Pasquale Sellitto
email: psellitto@lmd.ens.fr

Giuseppe Salerno

Pierre Briole

The EtnaPlumeLab (EPL) research cluster: advance the understanding of Mt. Etna plume, from source characterisation to downwind impacts

may come     The EtnaPlumeLab (EPL) research cluster: advance the understanding of Mt. Etna plume, from source characterisation to downwind impact

I. Introduction

M

ount Etna is a persistent source of gas and aerosols in the lower and upper Mediterranean troposphere. Its sulphur emission has been estimated to be about 10 times larger than anthropogenic emissions in the same area [Graf et al., 1997]. Recently, it has been shown that Mt. Etna has the potential to modulate the downwind aerosol properties, e.g. by the production of highlyreflective secondary sulphate aerosols from the conversion of sulphur dioxide emission [Villani et al., 2006, Sellitto et al., 2016]. In its turn, this has revealed the potential to exert a significant direct regional climatic forcing, depending on the environmental conditions and the optical properties of the produced aerosols [Sellitto andBriole, 2015, Sellitto et al., 2016].

Though Mt.

Etna is one of the most studied volcanoes in the world, e.g. [Bonaccorso et al., 2004], its atmospheric impact, from small-range air quality perturbation to far-range climatic forcing is not yet well understood, or even, its influence might be potentially underestimated. This is due to the multidisciplinary ancestry of this argument, which require a synoptic investigation framework from inner-volcanology to atmospheric sciences and climatology. This integrated framework would allow to better characterise the volcanic degassing and to refine and tune the atmospheric chemistry/physics and climate sciences modelling, to better understand both tropospheric and stratospheric downwind impacts. Based on this undeniable objects, in 2013 a multidisciplinary research cluster named EtnaPlumeLab (EPL) was kicked-off. This cluster results from the combination of three different interacting projects.

Here, we will detail the scopes, rationale and perspectives of EPL. The paper is structured as follows: Sect. II reports the general structure of EPL, in Sect. III the three different modules that constitute the EPL cluster are detailed and finally in Sect. IV conclusions and perspectives are given.

II. A multidisciplinary and multiscale modular approach for the characterisation of volcanic plumes

The EPL research cluster aims at a better understanding of Mount Etna's gas and particulate emissions, their generation, evolution and impacts. The target area is the Mediterranean basin. The cluster arises from the interacting combination of three different modules/projects:

• EPL-RADIO (Radioactive Aerosols and other source parameters for better atmospheric Dispersion and Impact estima-tiOns);

• SMED (Sulfur MEditerranean Dispersion);

• Med-SuV (MEDiterranean SUpersite Volcanoes) WP (work package) 5.

Figure 1 shows the spatial extent and the volcanic and atmospheric processes tackled by the three EPL modules, as well as their interactions. The EPL-RADIO project is specifically devoted to source characterisation, in terms of geochemistry and degassing processes, and the near-source plume characterisation. The SMED project investigates the dispersion of the SO 2 emissions in the Central Mediterranean region, whereas the Med-SUV WP5 project, though its multipack volcanology scope, also promotes the study of the impact of Mt. Etna's emissions at a broader regional scale. The SMED and the Med-SuV WP5 projects superpose to a certain geographical extent. Figure 2 shows the structure of the EPL research cluster into the three modules. The modules are interacting and knowledge gathered from one module (at smaller spatial scale) is fed to the following module (at larger spatial scale). For instance, EPL-RADIO has the potential to provide input source parameters for the dispersion modelling tackled by SMED; on the other hand SMED has the potential to give plume's parameters such as to define optical downwind aerosols properties essential for the climatic impact estimation explored by Med-SuV WP5. The different scientific targets and the methods employed in each of the three modules (e.g., modelling and observations) are also indicated and color-coded. The timeframes of the modules are also shown. The broad split into (a) source -EPL-RADIO , (b) evolution -SMED, and (c) impact -Med-SuV WP5, of Mt. Etna emissions follows from Fig. 1. The EPL research cluster relies on high quality direct and remote ground-and satellitebased observations of volcanic plume, together with chemistry, transport and radiative transfer modelling. While for the SMED and Med-SuV the employ of both observations and modelling is needed, EPL-RADIO is only based on ground-and satellite-based measurements for characterising the aerosol source (including gaseous precursors of secondary aerosols). The Med-SuV project ended in December 2015, but results are still available for further investigation. The SMED project runs from November 2015 to end 2016. The EPL-RADIO activities are foreseen until summer 2017.

Troposphere

III. The different modules

III.1 EPL-RADIO

The EPL-RADIO project aims at improving the characterisation of Mt. Etna as atmospheric aerosols source, gathering multi-domain (solid Earth and atmospheric sciences) information [Sellitto, 2016]. Object of EPL-Radio is to advance our knowledge on Mt. Etna's emissions, targeting emission processes, from inner degassing mechanisms to aerosol near-source characterisation. This project brings together different and original information, such as observations of (a) radioactive disequilibria of radon daughters, (b) coupled direct/remote size-resolved aerosols distribution, (c) primary/secondary near-source sulfate aerosols partitioning. This volcanic aerosols decription allows a detailed size-resolved physicochemical characterisation of the emitted aerosols. The point (a) is more thoroughly discussed by Terray et al. (2016).

The volcanic aerosol source is characterised by determining the size-resolved aerosol emissions, with a cascade impactor and a Microtops-II optical spectrometers (see, e.g. [Mather et al., 2004]), and the primary fraction of the emitted sulfate aerosols (with respect to the secondary sulfate aerosols produced by in-plume conversion of SO 2 emissions) by FTIR (Fourier Transform Infrared) spectroscopy [La Spina et al., 2010]. Complementary three-dimensional aerosols information is obtained with a scanning AMPLE Li-DAR (Aerosol Multi-wavelength Polarization Lidar Experiment -Light Detection And Ranging) system [Scollo et al., 2015]. These parameters are crucial to assess the size-dependent plume dispersion and downwind climatic impacts, and are then an input knowledge for the SMED and Med-SuV WP5 modules.

The EPL-RADIO project is linked to an extended transnational access at Mt. Etna's summit craters area, partially funded by the European Union's Horizon 2020 research and innovation programme (ENVRI+ TNA). Two dedicated campaigns have been scheduled in 2016 (C1: aerosol sampling at the summit craters, C2: distal observations in an area of about 30 km from the craters) and 2 in 2017 (C3 and C4). Highest priority is given to C1 and C3, i.e. the field experiments at the summit craters.

First, preliminary results of campaigns C1 and C2 have shown how near-source processes (in the first few km downwind the emitting craters) play a major role in determining the volcanic aerosol properties downwind, thus influencing the dispersion and impacts of the plume.

III.2 SMED

The SMED project [Salerno et al., 2016] aims at assessing: (a) the spatial distribution of the regional atmospheric dispersion of Mt. Etna's volcanic emissions, (b) the impact of moderate volcanic activity and long-lived passive degassing, and (c) lower to upper tropospheric sulphate aerosol formation processes and lifecycle, in the regional area of Central-Eastern sector of the Mediterranean basin.

While the project is mostly target to the Central-Eastern Mediterranean, where up to 80% of the Etnean plume disperses [Sellitto et al., 2017], the less downwindventilated Central-Southern Mediterranean region is also explored [Sellitto et al., 2016].

