REF] Archer | The economic impact of tourism in the Seychelles[END_REF], Valenghi 2004, Pratt 2015). The second most important source of income comes from the fishing and tuna processing industries 10% to 15% of the GDP directly and indirectly [START_REF] Rassool | An assessment of the socio-economic impact of Covid-19 in Seychelles, UNDP Seychelles and Government of Seychelles report[END_REF]. A new prevailing role was assigned to fishery-related industries and the "blue economy" after the COVID-19 shock. In the recent years, Seychelles has developed an ambitious and innovating programme for the development of the blue economy (Rep. of Seychelles 2018). The blue economy is hard to define and delineate accurately, though it follows the overarching objective of UN-SDG14 ("Conserve and sustainably use the oceans, seas and marine resources for sustainable development"). However, these two industries (tourism and fishing) are both highly energydependent, hence responsible for a great deal of CO2 emissions in Seychelles. How is the new deal of economic activity after the pandemic changing the rules and re-allocate carbon emissions between industries? Will the economy come back to the current specialization after the shock, or take this opportunity to re-direct its resources to a "greener" (or "bluer") economy? A thorough analysis of carbon emissions resulting from domestically-produced and imported commodities might help the government to decide what the best options are for the country.

The following article starts in Section 1 with a first glance at Seychelles CO2 emissions as reported by some international agencies. An original EE-IO model is introduced in the second section, following the international methodological standard used by the EU-27 and OECD countries to allow for comparisons. A numerical application, including the estimated impact of the COVID-19 pandemic on carbon emissions, is proposed in Section 3 on the basis of a Supply-Use Table recently released by the National Bureau of Statistics for the first time. Finally, a last section will discuss the results in terms of international specialization for Seychelles.

The overall CO2 emissions in the Republic of Seychelles

Data about emissions from fuel combustion are made available, among others, by the International Energy Agency (IEA, www.iea.org). Carbon dioxide emissions are calculated using IEA energy databases and the default methods and emission factors given in the 2006 IPCC Guidelines for National Greenhouse Gas Inventories 2 . From IPCC sources, we learn that carbon dioxide emission factors for electricity and heat have been derived as "the ratio of CO2 emissions from fuel inputs of power plants relative to the electricity and heat delivered" (www.ipcc.ch). The calculation is conducted at the country level, indirect emissions being derived by multiplying the amount of energy and heat consumed by CO2 emission factors at the sectoral level. The Emission Database for Global Atmospheric Research (EDGAR) developed by both the European Commission Joint Research Centre (JRC) and the Dutch environmental assessment agency provides an independent database of atmospheric emissions for all countries in the world since 1970. The calculation of emissions uses a technology based emission factor approach combining data from the International Energy Agency (IEA), the Food and Agricultural Organization (FAO) and additional sources [START_REF] Crippa | Fossil CO2 and GHG emissions of all world countries[END_REF] The amount of emissions can vary between sources, for instance between the IEA and the World Bank indicators based on data issued by the Carbon Dioxide Information Analysis Center (CDIAC). The IEA appears to be the most cited reference when dealing with CO2 emissions from heat and energy. The level of air emissions in the Republic of Seychelles has tremendously grown since the early 1970s, from 20,000 tonnes annually to more than 500,000 tonnes in 2016 (Fig. 1). A large proportion (82%) of these emissions is released by the power industry itself, based on fossil fuel, and the remaining part by the other industrial combustions, including transport and buildings. The increasing trend is obviously explained by the growing population but also by the economic development of the country, as illustrated by the steadily increasing level of emissions per capita, from less than half a tonne to more than 5.4 tonnes over the same period of time. We can compare this last figure to the average worldwide carbon emissions of 1.35 tonnes per capita in 2014, meaning that Seychelles is well above average and very close to the EU-27 level of 5.70 tonnes according to Eurostat4 . However, the above figures do not tell anything about the import content of emissions relatively to the domestic production, nor the details by activity. This is why we suggest to use the standard Input-Output approach extended to environment proposed by [START_REF] Leontief | Environmental repercussions and the economic structure: an inputoutput approach[END_REF] and introduced in greater details in Suh (2009) or [START_REF] Miller | Input-Output Analysis, Foundations and extensions[END_REF]. In their 10 th Chapter, [START_REF] Miller | Input-Output Analysis, Foundations and extensions[END_REF] explain how to extend the famous Leontief model to deal with energy flows or any environmental factor measured in monetary or physical terms. It can be done in a single-country or multi-regional framework, with respect to the level of available information about emissions, inter-industry transactions and international trade [START_REF] Hertwich | Carbon footprint of nations: A global, trade-linked analysis[END_REF][START_REF] Moran | Convergence between the Eora, WIOD, EXIOBASE, and Open EU's consumption-based carbon accounts[END_REF][START_REF] Wiebe | Estimating CO2 Emissions Embodied in Final Demand and Trade Using the OECD ICIO 2015: Methodology and Results[END_REF][START_REF] Wu | Carbon emissions embodied in the global supply chain: Intermediate and final trade imbalances[END_REF]. A good starting point for the Republic of Seychelles would consist in developing a single environmentally-extended input-output (EE-IO) model taking into consideration the import content of emissions. This would further serve for other types of footprint (water, energy, raw materials and waste) that could be helpful to minimize the impact of the economy on the natural environment.

The environmentally-extended input-output approach

In an Environmentally-Extended Input-Output model (EE-IO), the indirect emissions resulting from the inter-industry linkages along the supply chains are considered in addition to the direct emissions produced by each industry and by the end users (households, government, investors and rest of the world). A symmetric input-output table is therefore needed to calculate the input coefficients and the multipliers. Starting from the classical Leontief model:

𝑋 = (𝐼 -𝐴) . 𝐹 (1)
Where 𝑋 is a column vector of output, 𝐴 is the matrix of technical (or input) coefficients (the industry's requirements in every input to produce one unit of output in monetary terms), 𝐼 is the identity matrix and 𝐹 is a column vector of final uses (internal and external, see below).