The study is pursued by a comprehensive set of remote ground-and satellite-based measurements, such as: (a) the INGV -Osservatorio Etneo FLAME (FLux Automatic MEasurements) ultraviolet scanning spectrom-eter network [Salerno et al., 2009] and FTIR [La Spina et al., 2010] The AERONET stations routinely measure column optical and microphysical properties of the aerosol layer, which are used to assess the downwind impact of Mt. Etna's sulphur emissions. Additionally, the Lampedusa station is equipped with complementary instrumentation, including a LiDAR which provides height resolved aerosol and clouds information. These data allow the characterization of aerosol layer and their precursors, which is fundamental for identifying downwind volcanic contribution. Ground and space observations are complemented with an atmospheric Lagrangian model (FLEXPART -FLEXible PARTicle dispersion model, https://flexpart.eu/) for inspecting the transport and dispersion of volcanic plumes over the Mediterranean region.

First results have allowed the characterization of the Etnean plumes dispersion at the decadal timescale, thus quantifying the frequency of ventilation of different mediterranean sub-sectors and confirming the prevalence (about 80% frequency) of the dispersion of the plumes towards the eastern quadrant (i.e., towards Greece) [Sellitto et al., 2017].

III.3 Med-SuV WP5

The European Union's 7th Framework Program Med-SuV project aims at improving the assessment capacity of volcanic hazards in selected supersites (i.e., well instrumented sites) of Southern Italy. The main targets of this project are the optimisation and integration of existing and new observation/monitoring systems, the systematic study of volcanic pro-cesses and the enhancement of the coordination between the scientific and end-user communities.

Work package (WP) 5 of Med-SuV is devoted to improve the knowledge of processes occurring at Mt. Etna supersite. The primary aim of Med-SuV WP5 is the study of inner volcanology processes. Nevertheless, the degassing, transport and atmospheric impacts of Mt. Etna are also addressed. One specific target of Med-SuV WP5 is to extend the scopes of the Vamos Seguro (Volcanic Ash Monitoring and fOrecaSting between Sicilia and malta arEa and sharinG of the resUlts foR aviatiOn safety) project (http://www.vamosseguro.eu/) and to study the downwind impacts of Mount Etna's ash and SO 2 emissions in the central Mediterranean. In such a sense, Med-SuV WP5 extends geographically the scopes of SMED (see Fig. 1), allowing to investigate the regional impacts of volcanic emissions in the Central Mediterranean.

These interacting studies have shown, for the first time, that even relatively weak eruptions have the potential to modulate the optical and micro-physical properties of the aerosol layer at hundreds of km from Mt. Etna, thus exerting a significant, episodic radiative forcing [Sellitto et al., 2016], which is critically dependent from chemical/micro-physical processes in atmosphere [START_REF] Sellitto | On the radiative forcing of volcanic plumes: modelling the impact of Mount Etna in the Mediterranean[END_REF]. The radiative impact is studied with both radiative transfer modelling and satellite observations of radiative fluxes. Longer-term analyses have shown that Mt. Etna's sulphur emissions can have a significant decadal impact on aerosol optical properties even in weakly ventilated subsectors, like observed at Lampedusa (southwestern quadrant) [Sellitto et al., 2017].

IV. Conclusion and Perspectives

Emissions of volcanic gases and particles profoundly impact on terrestrial environment, atmospheric composition and climate, at various temporal and spatial scales. Early research results suggested that the regional impact of Mt. Etna in the Mediterranean might be poten-tially underestimated. A novel and vigorous multidisciplinary approach is then required to explore this subject. Hence, in 2013, the EPL multidisciplinary and multiscale research cluster has been established, aiming at a better characterisation of Mt. Etna's emissions, from source to downwind impacts.

First results have shown pioneering evidences of Mt. Etna's impacts in the Mediterranean, in terms of the modulation of the aerosol optical properties and of the potential direct radiative forcing at selected locations. The near-craters and near-source distal EPL-RADIO campaigns, foreseen for the summers 2016 and 2017, will provide further insights into the degassing and emission processes, and near-source plume characterisation. This will allow to refine the regional dispersion modelling and help the interpretation of satellite data at the regional scale. On the other hand, the SMED activities will provide an overall picture of the dispersion and regional distribution of volcanic effluents in the Mediterranean, that will give better direct climatic forcing estimations.

Further important topics, not addressed at present by EPL are the modelling and observation of the chemical and micro-physical evolution of the plume, the role of volcanic aerosol in the formation, determination of optical properties and climatic forcing of low and high clouds and, at a smaller spatial scale, the impact of volcanic emissions on local and regional air quality (see Fig. 2). Further research is ongoing to include these aspects in the research cluster.

Figure 1 :

 1 Figure 1: Spatial extents and interactions of the three modules establishing the EPL reserch cluster.

Figure 2 :

 2 Figure 2: Breakdown of the EPL research cluster, with indication of scientific targets, methods and timeframes of the three individual modules of the cluster.

Acknowledgements

All participants of the EPL-RADIO, SMED and MedSuV WP5 projets are gratefully acknowledged, especially: T. Caltabiano, E. Carboni, S. Corradini, A. di Sarra, P.-J. Gauthier, A. La Spina, B. Legras, L. Merucci, G. Puglisi, S. Scollo, L. Terray.

* This work has been partially supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No 654182 (ENVRI+), the European Union's 7th Framework Program under the grant No 308665 (Med-SuV) and the INGV under the grant SMED.


Jean-Baptiste Bonnet

Ilan Szwarc3

Antoine Avignon1

Sébastien Jugant4

Ariane Sultan1

Ariane Sultan
email: a-sultan@chu-montpellier.fr

Renal function is highly associated with podiatric risk in diabetic patients

Keywords: chronic kidney disease, diabetes, diabetic foot ulcer, dialysis, prevention --

Background.

Chronic kidney disease (CKD) is correlated with the incidence of diabetic foot ulcer (DFU). Furthermore, the International Working Group on the Diabetic Foot (IWGDF) has proposed a classification of the risk factors for DFU. The purpose of this study was to investigate the relationship between the IWGDF risk classification and the glomerular filtration rate level estimated by the CKD Epidemiology Collaboration formula (eGFR).

Methods.

We conducted a prospective multicentric study. Patients were recruited from either diabetology or nephrology departments. The secondary objectives were to determine this relationship after excluding people on dialysis and to identify the factors associated with podiatric risk.

Results.

Four hundred and eighty-six patients were included, with a mean age of 64.2 years +/-15.7) and a mean diabetes duration of 15.7 years +/-12.1). Based on the IWGDF classification, 53.5% of the population were in podiatric stage 0, 11.7% in stage 1 and 34.8% in stage 2 or 3. The mean eGFR level was significantly lower in patients with podiatric risk ≥2 (36.8 +/-33.9 mL/min/1.73 m 2 vs 71.9 +/-35.3 mL/min/1.73 m 2 , P < .0001) and a significant association was found between the eGFR and the podiatric risk. This association remained significant after the exclusion of the hemodialysis patients. After receiver operating characteristic analysis, a cutoff of 45 •+/-11 mL/min/1.73 m 2 (area under the curve 0.76) was found discriminant to define a group of CKD patients at higher risk for podiatric stage ≥2.