The impact factors need to be expressed as intensity rates in physical terms (e.g. Gigagrams of CO2) per monetary unit of output 𝑒 = 𝐸 𝑋 ⁄ , where 𝐸 denotes the industry j's CO2 emissions and 𝑒 the intensity (or direct impact) coefficient, i.e. the amount of pollutant type per unit of industry j's output value. The industry j's CO2 emission is therefore 𝐸 = 𝑒 . 𝑋 . Now substituting 𝑋 by its value in (1) gives the level of emissions directly and indirectly resulting from industry j's activity:

𝐸 = 𝑒 . (𝐼 -𝐴) . 𝐹 (2)
The interpretation is easy: any change by one unit of final demand for commodity j will result in a 𝑒 . (𝐼 -𝐴) shift of direct (the industry and its first suppliers) and indirect emissions (the suppliers' suppliers in a chain of cascading effects). What matters then is to separate the emissions caused by the sole domestic production from those linked with the imported inputs used by the industries and those associated with the imports of final goods and services. To do so, we followed the stepwise methodology to calculate the carbon footprint in France and in the rest of Europe or OECD countries [START_REF] Pasquier | Méthodologie de calcul de l'empreinte carbone de la demande finale intérieure française[END_REF][START_REF] Wiebe | Estimating CO2 Emissions Embodied in Final Demand and Trade Using the OECD ICIO 2015: Methodology and Results[END_REF][START_REF] Brown | Producing environmental accounts with environmentally extended input output analysis[END_REF]). Let's try first to measure the domestic production-based emissions

𝐸 = 𝑒 . (𝐼 -𝐴 ) . 𝐹 (3)
The capital letters denote the column-vectors, the hat symbol stands for an operator transforming a column-vector into a diagonal square matrix, the superscript d means domestic.

The import vector of the country must be split into the matrix of imported inputs (𝐴 ) used by the domestic industries and the vector of imported commodities and services for final uses (𝐹 ):

𝑀 = 𝐴 𝑋 + 𝐹 (4)
Substituting 𝑋 by its value in (1), we obtain:

𝑀 = 𝐴 . (𝐼 -𝐴 ) . 𝐹 + 𝐹 (5)
The CO2 emissions embodied in imports from a country c will depend both on the share this exporting country holds in the domestic imports, on its own emission intensity coefficients by industry, but also on the technology (combination of inputs) used to produce a commodity. For instance, producing electricity through renewable energy or fossil-based inputs will not result in the same level of carbon dioxide emissions. Consequently, the emissions embodied in imports can be written:

𝐸 , = 𝑒 . (𝐼 -𝐴 ) . 𝑀 (6)
We can replace 𝑀 in Eq. ( 6) by its value in (5):

𝐸 , = 𝑒 . (𝐼 -𝐴 ) . [𝐴 . (𝐼 -𝐴 ) . 𝐹 + 𝐹 ] (7)
By doing so, we can isolate the emissions embodied in imports of intermediate inputs by the domestic industries: 𝐸 , = 𝑒 . (𝐼 -𝐴 ) . [𝐴 . (𝐼 -𝐴 ) . 𝐹 ], and the emissions embodied in imports of commodities for final uses, 𝐸 , = 𝑒 . (𝐼 -𝐴 ) . 𝐹 By assuming a certain stability of input coefficients throughout time, both for domestic or imported input requirements of industries, we can estimate the current level of carbon dioxide emissions for the year of reference, but also calculate the percent change of emissions after an economic shock like the Covid-19 pandemic, by substituting 𝛥𝐹 + 𝛥𝐹 for (𝐹 + 𝐹 ) .

Eq. ( 7) represents the footprint which is consumption-based, i.e. including the domestic demand in 𝐹 but excluding the foreign demand (exports) of domestically-produced goods and services. It can be distinguished from a national inventory of production-based emissions, where the column-vector of exports will be re-integrated into 𝐹 but where also 𝐹 will not be considered in Eq. ( 7).

One further objective to improve the estimation is to use a Multi-regional input-output (MRIO) model [START_REF] Hertwich | Multiregional Input-Output Database, One Planet Economy Network[END_REF]. Although foreign trade data in Seychelles are poorly detailed by commodity, origin and destination, it remains possible to use a simple MRIO model such as the one introduced by Miller and Blair (2009, p. 264), where two countries would be considered (e.g. Seychelles and EU-27):

𝐸 = 𝑒. (𝐼 -𝐶𝐴) . 𝐶𝐹 (8)
Where E represents the vector of emissions resulting from the final demand, wherever the goods are produced, 𝑒̂ denotes a diagonal vector of emission intensity per unit of output in the domestic and trade partner countries, the two matrices A and C are, respectively:

𝐴 = 𝐴 0 0 𝐴 𝐶 = 𝑐 𝑐 𝑐 𝑐 (9)
A is a block diagonal matrix whose submatrices represent regional technology structures (technical coefficient matrices). Matrix C represents trade flows within and between regions as a proportion (i.e. market share) of total trade for every commodity, including the domestic region which is considered (i.e. the sum in column is equal to 1). Finally, F is the demand for exogenous sectors in the domestic country and trade partners. The multipliers obtained by [𝑒 . (𝐼 -𝐶𝐴) . 𝐶] would give not only the direct and indirect emissions due to a level or a change of exogenous final demand in the country, but would distribute the emission effects across supplying regions according to the percentages embodied in the components of block matrix C (Blair and Miller 2009).

Application to the Seychelles case

Data

The Miller and Blair 2009, p. 151, for an explanation of the method). The domestic IOT is shown in Appendix (Table A1).

From that point, we were in a position to calculate both 𝐴 and 𝐴 matrices, the sum of which representing the total technical coefficient matrix (𝐴 = 𝐴 + 𝐴 ). The 𝐴-matrix was used as a proxy of the foreign country technology structure, assuming that both national and foreign technologies were identical. This is not unrealistic because the biggest domestic industries in Seychelles (such as the fish cannery, Indian Ocean Ltd) are foreign capital-owned, compete internationally and export most of their output to European Union countries (France, UK, Italy, Germany…), hence having the highest technology standards to produce and export commodities.

When it comes to emission intensity coefficients, we used the EU-27 CO2 emissions, adjusted for African countries5 . The IEA suggests to apply coefficients of deviation to the EU-27 (air emission accounts) database at Eurostat of air pollutant emissions, with a different coefficient for electricity and non-electricity industries [START_REF] Pasquier | Méthodologie de calcul de l'empreinte carbone de la demande finale intérieure française[END_REF]. The emission coefficients 𝑒 are expressed in thousand tonnes of CO2 per million Euro, and thus need to be converted by the exchange rate in thousand tonnes per million SCR (the Central Bank rate was SCR16.96 per Euro in 2014). The coefficients were then averaged for some industries to be adjusted from a 65-row vector to a 23-row vector. Note that a single CO2 emission coefficient of the Manufacture of food, fishery, beverage and tobacco products from the EU-27 classification in 65 industries was used for the three distinct industrial categories found in the Seychelles industrial classification. The table of emission coefficients by activity of the Seychelles Industry Classification (SIC), corresponding to a selection drawn from the ISIC (UN International Standard Industrial Classification of all economic activities), is left in Appendix (Table A2).

The last missing information concerned the direct emissions released by domestic households: CO2 emissions resulting from the use of personal vehicles, air-conditioning, cooker and other devices… They are available for the EU countries, such as France (1.76 tonnes of CO2 equivalent per capita in 2014) or the whole EU-27 (1.59 t CO2e p.c. in 2014). At the worldwide level, the CDIAC provides the figure of 1.35 tCO2e p.c., which may be used as a proxy value for the country [START_REF] Boden | Global, Regional, and National Fossil-Fuel CO2 Emissions. Carbon Dioxide Information Analysis Center[END_REF].

Results

Beyond the direct emissions of households from combustion (personal vehicles and other equipment) representing 127 kt, the indirect emission content of final demand resulting from domestic production was calculated with Eq. (3). The CO2 footprint of Seychelles (direct + indirect emissions) reached 640 ktCO2 in 2014, i.e. 3.67 tonnes per capita. Such a figure may look quite high if we consider that imports represent 25% of final domestic consumption, but the local industry relies mostly on fossil fuel energy which is highly CO2-emitting. We can identify the key industries responsible for the highest level of CO2 emissions with regard to the domestic or foreign origin (Fig. 2). As far as the domestic GHG emissions resulting from the mere domestic demand (households' consumption + firms' investment + government expenditure) are concerned, the domestic output generates two thirds of the CO2 footprint, and another third is for imports of intermediate and final commodities. A large proportion of the territorial emissions (21%) originates from the electricity & gas sector because of the fossil fuel origin of energy, but the public administration & defence activity stands just behind (14%), followed by Transportation & storage, Wholesale & retail, Administrative & support services, and construction with very close shares (10%, 9%, 8% and 8%, resp.).

The picture is rather different when looking at the import content of emissions associated with the domestic demand (Fig. 2). First, the 'Other Manufacturing' industry produces two thirds of the emissions from final good imports (cars, electric and electronic equipment, etc.), the remaining share resulting mostly from three industries ( In the national inventory, only the emissions issued by the domestic output to meet the foreign demand for exports is included, for a total amount of 427 kt. However, exported commodities such as canned fish also need imports (e.g. of frozen fish) whose production process is CO2-emitting. When adding the emissions embodied in imports for external demand served by the Seychellois industries, the cumulative amount of emissions would reach 617 kt. Three industries are particularly involved in CO2 emissions for exports, reflecting the economic of the country. Transportation and storage is the first one, in particular to meet the external demand for tourism services. The second pillar of the economy, namely the manufacture of fishery products, comes in a second position, and finally, accommodation and food services (still for tourism purposes) is the third most polluting industry.

In order to summarize the previous values, the total CO2 of Seychelles are displayed in Table 1 This first assessment of a detailed CO2 footprint for Seychelles is consumption-based, i.e. not looking at the origin of commodities that are domestically consumed. This means that emissions related to exports are excluded. Re-assessing the national emissions on the mere basis of domestic production, whether the output is consumed internally or in foreign countries, would give a totally different picture. When subtracting the imported emissions for final use and including the CO2 content of exports, the total CO2 emissions related to domestic production (i.e. the national inventory) would reach 900 kt (or 9.55 tCO2 per capita), far above carbon footprint in Table 1.

One should also note that the total CO2 footprint per capita is estimated to 6.79 tonnes but this figure encompasses the direct household emissions linked with the gas combustion of personal vehicles, air conditioning and other devices estimated from the total emissions per capita at the worldwide level (Table 1). This latter amount may therefore include also the industrial emissions and be counted twice. When excluding the households' direct emissions, the footprint falls down to 5.44 tCO2, a level which is quite close to the one estimated by the IEA, 5.33 t CO2 (worldometers.info). Let's try to figure out why the total footprint in Table 1 is perhaps over-estimated through a sensitivity analysis (sub-section 3.3 below).

We can first compare these results with a different approach based on a MRIO model (Eq. 8 and Eq. 9 in previous section). Because NBS does not provide any details about foreign trade combining commodities and origin or destination, it was not possible to distribute the regional effects between partner countries. We chose to integrate the trade between Seychelles and the EU-27 block for which we know the A-matrix of technical coefficients and the e-vector of emission intensity by sector. We therefore assumed that all Seychelles imports originate from the EU-27 and that all Seychelles exports are directed to EU-27 partners. Such an assumption remains fairly realistic because EU countries supplied 34.4% of Seychelles imports and absorbed 88% of Seychelles exports in 2020 for the sole merchandise goods. The other major suppliers from which Seychelles import commodities are the United Arab Emirates (UAE, 26.5% in 2020), South Africa (6.5%), India (5%), China and Mauritius (3.5% each) (NBS 2021). EU visitors represent also 57% of foreign tourism visits between January 2010 and March 2020 (source: Central Bank of Seychelles). In our estimation, the mere domestic demand (excluding exports) was retained as the exogenous final demand vector. The results obtained are slightly lower than the results of Table 1: the domestic CO2 emissions resulting directly or indirectly domestic output amounted to 307 kt (i.e. 3.26 t per capita). The emissions in other countries proxied by the EU-27 block and resulting from the Seychelles final demand reached 187 kt (or 1.98 t p. c.), i.e. 494 kt in overall (5.24 t per c.p.), not including the direct emissions by local households. Consequently, this other way of assessing CO2 emissions gave decent results, only 4% lower than the previous EE-IO model, though difficult to interpret at the industry-specific level.

Sensitivity analysis

The previous estimations of CO2 emissions, resulting from the domestic production for internal final uses, or from CO2 embodied in imported commodities for final or intermediate domestic uses, were based on UE-27 intensity coefficients adjusted for Africa by the IEA coefficients of deviation (CD). Let's assume now that the technology used by Seychelles to produce goods and services is exactly identical to the EU technology, hence emission coefficients by industry equivalent to the EU-27 ones in Appendix Table A2. The sensitivity of the CO2 per capita outcome as a function of the CD varying from 1 (equivalent to EU-27 technology) and 2.47 (or 2.00 for the electricity & gas sector) is shown in Fig. 4. ). Adding this amount to the households' direct emissions of Table 1 would then give a total amount of 5.58 tCO2, in better connection with the overall IEA estimation of 5.33 tCO2 in 2014. Let's now estimate the influence of the CD, industry by industry (Fig. 5).