Conclusion.

eGFR levels are linked to podiatric stages in diabetes mellitus. Patients with eGFR <45 mL/min/1.73 m 2 and dialysis patients should be carefully managed in collaboration with diabetic foot specialized centers.

LAY SUMMARY

LAY SUMMARY

Chronic kidney disease has been correlated with an increased incidence of diabetic foot ulcer (DFU). However, there are no data describing the relationship between the risk score for DFU, which has been built in a logic of prevention, and the level of renal function. Therefore, the purpose of this prospective multicentric study was to describe the association between renal function and the risk of DFU using the International Working Group on the Diabetic Foot classification. Four hundred and eighty-six patients were included. In multivariate analysis, the podiatric risk stage was associated with eGFR and albuminemia. After receiver operating characteristic analysis, the Youden index best estimating the risk of reaching a high podiatric stage ≥2 was 45 +/-11 mL/min/1.73 m 2 [area under the curve (AUC) 0.76] for the entire population and 68 +/-6 mL/min/1.73 m 2 (AUC 0.70) after excluding hemodialysis patients. This study reinforces the arguments for joint work between nephrologists and diabetologists, and setting up alert thresholds that can be used in routine clinical practice.

INTRODUCTION

Diabetes mellitus (DM) is the main cause of chronic kidney disease (CKD) and non-traumatic lower limb amputation [START_REF] Bakker | The 2015 IWGDF guidance documents on prevention and management of foot problems in diabetes: development of an evidencebased global consensus[END_REF]. Moreover, 19%-34% of diabetic patients will develop a diabetic foot ulcer (DFU) in their lifetimes [START_REF] Armstrong | Diabetic foot ulcers and their recurrence[END_REF]. In 2017, people living with diabetes made up 49.6% of the dialysis population in the USA [START_REF]DPM Hemodialysis United States: Featured Measures[END_REF]. Diabetic patients with CKD and/or DFU are therefore a major public health concern. Furthermore, in 2017, amputation and end-stage renal disease accounted for a Medicare expenditure of 36 billion US dollars [4].

CKD has been correlated with an increased incidence of DFU [START_REF] Margolis | Association between renal failure and foot ulcer or lower-extremity amputation in patients with diabetes[END_REF], and the initiation of dialysis seems to be a consequential step [START_REF] Game | Temporal association between the incidence of foot ulceration and the start of dialysis in diabetes mellitus[END_REF]. Furthermore, the prognosis for DFU in the context of CKD is quite poor [START_REF] Ghanassia | Longterm outcome and disability of diabetic patients hospitalized for diabetic foot ulcers[END_REF][START_REF] Bonnet | Narrative review of the relationship between chronic kidney disease and diabetic foot ulcer[END_REF][START_REF] Lee | Risk factors affecting amputation in diabetic foot[END_REF]. Healing, amputation and mortality are all affected by CKD with an estimated glomerular filtration rate (eGFR) <90 mL/min/1.73 m 2 [START_REF] He | Association between estimated glomerular filtration rate and outcomes in patients with diabetic foot ulcers: a 3-year follow-up study[END_REF]. Further, the mortality relative risk for patients with DFU and CKD is dramatically higher than that for people without CKD, with an increased risk between 1.22 [START_REF] Jeyaraman | Mortality in patients with diabetic foot ulcer: a retrospective study of 513 cases from a single centre in the Northern Territory of Australia[END_REF] and 3.54 [START_REF] He | Association between estimated glomerular filtration rate and outcomes in patients with diabetic foot ulcers: a 3-year follow-up study[END_REF]. Last, CKD G5D (dialysis patients) is associated with DFU incidence and prognosis [START_REF] Akturk | Ulcer-free survival days and ulcer healing in patients with diabetic foot ulcers: a prospective cohort study[END_REF], notably in terms of amputation risk and mortality after amputation, with a mortality rate of 50%-70% 1 year after amputation [START_REF] Eggers | Nontraumatic lower extremity amputations in the Medicare end-stage renal disease population[END_REF].

In 1999, the International Working Group on the Diabetic Foot (IWGDF) suggested a way to assess the risk of DFU in diabetic patients, which is today in routine use. Since 2019, the IWGDF has included CKD G5D in its classification [START_REF] Bus | Guidelines on the prevention of foot ulcers in persons with diabetes (IWGDF 2019 update)[END_REF]. This update paralleled a similar decision of the National Institute for Health and Care Excellence (NICE) [START_REF]Diabetic foot problems: prevention and management | Guidance | NICE[END_REF], which added CKD G5D to its own podiatric risk classification in 2015. To our knowledge, however, there are no data describing the relationship between the risk score for DFU and the level of renal function, yet this information is essential for implementing adapted DFU prevention measures.

Therefore, the twofold purpose of this study was to describe the association between renal function and the risk of DFU using the IWGDF classification, and to determine whether there is a threshold effect in this relationship. Predictive factors associated with the podiatric risk were also analyzed.

MATERIALS AND METHODS

We conducted a prospective observational multicenter study. All patients received written information about the study and none presented opposition. The study was reviewed and approved by the Institutional Review Board of the University Hospital of Montpellier, France (identification number: 2019_IRB-MTP_09-12; clinical trial: NCT04100551).

Patients

The study was proposed to all outpatients with DM followed by the diabetology department of Montpellier University Hospital. Only consulting patients were recruited in order to maintain our focus on stable CKD. Patients on dialysis were recruited in the dialysis centers of the University Hospital of Montpellier and in Nephrocare, a private hospital in Montpellier. All patients were over 18 years old and none was under conservatorship. Patients with a previous solid organ transplant of any kind were excluded.

Data collection

Basic characteristics such as gender, age, height, weight, body mass index (BMI), type and length of diabetes, diabetic complications and current treatments were collected. Diabetes balance was evaluated according to HbA1c. Blood pressure was classified as: high blood pressure but at or under the objective while under treatment according to the European Society of Hypertension, high blood pressure exceeding the objective, and no high blood pressure [START_REF] Williams | ESC/ESH Guidelines for the management of arterial hypertension: The Task Force for the management of arterial hypertension of the European Society of Cardiology (ESC) and the European Society of Hypertension (ESH)[END_REF]. Albuminuria was expressed by the albuminuria/ creatinuria ratio.

Routine biological explorations were made concomitant to creatinine measurement: hemoglobin (Hb), C-reactive protein (CRP), urea, magnesium, uric acid, sodium, potassium, bicarbonate (HCO3 -), albumin, calcium, phosphorus, 25-OH-vitamin D, parathyroid hormone and blood lipid parameters [triglycerides, total cholesterol, calculated or measured lowdensity lipoprotein (LDL) and high-density lipoprotein (HDL)].

Podiatric risk classification

Podiatric risk classification followed the 2019 IWGDF guidelines as follows: stage 0, no neuropathy; stage 1, peripheral neuropathy or peripheral artery disease (PAD); stage 2, peripheral neuropathy (PND) with PAD and/or foot deformity; and stage 3, peripheral neuropathy or PAD with previous foot ulcer, lower-extremity amputation or CKD G5D (Supplementary data, Table S1) [START_REF] Bus | Guidelines on the prevention of foot ulcers in persons with diabetes (IWGDF 2019 update)[END_REF]. Given the study objective, we did not automatically categorize the patients with CKD G5D as having podiatric stage 3 in order to study the diversity and distribution of the stages within this population. This decision allowed us to exhaustively examine the whole population with CKD.