Fig. 5. Sensitivity of CO2 emissions per capita to the CD by industry

Source: own elaboration with SensIt 1.53 software In Fig. 5, the individual effect of each industry is calculated by using values of the emission coefficient (CO2 per unit of output) between the UE-27 (CD=1) and Africa-adjusted emission coefficients (UE-27 rates multiplied by a CD=2.47 for non-electrical industries or 2.00 for the electricity & gas industry) centred on the mean between the lowest and highest values of the intensity ratio by industry. For instance, the Electricity & gas CO2 coefficient fluctuates between 0.094 and 0.187 with a mean value of 0.140. Consequently the footprint per capita would be found between 2.32 and 3.14 tCO2, other things being equal (i.e. other ratios sticking to their mid-value). Obviously, the energy sector has the most prominent effect on the CO2 footprint, followed by the 'Transportation and storage' and 'Manufacturing, other' activities to a lesser extent. The rest of industries have a much smaller influence on the domestic emissions per capita. The impact of each industry is not only explained by the intensity of emissions per unit of output value, but also by the level of final demand for the various commodities and the technology of the country in terms of inputs to produce one unit of output (Leontief matrix). In addition to the intensive use of fossil fuel-based energy, the transport and energy sectors cumulate a high level of demand from all industries and the institutional end users, with strong backward and forward linkages across the economy. As a result, the footprint level is unsurprisingly linked with these two industries.

Impact of the COVID-19 pandemic

Like many other SIDS, the Republic of Seychelles was particularly affected by the COVID-19 crisis after March 2020, because of trade restrictions and new priorities set for passenger or freight transport [START_REF] Kontovas | Maritime Cargo Prioritisation during a Prolonged Pandemic Lockdown Using an Integrated TOPSIS-Knapsack Technique: A Case Study on Small Island Developing States-The Rodrigues Island[END_REF][START_REF] Rassool | An assessment of the socio-economic impact of Covid-19 in Seychelles, UNDP Seychelles and Government of Seychelles report[END_REF][START_REF] Imf | World Economic Outlook[END_REF][START_REF] Afdb | African Economic Outlook 2021, from debt resolution to growth: the road ahead for Africa, African Development Bank Group[END_REF]. A ban on international travel was implemented by the State with effect from 23 March 2020 until 30 July 2021. Cruise ship visits were also suspended indefinitely since February 2020 and extended until 2021 [START_REF] Rassool | An assessment of the socio-economic impact of Covid-19 in Seychelles, UNDP Seychelles and Government of Seychelles report[END_REF]. The second semester of 2020 was not better than the first one because most foreign countries where tourists originate had implemented a lockdown and restricted travelling abroad. The UNDP report estimated the impact through the decreasing earnings from tourism before and after the pandemic, taking year 2019 as the reference (Rassool et al. 2020, p.12).

We assumed that the structure of exports to non-resident tourists followed the contribution of tourism to GDP published by NBS (average 2014-2019[START_REF] Nbs | Supply and Use Table[END_REF]. Four exporting industries would therefore be impacted: Wholesale and retail trade, Transportation and storage, Administrative and support service activities, and Accommodation and food services. We used the NBS survey on the number of visitors and the CBS survey on the estimated earnings from tourism6 . The number of visitors fell by 70% in 2020 (114,858) compared to 2019 (384,204), with a corresponding 58% decrease of foreign currency revenues (Fig. 6). Assuming that the shock had exclusively affected the level of exports of the four above-mentioned industries, proportionately to their initial value, the new vector of final uses was introduced to simulate the new levels of CO2 emissions. The total exports fell down by 24%, a bit less than the IMF expectation which considered other affected industries. Obviously, the consumption-based CO2 did not change because it was assumed that none of the internal institutions had modified their consumption level. The mere emissions of the domestic output directed to exports declined by 35%. In overall, the CO2 emissions found in the national inventory were reduced by 16%, from 900 kt to 752 kt. As expected, nearly two thirds of the 148 tCO2 reduction of emissions were due to lower transportation, and 30% to the decreasing use of Accommodation and food services industry.

It is important to remind that the external shock concerned only the tourism industry in this simulation, not other sectors affected by the pandemic. However, when looking at the actual figures for 2020, it appears that a few other key industries such as the fish processing activity have been poorly affected by the pandemic, exports even increasing in 2020 both in quantity and value.

Discussion of the results

Why is this CO2 emission assessment useful for Seychelles?

Like many SIDS in the world, the Republic of Seychelles is particularly vulnerable to the effects of global warming and very concerned by the international commitments of reduction taken after the protocol of Kyoto in 1997 and the 2007 Bali Action Plan (UNDP 2008[START_REF] Burns | The changing landscape of climate change: NAMAs, SIDS and tourism[END_REF][START_REF] Nath | Climate Change, Sea Level Dynamics, and Mitigation[END_REF]. Considering the long-term effects of climate change as equivalent to an ENSO (El Niño Southern Oscillation) episode, global warming would cost several percent points of GDP in the long run [START_REF] Robinson | Impacts of climate variability on the tuna economy of Seychelles[END_REF]. Despite a very small contribution to the global carbon dioxide emissions, the country has to find pathways to reduce its ecological footprint by adapting its own patterns of production and consumption. First of all, this is the only means for the country to shift away from fossil fuel energy and transition to a more sustainable economy. Secondly, SIDS depend to a large extent on foreign trade: exports on the one hand for their foreign exchange income, and imports on the other for their intermediate and final consumptions.

However, the comparative advantages of developing countries tend to concentrate locally the carbon emissions while developed countries can specialize in less polluting activities [START_REF] Sannassee | Trade openness and CO2 emission: from a SIDS. In Handbook of environmental and sustainable finance[END_REF]. Trade openness might be wealth-creating and provide foreign currency resources to a SIDS, but often at the expense of higher environmental costs. For example, it was shown econometrically for Mauritius that "a 1% increase in trade openness is accompanied by an upshot of 0.60% in CO2 emission" (Sannassee and Seetanah 2016, p. 175). What is the need of exporting processed food products if the spillover effects are few for the country because of foreign shareholders, profits transferred to offshore companies, massive imports of inputs, and employment of foreign workers spending their income overseas? Why should final consumption of domestic households be based on imported processed products rather than local production? Even the steady growth of tourism, pillar of economic development for the Seychelles, is questionable in terms of sustainability (Pratt 2015). The increasing number of visitors (+120% between 2010 and 2019, i.e. an annual growth rate of +9%) results in greater air traffic and emissions, deforestation to build new resorts and accommodation capacity, privatization of land and restricted access to beaches for the local population, foreign investment resulting in outflows of capital income, imports of many intermediate and final goods, more income inequalities, etc. The assessment of carbon emissions may send the right signals to re-direct the country's international specialization [START_REF] Sannassee | Trade openness and CO2 emission: from a SIDS. In Handbook of environmental and sustainable finance[END_REF][START_REF] Roth | Blue Bonds: financing resilience of coastal ecosystems, Report prepared for IUCN and the Swedish Government by 4Climate[END_REF], Republic of Seychelles 2018).

4.2 Which methodology for a replicable carbon assessment on a routine basis?

Two strands of methodology are usually distinguished to calculate CO2 emissions from human activities. The first one is of microeconomic nature, with detailed information on household consumption and emission factors by type of commodity [START_REF] Pasquier | Méthodologie de calcul de l'empreinte carbone de la demande finale intérieure française[END_REF]). The second one is more macroeconomic and combines inter-industrial data with national accounts on air pollutant emissions (Suh 2009[START_REF] Hertwich | Carbon footprint of nations: A global, trade-linked analysis[END_REF][START_REF] Boden | Global, Regional, and National Fossil-Fuel CO2 Emissions. Carbon Dioxide Information Analysis Center[END_REF]. The latter are rather based on a multi-regional framework, but can also be developed in a single-country framework [START_REF] Miller | Input-Output Analysis, Foundations and extensions[END_REF][START_REF] Pasquier | Méthodologie de calcul de l'empreinte carbone de la demande finale intérieure française[END_REF], Boutaud and Gondrand 2018). The difference of methodology and the level of details may explain the difference between the two types of approaches. For instance, the World Bank, using the CDIAC approach based on the multiregional GTAP model, found a CO2 footprint per capita of 6.32 tonnes for Seychelles in 2016 (data.worldbank.org), quite different from the one estimated the same year by the IEA with 5.43 tonnes (worldometers.com). Moreover, the first source provides a time series of CO2 emissions with a very irregular pattern since 1971, following the booms and busts of the world economy, whereas the second one shows a more regular and increasing trend (see Fig. 1). There is no absolute truth when it comes to the estimation of CO2 emissions, but the least is to know how the calculus is made in order to simulate the effects resulting from major shocks affecting the economy, such as the Covid-19 pandemic for instance, or trade policy changes, and to reduce the overall level of air emissions.

In that respect, through the present contribution, we suggest to apply a widespread and transparent EE-IO methodology to the CO2 of Seychelles to serve as a reference basis for the coming years. Because there is no national account of air pollutant emissions yet, we used the European Union air emission accounts database, which has the twofold advantage of being published online on a yearly basis and to cover the wide variety of 27 EU member states across different levels of industrial development and technology. Moreover, the EU is a major trade partner of the country, weighting one third of imports and nearly 80% of merchandise exports, meaning that the technology structure materialized by the Leontief matrix and the air emission intensity could be fairly similar because the same environmental quality standards must be fulfilled by both partners.

Details about the CO2 emissions by origin and industry

The results show a carbon dioxide footprint of 513 ktCO2, not including the direct emissions of households' vehicles and air conditioning personal systems for which we have no specific data yet (640 kt otherwise). This would represent an average amount of 5.44 tCO2 per capita (6.79 tCO2, resp. -Fig. 7a). These amounts are quite close to the IEA estimation of 502 kt (i.e. 5.33 tCO2 p.c.), showing the reliability of the estimated outcome. They are also well above the worldwide average figure of 1.35 tCO2 p.c., a difference which is consistent with the economic status of Seychelles as high income country, with a GDP per capita of US$ 14,700 in 2014 (IMF 2021). However the national inventory, which includes the emission content of exports and excludes that embodied in imports, reaches 900 ktCO2 (i.e. 9.55 tCO2 p.c., Fig. 7b), exceeding the domestic footprint. As a comparison, the respective levels of national footprint and inventory in France are just opposite, showing a footprint greater than the inventory: 10.6 vs 7.3 tCO2 p.c., respectively [START_REF] Boutaud | L'empreinte écologique[END_REF]. Proportionately, France exports relatively less and imports relatively more industrial commodities than Seychelles. Another interest of the study is to disentangle the content of emissions by origin, either from imports or domestic production. This is important to show the gap between a productionbased national inventory of CO2 emissions and a consumption-based footprint where the responsibility of the country extends to the foreign nations where the commodities are produced [START_REF] Wiebe | Estimating CO2 Emissions Embodied in Final Demand and Trade Using the OECD ICIO 2015: Methodology and Results[END_REF][START_REF] Boden | Global, Regional, and National Fossil-Fuel CO2 Emissions. Carbon Dioxide Information Analysis Center[END_REF][START_REF] Wu | Carbon emissions embodied in the global supply chain: Intermediate and final trade imbalances[END_REF]. Fig. 7b shows a great deal of carbon dioxide emissions related to exports, explaining that the level of emissions of the national inventory (900 ktCO2, production-based) exceeds by far the footprint level (640 ktCO2, consumption-based). The country imports raw materials (e.g. frozen tuna) that are further processed to be re-exported as final goods (canned tuna

Conclusion: towards a CO2 abatement policy

On the basis of this first CO2 assessment, the government can take actions to reduce its footprint, mostly by substituting the fossil fuel origin of power used by the citizens and the local industries. It has started to do so, although at a slow pace, with an investment in renewable energy over the past decade. A 6 MW wind farm along with a 5.8 MW grid-tied roof top photovoltaic system supplied nearly 6% of total electricity produced in 2020, the rest being provided by fossil fuels, mostly heavy fuel oil (PUC 2021). The use of electric cars would be of little effect on direct household emissions as long as the electricity is carbon-based. A significant proportion of emissions related to the domestic supply also comes from imports, through the transportation of commodities (e.g. food and beverage products) and the emissions embodied in foreign production. One type of abatement would stimulate the primary sectors of agriculture and fisheries, or promote other domestic sources of supply such as aquaculture to cover the local food of the population. Once again, the government has launched several actions to promote the blue economy, developing coastal fisheries and aquaculture, or investing in blue bonds and marine renewable energy (Rep. of Seychelles 2018).