Renal function evaluation

Blood creatinine was measured by enzymatic techniques in all patients at least 3 months before and not more than 3 months after the podiatric classification. The GFR was estimated (eGFR) by the CKD Epidemiology Collaboration (CKDEPI) equation. The association between the eGFR and the IWGDF score was the main endpoint of this study and the patients were divided into groups according to the eGFR level used in the CKD classification (eGFR ≥90 mL/min/1.73 m 2 , between ≥60 and <90 mL/min/1.73 m 2 , between ≥45 and <60 mL/min/1.73 m 2 , between ≥30 and <45 mL/min/1.73 m 2 , between ≥15 and <30 mL/min/1.73 m 2 , <15 mL/min/1.73 m 2 with or without dialysis, and CKD G5D for those patients on dialysis). If a patient was on dialysis, we asked for the length of time on renal replacement therapy and the value given to the eGFR was "0 mL/min/1.73 m 2 ."

Statistical analysis

Non-normality of the distributions of the numerical variables was detected by the Q-Q plot and the Shapiro-Wilk test. The variables were described using mean +/-standard deviation (SD), the median and interquartile range, or percentage, depending on their distribution.

Prior to analysis, the association between missingness and baseline variables was tested in a logistic regression (data not shown). A significant association with eGFR indicated that our data were not missing completely at random, and missing at random (MAR) may be a more plausible assumption under which to conduct further analysis.

Patients were divided into groups according to their podiatric stage. The podiatric risk classification was highly skewed and, to limit the efficiency loss and bias on estimates of the regression coefficients, the main outcome was analyzed as an ordered categorical response. We assigned the numerical value of the classification scale for the ordered categories (c = 0, 1, 2, 3). Generalized linear mixed models (GLMM) for analyzing multilevel ordinal categorical responses were used. Under MAR condition, mixed models offer an alternative to handle missing data without requiring imputations. GLMM were constructed for each explanatory variable with a random intercept and an unstructured covariance matrix. Multivariate modeling was performed using the significant variables found at the univariate level. The models were compared by the likelihood ratio test in order to obtain the most simplified final model. Two final models were described, depending on whether patients on dialysis were kept or not. Pairwise comparisons were performed if necessary with the Tukey SHD.

We conducted a second analysis to compare the characteristics of the patients with high podiatric risk (stage ≥2) vs low podiatric risk (stage <2) [START_REF] Schaper | Practical guidelines on the prevention and management of diabetic foot disease (IWGDF 2019 update)[END_REF]. The comparisons between groups were performed by the T-test, the Wilcoxon rank sum test or the χ2 test as appropriate and GLMMs for analyzing binomial responses were used the most simplified model was obtained using the same methodology as previously described. Two final models were also constructed, depending on whether patients on dialysis were kept or not.

To assess the robustness of our findings, we finally conducted a sensitivity analysis in which the association between eGFR and the podiatric stage was tested in a complete case analysis (106 non-dialysis patients) using the same statistical procedure.

We finally used a receiver operating characteristic (ROC) analysis to evaluate the Youden index to account for the potential discriminating ability of eGFR to define a group of CKD patients at higher risk for podiatric stage ≥2.

All statistics were performed using Stata14 (Statacorp, College Station, TX, USA) and Prism 8 (Graphpad Software).A P < .05 was considered as significant.

RESULTS

Description of the population

Four hundred and eighty-six patients were included between 1 November 2019 and 31 May 2021. The clinical characteristics of the whole population are shown in Table 1. Mean age was 64.2 years (+/-15.7) and mean diabetes duration was 15.7 years (+/-12.1). Eighty-four percent of the population had type 2 diabetes, with 77.6% having BMI >25 kg/m 2 and 42.8% BMI >30 kg/m 2 . More than half the population (59.5%) was treated with insulin.

According to the podiatric classification, 53.5% of the patients were stage 0, 11.7% stage 1, 14.8% stage 2 and 20% stage 3.

According to eGFR levels, 143 (29.4%) presented an eGFR >90 mL/min/1.73 m 2 , 100 (20.6%) were between ≥60 and <90 mL/min/1.73 m2, 51 (10.5%) were ≥45 and <60 mL/min/ 1.73 m 2 , 70 (14.4%) were ≥30 and <45 mL/min/1.73 m2, 53 (10.9%) were ≥15 and <30 mL/min/1.73 m 2 , and 70 (14.4%) were <15 mL/min/1.73 m 2 . Fifty-six (11.5%) were on dialysis. All the clinical and biological characteristics of the whole group are presented in Table 1.

Association between eGFR level and podiatric risk

In non-dialysis patients, the mean eGFR was 76.4 •+/-33.7, 62.3 •=/-30.3, 48.2 •} 30.1 and 51.7 •+/-29.7 mL/min/1.73 m 2 , respectively, in podiatric risk stage 0, 1, 2 and 3. There was a significant difference between podiatric risk stage 0 and 1 (P = .02), 1 and 2 (P < .001), and 1 and 3 (P < .001), but not between 2 and 3 (Fig. 1). Furthermore, the prevalence of podiatric risk stage ≥2 increased with the worsening of the renal function (P = .0001) (Fig. 2).

Factors associated with the podiatric risk classification

In the univariate analysis, the risk stage was associated with creatininemia, eGFR, dialysis, dialysis duration, albuminuria/creatinuria ratio, BMI, biguanide use, DPP4-inhibitor or insulin treatment, the completion of a retinographic check-up within the year, the balance of high blood pressure ,magnesemia, HCO3 -, CRP and LDL-cholesterol (Table 2 and Supplementary data, Table S2). In the multivariate analysis, when dialysis patients were included, eGFR (β = -0.013 •+/-0.0017, P < .0001) and albuminemia ((β = -0.039 •+/-0.014, P = .007) were associated to podiatric risk. After exclusion of the dialysis patients, only eGFR (β = -0.012 •} 0.003, P < .0001) remained associated with podiatric risk (Table 3).

Factors associated with high podiatric risk

High podiatric risk (stage ≥2) was associated with creatininemia, eGFR, dialysis, dialysis duration, albuminuria/creatinuria ratio, BMI, biguanide use, DPP4-inhibitor or insulin treatment, the completion of a retinographic check-up within the year, the balance of high blood pressure, magnesemia, HCO3 -, CRP, LDL cholesterol and Hb (Table 4 and Supplementary data, Table S3).

In the multivariate analysis, when dialysis patients were considered, eGFR [odds ratio (OR) 0.98 (0.97; 0.99), P < .0001] was the only variable associated with high podiatric risk. After exclusion of the dialysis patients, eGFR [OR 0.98 (0.97; 0.98), P < .0001] and albuminemia [OR 0.92 (0.87; 0.98), P = .006] remained associated with podiatric risk (Table 5).

eGFR threshold predicting podiatric risk

After ROC analysis, the Youden index best estimating the risk of reaching podiatric stage ≥2 was 45 •+/-11 mL/min/1.73 m 2 [area under the curve (AUC) 0.76] for the entire population and 68 •+/-6 mL/min/1.73 m 2 (AUC 0.70) after excluding hemodialysis patients (Fig. 3a).