Another issue to deal with concerns the important content of emissions related to exports, whether they concern the tourism or the fish processing industries. We have shown that the national inventory (i.e. domestic emissions including the demand for exports) is greater than the national footprint. This is the case of many open economies like SIDS attempting to catch foreign exchange revenues, thus creating a livelihood for the population. The Republic of Seychelles has been quite successful over the past decades to create an endogenous development of its economy through tourism (accommodation and food services, air traffic, support services such as tour operators, banking, arts and entertainment activities, etc.). However, the environmental cost could be important for Seychelles because this highstandard activity relies to a great extent on imports and foreign investment, and the 2020 pandemic crisis has also demonstrated the vulnerability of islanders with the steady decline (-70% in 2020 / 2019) in the number of visitors [START_REF] Rassool | An assessment of the socio-economic impact of Covid-19 in Seychelles, UNDP Seychelles and Government of Seychelles report[END_REF]. In this research, we have attempted to estimate the post-Covid impact on CO2 emissions through a simulation of a 58% decrease of expenditure from tourism that has been observed by the Central Bank of Seychelles in 2020. This would be equivalent to a 24% reduction of total exports of goods and services for the country, resulting in a 16% reduction of CO2 emissions (Fig. 7c). Two thirds of this reduction of emissions would be obtained by the shrinking use of transportation services, and one third by the decreasing foreign demand for accommodation and food services.

This first attempt to estimate the carbon footprint of Seychelles by using a standard EE-IO approach has also several limitations, such as the lack of air emission data monitored by the country, or the incomplete input-output table with regard to the distinction between domestic and imported intermediate consumptions. Many efforts are under way to improve the data collection scheme, and a first Supply and Use Table is now available and can be used to develop IO-based modelling and planning models (NBS 2021). Further endeavour still needs to be made prior to achieve a bigger MRIO framework, a social accounting matrix, a material flow analysis or a computable general equilibrium model, in order to trace more accurately the sinks and sources of carbon emissions, to assess other types of ecological footprints (e.g. plastics) and find effective pathways of reducing outflows. At least this first estimation will serve as a transparent and replicable basis to follow the evolution of the carbon footprint and develop some mitigating projects for the country in a near future.

  3 . The results for Seychelles are presented in the Figure below (Fig.1).

Fig. 1 -

 1 Fig. 1 -CO2 emissions from fuel combustion in Seychelles Source: Own elaboration from IEA and EDGAR data released by www.worldometers.info (total emissions in thousand tonnes, blue solid line & left axis, and in tonnes per capita, dashed red line & right axis)

Fig. 2 -

 2 Fig. 2 -CO2 emissions resulting from domestic demand by industry and direct combustion. Direct_HH = Direct emissions of households from gas combustion, Domestic_output = emissions resulting from the domestic output process to meet the internal demand, import_FD and import_IC = imports of commodities and services for the final and intermediate domestic demand, respectively.

  Administrative & support services, Transportation & storage, and the Manufacturing of other food products). As far as the emissions from imported inputs (intermediate consumption) are concerned, the tonnes of emitted CO2 stem from a broader variety of industries (Construction, Public administration & defence, Administrative & support services, Wholesale & retail trade, etc.).

Fig. 3 -

 3 Fig. 3 -CO2 emissions resulting from Seychelles exports by industry. CO2_output_to_X = CO2 emissions resulting from domestic output to meet export demand;_impIC_to_X = emissions linked to imported intermediate consumption used for exports; impF_to_X = emissions linked to re-exports of imported final goods and services.

Fig. 4 .

 4 Fig. 4. Emissions in tCO2 per capita as a function of the coefficient of deviation (CD) Source: own elaboration with R software (R Core Team, 2021)

Figure 6 .

 6 Figure 6. Number of visitors (solid blue and left axis) and estimated tourism expenditure (Million SCR -dashed red and right axis) per month in Seychelles (January 2014 -August 2021) Source: NBS (number of visitors) & Central Bank of Seychelles (CBS, expenditure)

Fig. 7 .

 7 Fig. 7. CO2 footprint and inventory in the Seychelles (ktCO2) Source and notes: own elaboration. DHE = Direct household emissions, RTD = Emissions related to domestic production for domestic use, RTE = Emissions related to exports, EiM = Emissions embodied in imports. The emissions related to export (green layer) do not include the import content of production: only the territorial emissions matter in the national inventory. The Seychelles population was 94,215 inhabitants in 2014 (NBS 2021).

  application of the EE-IO model requires different types of data: input-output technology, CO2 emissions by industry, final uses... All data are not available yet for the Republic of Seychelles. However, the National Bureau of Statistics (NBS) has published for the first time in 2020 a Supply and Use Table for the year 2014, with a selection of 23 key industries for the domestic economy (NBS 2021). On that basis, an input-output table was constructed but with no distinction between domestic and imported inputs. Consequently, we had to estimate first the import (interindustry) matrix (𝐴 ) by allocating the vector of imports M proportionately to the distribution of intermediate and final output (see

  : Note: * the direct emissions per capita were extrapolated from[START_REF] Boden | Global, Regional, and National Fossil-Fuel CO2 Emissions. Carbon Dioxide Information Analysis Center[END_REF]. The size of the Seychelles population was 94,215 inhabitants in 2014 (NBS 2021).

	Table 1. CO2 footprint per capita in the Seychelles in 2014
		ktCO2	tCO2 per
			capita
	Direct emissions of households*	127	1.35
	Emissions resulting from domestic output	346	3.67
	Emissions embodied in imports for final use	85	0.90
	Emissions embodied in imports for	82	0.87
	intermediate use
	TOTAL CO2 footprint (excl. exports)	640	6.79

Table A2 .

 A2 The Seychelles CO2 emission coefficients for the 23 industries in 2014 (in ktCO2 per million SCR of output) Note: The EU-27 are the intensity coefficients of CO2 emissions per unit of output value, expressed in ktCO2 per million Euro (from Eurostat NAMEA data, https://ec.europa.eu/eurostat/) and converted in ktCO2 per million of Seychelles rupee with an exchange rate of 16.96 (CBS rate in 2014). The Seychelles emission rate is the EU-27 coefficient adjusted for Africa with regard to the IEA deviation coefficients of 2.47 for non-electrical industries and 2.00 for the electricity sector.