Further, the Youden index best estimating the risk of an error for the 10-g Semmes-Weinstein monofilament test was 46 •+/-14 mL/min/1.73m 2 (AUC 0.62) for the entire population and 76 •+/-13 mL/min/1.73 m 2 (AUC 0.59) after excluding hemodialysis patients (Fig. 3b). Results for eGFR levels are presented in the Supplementary data, Table S4.

Sensitivity analysis

In the sensitivity analysis, 106 patients without missing data were analyzed. We found a persistent association between the podiatric risk and the eGFR (β = -0.014 •+/-0.003, P < .0001) (Table 6).

DISCUSSION

To our knowledge, this is the first study that highlight a close association between the podiatric risk and eGFR level. Indeed, we found that a low eGFR and worse CKD stage were linked to a greater podiatric risk of DFU. However, we found no association between podiatric risk and diabetes type, duration of diabetes, and HbA1c.

Renal function and risk stage in the overall population

The main result of this study underscored that the podiatric stage progressively increased as eGFR level decreased, especially up to stage 2. This result persisted whether dialysis patients were considered or excluded. The optimal eGFR cut-point in screening for diabetic patients at risk of stage ≥2 in our study was 68 mL/min/1.73 m 2 when patients on dialysis were considered a specific population and excluded from the analysis. A threshold of 60 mL/min/1.73 m 2 was already shown to differentiate in terms of prognosis for people with active foot wounds [START_REF] Ghanassia | Longterm outcome and disability of diabetic patients hospitalized for diabetic foot ulcers[END_REF] and cardiovascular outcomes [START_REF] Gansevoort | Chronic kidney disease and cardiovascular risk: epidemiology, mechanisms, and prevention[END_REF]. The high cut-point value of eGFR may be a sign of renal vascular damage before the occurrence of more severe renal impairment. Indeed, arteriolar lesions found on renal biopsies do not correlate well with GFR at the time of the biopsy [START_REF] Misra | Renal histology in diabetic nephropathy predicts progression to end-stage kidney disease but not the rate of renal function decline[END_REF].

In any case, we did note an increasingly more significant relationship than previously described in studies focused on DFU [START_REF] Game | Temporal association between the incidence of foot ulceration and the start of dialysis in diabetes mellitus[END_REF][START_REF] Mcgrath | Recent commencement of dialysis is a risk factor for lower-extremity amputation in a high-risk diabetic population[END_REF]. This is further evidence of a joint evolution of the two complications. Moreover, this hypothesis was further reinforced by the association in the univariate analysis between the level of albuminuria (and glomerular damage) and the podiatric stage.

People on dialysis are a high-risk population and they require special attention, but an evaluation before they reach the CKD G5D-i.e. during the nephrological follow-up of the attempt to slow down the renal disease-seems equally important. According to several studies, efforts in dialysis centers have led to tangible improvements in patients' foot health [START_REF] Mcgrath | Recent commencement of dialysis is a risk factor for lower-extremity amputation in a high-risk diabetic population[END_REF][START_REF] Franz | Trends in rates of lower extremity amputation among patients with end-stage renal diseasewho receive dialysis[END_REF][START_REF] Harding | Trends of nontraumatic lower extremity amputation in end-stage renal disease and diabetes, United States, 2000-2015[END_REF][START_REF] Pernat | Implementation of routine foot check in patients with diabetes on hemodialysis: associations with outcomes[END_REF][START_REF] Foster | Reduction of gangrene and amputations in diabetic renal transplant patients: the role of a special foot clinic[END_REF], but we are not aware of any specific podiatric intervention study during the followup period prior to CKD G5D [START_REF] Bonnet | Narrative review of the relationship between chronic kidney disease and diabetic foot ulcer[END_REF]. Our results may be explained by our focus for the first time on the progressive markers of DFU risk and not on the event that is the ulcer. In this sense, we agree with previous reports on the progressive link between PND and creatininemia [START_REF] Hurley | A prospective study of risk factors for foot ulceration: the West of Ireland Diabetes Foot Study[END_REF].

Podiatric stage and CKD G5D

As expected, the patients on dialysis had high podiatric risk stages. However, although 73.6% of those with CKD G5D were in stage 2 or 3, 15.5% were in stage 0 and 9.9% in stage 1.Within this population with CKD G5D, with or without dialysis, it was not clear whether some patients were at greater risk than others. Further study is thus needed to determine whether patients in stage 0 or 1 have been newly diagnosed with diabetes. Indeed, it will be important to follow this population to quantify their actual risk of developing a DFU, especially since according to the IWGDF stratification, all patients on dialysis are automatically classed into stage 3. In any case, the population with CKD G5D was overall in a worse risk stage, although this did not appear to be a specific step in foot complications [START_REF] Game | Temporal association between the incidence of foot ulceration and the start of dialysis in diabetes mellitus[END_REF][START_REF] Mcgrath | Recent commencement of dialysis is a risk factor for lower-extremity amputation in a high-risk diabetic population[END_REF].

Notably, when we focused on dialysis itself, we found that dialysis duration was a strong predictor of worse DFU risk classification. However, given the wide distribution of stages among the patients on dialysis and the small number of them in stage 0 or 1, it seems that hundreds of patients on dialysis would need to be recruited in order to draw statistically significant conclusions. A study of DFU-free survival might be more appropriate in this population to evaluate the impact of the quality of dialysis.

HbA1c, DFU risk stage and renal function

We did not find a correlation between HbA1c and the podiatric stage in this population with a high prevalence of renal failure, but we did find a progressive decrease in HbA1c as eGFR deteriorated, as already described [START_REF] Abe | Is there a "burnt-out diabetes" phenomenon in patients on hemodialysis?[END_REF]. HbA1c had no impact on the risk stage, whereas intensive insulin therapy showed some impact on PND, retinopathy and nephropathy in diabetes type 1 [START_REF] Martin | Neuropathy and related findings in the diabetes control and complications trial/epidemiology of diabetes interventions and complications study[END_REF]. This remind us that a glycemic control strategy to prevent DFU must be built over time for optimal results [START_REF] Control | The effect of intensive treatment of diabetes on the development and progression of long-term complications in insulin dependent diabetes mellitus[END_REF][START_REF] Singh | Preventing foot ulcers in patients with diabetes[END_REF] and that intensive glycemic therapy might not necessarily be efficient to prevent DFU once complications have set in. We should note that we were unable to adjust our results on hypoglycemic events or glycemic variability, although it is known that hypoglycemic events and glycemic variability can worsen PND [START_REF] Pettus | Incidences of severe hypoglycemia and diabetic ketoacidosis and prevalence of microvascular complications stratified by age and glycemic control in U.S. adult patients with Type 1 diabetes: a realworld study[END_REF][START_REF] Flatt | Predictors of recurrent severe hypoglycemia in adults with Type 1 diabetes and impaired awareness of hypoglycemia during the HypoCOMPaSS study[END_REF][START_REF] Kwai | Association between glycemic variability and peripheral nerve dysfunction in type 1 diabetes[END_REF].

Clinical implications

A foot examination is not part of the nephrologist's usual clinical examination, although significant progress in foot-care has been made in dialysis centers. Studies are consistent in showing the effectiveness of therapeutic education or diabetic foot monitoring programs in dialysis centers, notably with regard to predicting wound evolution and even reducing the incidence of amputations [START_REF] Pernat | Implementation of routine foot check in patients with diabetes on hemodialysis: associations with outcomes[END_REF][START_REF] Mcmurray | Diabetes education and care management significantly improve patient outcomes in the dialysis unit[END_REF][START_REF] Rith-Najarian | Preventing amputations among patients with diabetes on dialysis[END_REF]. The impact of a systematic podiatrist consultation for eGFR values <60 mL/min/1.73 m2 should now be investigated.

It was quite surprising and unsettling that we were unable to find a sufficiently strong link to point to a threshold of predictability of a monofilament error. Renal function may in some way interact independently of this monofilament test result, and this possibility also deserves further study.

Limitations and strengths of the study

We recruited only outpatients with stable eGFR and to our knowledge, this is the first description of this type of population using the IWGDF stratification system.

Our prospective study included a recent creatinine blood test and a standardized clinical examination based on the international 2019 IWGDF risk stratification guidelines, which have been recognized by health authorities [START_REF] Schaper | Practical guidelines on the prevention and management of diabetic foot disease (IWGDF 2019 update)[END_REF]35]. Unfortunately, we were not able to define CKD stage 1 and 2 in all patients with an eGFR >60 given the lack of data on the necessary criteria requested by the KDIGO (albuminuria, hematuria and kidney morphology). As we found a rather high threshold of eGFR associated with the podiatric stage, it would be interesting to redo a study to analyze stages 1 and 2.

We took into account a maximum of biological confounding factors, and 79% of all patients with eGFR <60 mL/min/1.73 m2 had had a recent phosphocalcic and parathyroid hormone control. These missing data remain a limitation to the study, especially regarding the possibility of integrating them as a correction factor in a multivariate analysis. Although we had set up a systematic recruitment protocol, we had a good distribution of patients across all stages of CKD and with no CKD. Similarly, we recruited the same number of patients in risk stage 0 as in stage 1, 2 or 3. This balanced distribution of the patients of our study may have strengthened the results.

FUNDING

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

DATA AVAILABILITY STATEMENT

The data underlying this article will be shared on reasonable request to the corresponding author.

CONCLUSION

Poor renal function is associated with high podiatric risk and patients on dialysis have the higher risk. The effect of renal function decline on podiatric risk seems to appear early, and diabetic patients with CKD should be referred to foot specialist centers to benefit from enhanced education programs and careful foot management. This study reinforces the arguments for joint work between nephrologists and diabetologists.

CONFLICT OF INTEREST STATEMENT

The authors declare that they have no relevant financial interests.

AUTHORS' CONTRIBUTIONS

Research idea and study design: J.-B.B., I.S., A.S., S.J., A.A.; data acquisition: J.-B.B.; data analysis/interpretation: J.-B.B., I.S., A.S.; statistical analysis: I.S.; supervision or mentorship: I.S.,A.S., A.A. Each author contributed important intellectual content during manuscript drafting or revision and agrees to be personally accountable for the individual's own contributions and to ensure that questions pertaining to the accuracy or integrity of any portion of the work, even one in which the author was not directly involved, are appropriately investigated and resolved, including with documentation in the literature if appropriate.


Zakhar Kabluchko
email: zakhar.kabluchko@uni-muenster.de

AND Alexander Marynych
email: marynych@unicyb.kiev.ua

Kilian Raschel
email: raschel@math.cnrs.fr

RANDOM WALKS IN THE HIGH-DIMENSIONAL LIMIT II: THE CRINKLED SUBORDINATOR

Keywords: 2020 Mathematics Subject Classification. Primary: 60F05, 60G50; Secondary: 60D05, 60G51 crinkled arc, Gromov-Hausdorff convergence, Hausdorff distance up to isometry, high-dimensional limit, random metric space, random walk, subordinator, Wiener spiral

A crinkled subordinator is an ℓ 2 -valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. We show that the path of a d-dimensional random walk with n independent identically distributed steps with heavy-tailed distribution of the radial components and asymptotically orthogonal angular components converges in distribution in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as

Let (X (d) ) d∈N be a sequence of random variables defined on a common probability space (Ω, F , P) such that X (d) takes values in R d (identified with the linear span of (e 1 , e 2 , . . . , e d ) in ℓ 2 ), d ∈ N. Assume that the

INTRODUCTION

Let ℓ 2 be the infinite-dimensional (real) Hilbert space of square-summable sequences endowed with the standard Hilbert norm

(x 1 , x 2 , . . .) 2 := ∞ ∑ k=1 x 2 k , (x 1 , x 2 , . . .) ∈ ℓ 2 ,
and the standard inner product (x 1 , x 2 , . . .), (y 1 , y 2 , . . .) 2 := ∞ ∑ k=1

x k y k , (x 1 , x 2 , . . .), (y 1 , y 2 , . . .) ∈ ℓ 2 .

Fix the standard orthonormal basis (e k ) k∈N of ℓ 2 and consider the natural embeddings

R ⊂ R 2 ⊂ • • • ⊂ R d ⊂ • • • ⊂ ℓ 2 ,
obtained by identifying R d with the linear span of (e 1 , e 2 , . . . , e d ), d ∈ N. This will allow us throughout the paper to treat elements of R d as elements of ℓ 2 and use the notation x 2 (respectively, x, y 2 ) for the usual Euclidean norm of x ∈ R d (respectively, the standard inner product of x, y ∈ R d ). Denote by ρ 2 (x, y) := x -y 2 the metric on ℓ 2 induced by the norm • 2 .

space (Ω, F , P) is reach enough to accommodate a sequence (X (d) i ) i∈N of independent copies of X (d) , for each d ∈ N. Consider a family of random walks defined via S (d) can be regarded as a continuous piecewise-linear curve in ℓ 2 starting at the origin and living in the finite-

0 := 0, S (d) k := X (d) 1 + X (d) 2 + • • • + X (d) k , k ∈ N, (1)
dimensional subspace R d , d ∈ N.
This paper is a continuation of [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF] and devoted to finding an answer to the question: How does the curve

S (d)
n (after an appropriate renormalization and up to isometries in ℓ 2 ) look like when d and, therefore, n tend to infinity? Does it approach some deterministic or random curve in ℓ 2 ? Under the assumptions EX (d) = 0, E X (d) 2 = 1 and the components of X (d) are uncorrelated (plus some mild technical assumptions), it was proved in [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF] that ( S

(d) n /
√ n, ρ 2 ), regarded as a compact metric space, converges in probability in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense, see Section 2.5 below for the definitions, to a deterministic metric space called the Wiener spiral. An isometric copy of the Wiener Spiral in the space ℓ 2 is given by a continuous curve W := ( w t ) t∈ [0,[START_REF] Abraham | A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces[END_REF] , where

w t := 2 √ 2 π ∞ ∑ k=1 sin(π(k -1/2)t) 2k -1 e k , t ∈ [0, 1], (2)
which possesses peculiar properties

w t -w s 2 = |t -s|, 0 ≤ s,t ≤ 1, (3)
and also

w t -w s , w u -w v 2 = 0, 0 ≤ v ≤ u ≤ s ≤ t ≤ 1. (4)
Note that (3) means that, as a metric space, (W, ρ 2 ) is isometric to the interval [0, 1] endowed with the distance |t -s|. The Wiener spiral is also isometric to a continuous curve (w t ) t∈ [0,[START_REF] Abraham | A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces[END_REF] in L 2 ([0, 1]), given by

w t = ½ [0,t] (•) ∈ L 2 ([0, 1]), t ∈ [0, 1]
. This can be seen by noting that

w t , w s L 2 ([0,1]) = min(t, s) = w t , w s 2 .
It is worth mentioning that replacing e k in (2) by N k , where (N k ) k∈N are independent identically distributed (i.i.d.) standard normal random variables, gives the Karhunen-Loéve expansion of a standard Brownian motion.