		𝑒 EU-27	𝑒 Seychelles
	Agriculture	0,0135	0,0335
	Fishing	0,0323	0,0798
	Manufacture of fishery products	0,0031	0,0075
	Manufacture of other food	0,0031	0,0075
	Manufacture of beverage and tobacco	0,0031	0,0075
	Manufacturing, other	0,0110	0,0272
	Electricity, gas, steam and air	0,0936	0,1870
	conditioning supply
	Water supply; sewerage, waste	0,0067	0,0164
	management
	Construction	0,0018	0,0045
	Wholesale & retail trade; repair of motor	0,0016	0,0040
	vehicles
	Transportation and storage	0,0325	0,0803
	Accommodation and food service	0,0014	0,0035
	activities
	Information and communication	0,0005	0,0013
	Financial and insurance activities	0,0003	0,0007
	Real estate activities	0,0002	0,0004
	Owner occupied dwellings	0,0000	0,0000
	Professional, scientific and technical	0,0006	0,0015
	activities
	Administrative and support service	0,0009	0,0022
	activities
	Public administration and defence	0,0014	0,0035
	Education	0,0013	0,0031
	Human health and social work activities	0,0011	0,0028
	Arts, entertainment and recreation	0,0013	0,0033
	Other service activities	0,0012	0,0029

The CDIAC is a research centre of the US Department of Energy calculating the carbon dioxide footprint through the multi-regional input-output model GTAP (Global Trade and Analysis Project). Their estimations can be found on the World Bank website(Hertwich and Peters

2009).

https://www.worldometers.info/co2-emissions/seychelles-co2-emissions/

Emission Database for Global Atmospheric Research EDGARv6.0_GHG websites: https://edgar.jrc.ec.europa.eu/dataset_ghg60; https://data.europa.eu/doi/10.2904/JRC_DATASET_EDGAR; More information about the methodology can be found in[START_REF] Crippa | Fossil CO2 and GHG emissions of all world countries[END_REF]

See the Eurostat portal https://ec.europa.eu/eurostat/, including the Air emissions accounts (AEA) by NACE Rev. 2 activity.

Available from the Eurostat portal: https://ec.europa.eu/eurostat/, file of air emission accounts for

industries + direct emissions [env_ac_ainah_r2]. The coefficient for Africa in 2014 was 2.47 for non-electrical industries, and 2.00 for the electricity industry.

On a monthly basis, the Central Bank of Seychelles collects accurate and comprehensive information about transactions captured from credit card systems in hotels, online bookings, restaurants, tourism-related transactions, destination management companies, ferries, etc.

Acknowledgement

The authors are grateful to Manuel Baude, Service des données et études statistiques (SDES), French Ministry of Ecological Transition / CGDD, for his support regarding the methodology of carbon footprint assessment, Prof. Paul Smith (University of Southampton, UK) for his sound advice, two anonymous reviewers for their very helpful comments and to Laurent SAM (PUC, Seychelles) for his helpful contribution. PG acknowledges the financial support of the French National Research Agency program CIGOEF (ANR-17-CE32-0008).

APPENDICES


Tuffin Clts

Rqmc Kaust

Review: Monte Carlo (MC) MC: random sampling to estimate µ = E[ h(U) ]

How to choose RQMC Allocation (m n , r n ) with m n × r n ≈ n ? AVCI relies on CLT: not established for many RQMC settings.

Nested scrambling of digital nets: CLT as

Main Issue: How to choose c? More general allocation (m n , r n ):

Assumption 2. σ 2 mn ≡ Var[X n,1 ] > 0 for all n large enough.

Results based on CLT for triangular arrays. For a triabgular array, the r n variables within each row n are independent, but variables in different rows may be dependent. Here:

X n,1 , X n,2 , . . . , X n,rn are i.i.d., each with some distribution F n .

Review: MC -Error Estimation Easy, But Slow Convergence MC estimator:

µ MC n = 1 n n i=1 h(U i ) CLT: If ψ 2 ≡ Var[ h(U) ] ∈ (0, ∞), then [Billingsley 1995] n ψ 2 µ MC n -µ ⇒ N (0, 1) as n → ∞
Approximate 100γ% confidence interval (CI) for µ:

I MC n,γ ≡ µ MC n ± z γ ψ n √ n ψ 2 n = 1 n-1 i=1 h(U i ) -µ MC n 2 and Φ(z γ ) = 1 -(1 -γ)/2.
Asymptotically valid CI (AVCI):

P( µ ∈ I MC n,γ ) → γ,
µ Q n = 1 n n i=1 h(ξ i ) Low-discrepancy sequence Ξ = (ξ i : i = 1, 2, . . .)
Ξ is deterministic and evenly fill [0, 1] s lattices (e.g., Korobov, . . . ),

Review: QMC -Fast Convergence QMC: deterministic points to estimate µ = E[ h(U) ] µ Q n = 1 n n i=1 h(ξ i ), Ξ = (ξ i : i = 1, 2, . . .) Koksma-Hlawka (K-H) inequality [Niederreiter 1992]: for each n > 1, | µ Q n -µ | ≤ V HK (h) D * n (Ξ) Hardy-Krause variation V HK (h) ∈ [0, ∞]: "roughness" of h Star-discrepancy D * n (Ξ) ∈ [0, 1]: how unevenly first n points of Ξ fill [0, 1] s D * n (Ξ) = O n -1 (ln n) s ≈ O n -1 , n → ∞. If V HK (h) < ∞ (BVHK), then K-H bound shrinks at faster rate than MC rate Θ(n -1/2 ) | µ Q n -µ | ≈ O n -1 .
BVHK: "bounded variation in sense of Hardy and Krause"

Review: QMC -But Error Estimation Difficult Two components: V HK (h) and D * n (Ξ). Variation Difficult to compute: sum of 2 s -1 variations in the sense of Vitali (each difficult to estimate). V HK (h) = ∞ often, even for simple functions such as

h(x 1 , x 2 , x 3 ) = min(x 1 + x 2 + x 3 , 1)
for which QMC converges (faster than for MC).

Discrepancy Expression

D * n (Ξ) = sup x∈[0,1] s n i=1 1 s k=1 [0,x k ] (ξ i ) n - s k=1 x k .
We only have bounds with asymptotic usefulness, not for practice use. Ex: bounds for the (0, s)-Niederreiter sequence (while we know D * n (Ξ) < 1!) Review: Randomized Quasi-Monte Carlo (RQMC)

RQMC computation budget of n evaluations of h (as for MC)

allocation (m n , r n ) with m n × r n ≈ n r n = # i.i.d. randomizations m n = # points used from jth randomized sequence Ξ j = (U i,j : i ≥ 1), j = 1, 2, . . . , r n RQMC: r n ≥ 2 i.i.d. randomizations to estimate µ = E[ h(U) ] µ RQ mn,rn = 1 r n rn j=1 X n,j , where X n,j = 1 m n mn i=1 h(U i,j ) X n,1 , X n,2 , . . . , X n,rn i.i.d.: estimate σ 2 mn ≡ Var[X n,1 ] by σ 2 mn,rn = 1 r n -1 rn j=1 X n,j -µ RQ mn,rn 2 .
Approx γ-level CI for µ

I RQ mn,rn,γ ≡ µ RQ mn,rn ± z γ σ mn,rn √ r n X n,1 , X n,2 , . . . ,
E (X n,1 -µ) 2 ; |X n,1 -µ| > t r n σ 2 mn E (X n,1 -µ) 2 → 0, as n → ∞, ∀t > 0;
or Lyapounov condition:

E |X n,1 -µ| 2+b r b /2 n σ 2+b mn → 0, as n → ∞, for some b > 0. σ 2 mn = E[ (X n,1 -µ) 2 ] = variance of estimator X n,1 from single randomization of m n points.