The aforementioned result of [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF] can be compared with the classical functional weak law of large numbers for one-dimensional random walks. Recall that the latter tells us that finiteness of the first moment of a generic step implies uniform convergence of the path of the rescaled random walk to a deterministic linear function in probability; see [START_REF] Glynn | Extensions of the queueing relations L = λW and H = λ G[END_REF] for example. In the setting of [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF], the authors show that the finiteness of E X (d)2 = 1 also implies convergence to a deterministic limit. A natural question arising from this comparison is the following. It is known that if a generic step of a one-dimensional random walk is a.s. positive, has infinite mean and its distribution has a regularly varying at infinity tail, then the path of the rescaled random walk converges to a random limit being the path of a subordinator. Thus, suppose now that the distribution of X (d) 2 is regularly varying at infinity. Keeping in mind the above analogy with one-dimensional random walks, it is natural to expect that in this scenario, S

n converges as d → ∞, after an appropriate rescaling, to a genuinely random curve in ℓ 2 , a path of a certain ℓ 2 -valued random process derived from a subordinator.

The main result of our paper confirms these expectations.

ASSUMPTIONS, DEFINITIONS AND MAIN RESULTS

2.1. Assumptions. We shall now present our assumptions on the distributions of (X (d) ) d∈N which will be used throughout the paper. The components of the vectors X (d) i (independent copies of X (d) ), and

S (d) i are denoted by X (d) i = (X (d) i,1 , . . . , X (d) i,d ) and S (d) i = (S (d) i,1 , . . . , S (d) i,d ), respectively. Furthermore, let Θ (d) := X (d) X (d) 2 , Θ (d) i := X (d) i X (d) i 2 , i ∈ N, d ∈ N,
denote the angular components of X (d) and X (d) i 's. Suppose that the following hypotheses hold:

(a) There exist constants (a(k)) k∈N and a Lévy measure ν on (0, ∞] satisfying

(0,∞) min(1, x)ν(dx) < ∞, ν(∞) = 0, (5)
and such that

nP{(a(n)) -1 X (d) 2 2 ∈ •} v -→ d→∞ ν(•), (6)
where v -→ d→∞ stands for the vague convergence of measures on (0, ∞]. Suppose, further, that lim

s→0+ lim sup d→∞ n a(n) E X (d) 2 2 ½ { X (d) 2 2 ≤sa(n)} = 0. (7)
(b) With the sequence (a(k)) k∈N defined in part (a), the following relation holds true, for all fixed s > 0 and ε > 0:

lim d→∞ P Θ (d) 1 , Θ (d) 2 2 > ε X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n) = 0. ( 8
EX (d) ½ { X (d) 2
As we shall now see, there are several cases in which the assumptions (a)-(c) can be significantly simplified.

IDENTICALLY DISTRIBUTED X (d) 2 , d ∈ N. Under the assumption that the distribution of X (d) 2 does not depend on d, condition ( 6) is equivalent to the following one. There exist α ∈ (0, 1) and a function L slowly varying at infinity such that, for all d ∈ N,

P{ X (d) 2 > t} = t -α L(t), t > 0. ( 10
)
In this case ν(x, ∞) = x -α for all x > 0, and (a(k)) k∈N can be any positive sequence satisfying

lim n→∞ nP{ X (d) 2 > a(n)} = lim n→∞ nL(a(n)) a(n) α = 1.
The existence of such a sequence follows by a standard argument involving de Bruijn conjugates; see Chapter 1.5.7 in [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]. Note that the restriction α ∈ (0, 1) comes from the fact that we require the Lévy measure to satisfy the integrability condition [START_REF] Bingham | Regular variation, volume 27 of Encyclopedia of Mathematics and its Applications[END_REF]. Furthermore, in this case the condition (7) holds automatically; see (31) below.

INDEPENDENT RADIAL AND ANGULAR COMPONENTS OF X (d) . The conditions in parts (b) and (c) take a particularly simple form if the radial and angular components of X (d) are independent. More precisely, if

X (d)
2 and Θ (d) are independent, then ( 8) is equivalent to

Θ (d) 1 , Θ (d) 2 2 P
-→ 0, d → ∞. (11)
That is to say, the condition of part (b) is equivalent to saying that two independent copies of X (d) are asymptotically orthogonal in probability, as d → ∞. If also EΘ (d) = 0, then (9) holds automatically, since the truncated expectation in ( 9) is equal to zero for independent X (d) 2 and Θ (d) .

SYMMETRIC DISTRIBUTION OF X (d) . Assume that the law of X (d) is the same as that of -X (d) , for all d ∈ N. In this case the truncated expectation in ( 9) is equal to zero, since the function of X (d) under E in [START_REF] Glynn | Extensions of the queueing relations L = λW and H = λ G[END_REF] is odd. Thus, (c) holds automatically.

2.2. The crinkled subordinator. Fix T > 0. It is known, see Theorem 7.1 on p. 214 in [START_REF] Resnick | Heavy-tail phenomena: Probabilistic and statistical modeling[END_REF], that the assumptions in part (a) imply   X

(d) 1 2 + X (d) 2 2 + • • • + X (d) ⌊nt⌋ 2 a(n)   t∈[0,T ] =⇒ (S ν (t)) t∈[0,T ] , n → ∞, (12)
in the Skorokhdod space of càdlàg functions defined on [0, T ] and endowed with the Skorokhod J 1 -topology.

Here, S ν is a subordinator, whose construction, which we are now going to recall, is of major importance for everything to follow.

A subordinator is an increasing stochastic process that has independent and homogeneous increments.

For the purposes of the present paper, the following definition (called Itô's decomposition) serves best. Let P := ∑ k δ (x k ,y k ) be a Poisson random measure on [0, ∞) × (0, ∞] with the intensity measure LEB × ν, where LEB denotes Lebesgue measure and ν is the Lévy measure as in (a). Here and in what follows, δ x denotes a Dirac measure at x. The stochastic process

S ν (t) := ∑ k: x k ≤t y k = [0,t]×(0,∞]
yP(dx, dy), t ≥ 0, is called a drift-free subordinator with Lévy measure ν. Condition (5) ensures that the sum above is a.s. finite for all t ≥ 0. In case when ν(x, ∞) = x -α , x > 0, for some α ∈ (0, 1), the subordinator S ν is called α-stable.