RQMC Asymptotically Valid CI (AVCI)

Recall Lyapounov condition:

E |X n,1 -µ| 2+b r b /2 n σ 2+b mn → 0, as n → ∞, for some b > 0. σ 2 mn,rn = 1 rn-1 rn j=1 X n,j -µ RQ mn,rn 2 is unbiased estimator of σ 2 mn = Var[X n,1 ]. Approx. γ-level CI for µ I RQ mn,rn,γ = µ RQ mn,rn ± z γ σ mn,rn √ r n Theorem 2
If Assumptions 1 and 2 hold, along with Lyapounov condition for b = 2, then CLT

r n σ 2 mn,rn µ RQ mn,rn -µ ⇒ N (0, 1), as n → ∞
and AVCI P(µ ∈ I RQ mn,rn,γ ) → γ, as n → ∞.

Ensuring CLT (with σ 2 m > 0) Instead of condition (1), impose alternative conditions on integrand h Assumption 3.A:

V HK (h) < ∞ (BVHK) Assumption 3.B: h is bounded Assumption 3.C: E[ |h(U) -µ| 2+b ] < ∞ for some b > 0, where U ∼ U[0, 1] s . Proposition 1 Under Assumption 3.A, if ∃λ ∈ (0, 1) r 1-λ n mnσm n (ln mn) s 2 → ∞ as n → ∞, Under Assumption 3.B, if s 2 n = r n σ 2 mn → ∞ as n → ∞, Under Assumption 3.C, if r b/(2+b) n σ 2 mn → ∞ asn → ∞,
E |X m,1 -µ| 2+b σ 2+b m ≤ k 1 with b = 2,

then AVCI holds

Corollaries Ensuring CLT or AVCI under Assumption 1.B

Assumption

For estimator X n,1 from single randomization of m n points,

σ mn ≡ Var[ X n,1 ] ≈ Θ(m -α * n ) as m n → ∞, where α * ≡ -lim mn→∞ ln(σ mn ) ln(m n ) > α * ≥ 1 when V HK (h) < ∞ (BVHK). Under Assumption 1.B [ (m n , r n ) = (n c , n 1-c ), c ∈ (0, 1) ], RMSE µ RQ mn,rn = σ mn √ r n ≈ Θ n -v (α * ,c) as n → ∞, with v (α * , c) ≡ c α * -1 2 + .
Corollaries

  as n → ∞ Root mean-squared error: RMSE µ MC n = ψ √ n Review: Quasi-Monte Carlo (QMC) QMC: deterministic points to estimate µ = E[ h(U) ]

  i.i.d. randomizations of Ξ = (ξ i : i ≥ 1), each yielding Ξ = (U i : i ≥ 1)Each U i ∼ U[0, 1] s Ξ retainslow-discrepancy properties of Ξ Lattice: random shift [Cranley & Patterson 1976] Digital net: nested scrambling [Owen 1995], digital shift [L'Ecuyer & Lemieux 2002], . . .

  Assumption 3.A =⇒ 3.B =⇒ 3.C, leading reversely strict conditions in the above proposition.Ensuring AVCI (with σ 2 m > 0)Instead of condition (1), impose alternative conditions on integrand h Assumption 3.A:V HK (h) < ∞ (BVHK) Assumption 3.B: h is bounded Assumption 3.C: E[ |h(U) -µ| 2+b ] < ∞ for some b > 0, where U ∼ U[0, 1] s .

  =⇒ 3.B =⇒ 3.C Assumption 3.A: VHK(h) < ∞ (BVHK: =⇒ α * ≥ 1) Assumption 3.B: h is bounded. Assumption 3.C: E[ |h(U) -µ| 2+b ] < ∞ for some b > 0, where U ∼ U [0, 1] s . Comparisons for fixed α * > 1/2 (mn, rn) = (n c , n 1-c ), c < c k (α * ), opt RMSE ≈ Θ(n -v k (α * ) ).

  How to choose RQMC Allocation (m n , r n ) with m n × r n ≈ n ? Heuristic: For given budget n, choose r n small and m n ≈ n/r n large to exploit QMC.

	CI: I RQ mn,rn,γ ≡ µ RQ mn,rn ± z γ r n = # i.i.d. randomizations m n = # points used from each randomized sequence σm n ,rn √ rn But heuristic lacks rigorous justification. Nakayama & Tuffin CLTs for RQMC Theorem 1 If Assumptions 1 and 2 hold, then RQMC estimator µ RQ mn,rn satisfies CLT r n σ 2 mn µ RQ mn,rn -µ ⇒ N (0, 1), as n → ∞ under either KAUST 2022 RQMC CLT Lindeberg condition:

X n,rn i.i.d., but distn of each X n,j depends on n: Triangular array.

  CLT or AVCI: Tradeoffs Cor. k Ensures Assumption on h c upper bd c k (α * ) RMSE rate exp v k (α * )

	2	CLT	3.A (BVHK)	1 2α * -1	>	1	>
	3	CLT	3.B (h bdd)	1 2α * +1

Nakayama & TuffinCLTs for RQMC KAUST 2022

Question and Goal

Is this heuristic theoretically valid?

Goal: Obtain sufficient conditions to ensure CLT and AVCI (as n ≈ m n × r n → ∞).

Nakayama & Tuffin CLTs for RQMC KAUST 2022 12 / 23

Corollaries Ensuring CLT or AVCI under Assumption 1.B

Assumption

For estimator X n,1 from single randomization of m n points,

=⇒ RQMC better than MC.

Nakayama & Tuffin CLTs for RQMC KAUST 2022

Corollaries Ensuring CLT or AVCI Corollary 1

Suppose that Assumptions 1 and 2 hold, and ∃ b > 0 and k 1 ∈ (0, ∞) such that

(1)

and optimal RMSE ≈ Θ(n

If

Assumption 1: (mn, rn) = (n c , n 1-c ), c ∈ (0, 1)

Conditions Ensuring CLT or AVCI: Tradeoffs

Concluding Remarks

Sufficient conditions for RQMC CLT and AVCI for µ