We allow ν to be a finite measure, in which case S ν is a compound Poisson process.

Definition 2.1. A crinkled subordinator with Lévy measure ν (and with respect to the chosen basis of ℓ 2 ) is an ℓ 2 -valued stochastic process (C ν (t)) t≥0 defined by

C ν (t) := ∑ k: x k ≤t e k √ y k , t ≥ 0,
where

P = ∑ k δ (x k ,y k ) is the Poisson process on [0, ∞) × (0, ∞) with intensity measure LEB × ν.
Note that for every fixed t ≥ 0,

C ν (t) 2 2 = ∑ k: x k ≤t y k = S ν (t) ∈ [0, ∞)
and therefore C ν (t) is a random element of ℓ 2 a.s., for every t ≥ 0. Note also that, as a curve,

t → C ν (t) is not ℓ 2 -continuous but is a.s. càdlàg.
In a similar way as the closed range of a subordinator is defined, see [3, Section 1.4], we define the closed range of a crinkled subordinator.

Definition 2.2. Fix T > 0. The range R ν (T ) of a crinkled subordinator on [0, T ] is a random closed subset of ℓ 2 defined as the closure in ℓ 2 of the image of t → C ν (t), t ∈ [0, T ]. Thus, R ν (T ) = cl ({C ν (t) : 0 ≤ t ≤ T }) = {C ν (t) : 0 ≤ t ≤ T } ∪ {C ν (t-) : 0 ≤ t ≤ T }.
According to Lemma 4.1 below, the set R ν (T ) is a.s. compact in ℓ 2 for every fixed T > 0. In particular, (R ν (T ), • 2 ) is a compact metric space. Moreover, this space (up to isometries of ℓ 2 ) does not depend on the choice of an orthonormal basis of ℓ 2 , whereas the crinkled subordinator itself does depend on the basis.

As a metric space, (R ν (T ), ρ 2 ) is isometric to the closed range of the subordinator (S ν (t)) t∈[0,T ] given by

R ν (T ) := cl ({S ν (t) : 0 ≤ t ≤ T }) = {S ν (t) : 0 ≤ t ≤ T } ∪ {S ν (t-) : 0 ≤ t ≤ T }, (13)
and endowed with the metric

(t, s) → |t -s|. An isometry ϕ : R ν (T ) → R ν (T ) is given by ϕ(C ν (t)) = C ν (t) 2 2 = S ν (t), ϕ(C ν (t-)) = C ν (t-) 2 2 = S ν (t-), t ∈ [0, T ]. (14)
Among other things, this implies that the Hausdorff dimension of R ν (T ) is equal to twice the Hausdorff

k := S (d) 0 a(n) , S (d) 1 a(n) , . . . , S (d) k a(n) , d ∈ N, k ∈ N. Each M (d) k
is endowed with the induced ℓ 2 -metric. Equivalently, since M (d) k lives in R d , which we assume to be naturally embedded into ℓ 2 , this induced metric coincides with the standard Euclidean metric.

Here is our main result.

M (d) ⌊nT ⌋ , ρ 2 =⇒ (R ν (T ), ρ 2 ), d → ∞. ( 15
)
Remark 2.4. A reminder on the Gromov-Hausdorff space will be given in Section 2.5.

Corollary 2.5. Assume that the distribution of X (d) does not depend on d and satisfies [START_REF] Greven | Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)[END_REF]. Suppose further that Θ (d) and X (d) are independent, EΘ (d) = 0 and (11) holds. Then (15) holds.

For a compact metric space M, denote by diam(M) its diameter. Since the mapping M → diam(M) is continuous with respect to the Gromov-Hausdorff metric, see Exercise 7.3.14 in [START_REF] Burago | A course in metric geometry[END_REF], we immediately obtain the following corollary of Theorem 2.3.

Corollary 2.6. Under the same assumptions as in Theorem 2.3, we have Example 1 (Rotationally invariant distributions). Let X (d) be a random vector in R d with a rotationally invariant distribution. This means that Θ (d) is uniformly distributed on the unit sphere in R d , and X (d)

(diam(M (d) ⌊nT ⌋ )) 2 = max 0≤i,k≤⌊nT ⌋ S (d) i -S (d) k 2 2 a(n) =⇒ (diam(R ν (T )) 2 = S ν (T ), d → ∞.
2
and Θ (d) are independent. Assume that the distribution of X (d) 2 2 satisfies ( 6) and [START_REF] Burago | A course in metric geometry[END_REF]. Condition (b) follows from Remark 3.2.5 in [START_REF] Vershynin | High-dimensional probability: An introduction with applications in data science[END_REF], whereas condition (c) is a consequence of EΘ (d) = 0 and independence of X (d) 2 and Θ (d) .

Example 2 (Random walks jumping along the coordinate axes). The following model is similar to the simple random walk on Z d . Let V (d) be a random vector distributed uniformly on the set {±e 1 , . . . , ±e d }, that is

P{ V (d) = e j } = P{ V (d) = -e j } = 1/(2d) for all j ∈ {1, . . . , d}. For every d ∈ N, put X (d) := R (d) • V (d) ,
where R (d) is a positive random variable which is independent of V (d) . Assume that the distribution of 6) and [START_REF] Burago | A course in metric geometry[END_REF]. Condition (b) holds automatically, since Θ (d) = V (d) , and therefore

X (d) 2 2 = (R (d) ) 2 satisfies (
P Θ (d) 1 , Θ (d) 2 = 0 = P V (d) 1 , V (d) 2 = 0 = 1 d → 0, d → ∞.
Condition (c) also holds automatically since E V (d) = 0.

Example 3 (Random walks with i.i.d. symmetric heavy-tailed components). Let ξ be a symmetric random variable such that for some α ∈ (0, 1),

P{ξ 2 > t} ∼ t -α , t → ∞. ( 16
)
For an array (ξ i, j ) i, j∈N of independent copies of ξ , put

X (d) i := d -1/(2α) (ξ i,1 , ξ i,2 , . . . , ξ i,d ), i ∈ N, d ∈ N,
and let X (d) be a generic copy of X (d)

i . According to Eq. ( 4) in [START_REF] Heyde | On large deviation probabilities in the case of attraction to a non-normal stable law[END_REF], for every fixed x > 0,

lim d→∞ nP{ X (d) 2 2 > xn 1/α } = lim d→∞ n • d • P{ξ 2 > xd 1/α n 1/α } = x -α . (17)
Thus, ( 6) holds with ν(x, ∞) = x -α and a(n) = n 1/α . Condition [START_REF] Burago | A course in metric geometry[END_REF] follows from the following chain of estimates:

n a(n) E X (d) 1 2 2 ½ { X (d) 1 2 2 ≤sa(n)} = n (nd) 1/α E d ∑ j=1 ξ 2 1, j ½ {∑ d j=1 ξ 2 1, j ≤s(nd) 1/α } ≤ (nd) 1-1/α E ξ 2 1,1 ½ {ξ 2 1,1 ≤s(nd) 1/α } ∼ α 1 -α s 1-α , d → ∞,
where the last asymptotic equivalence is a consequence of [START_REF] Resnick | Heavy-tail phenomena: Probabilistic and statistical modeling[END_REF]; see p. 579 in [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. The right-hand side of the last display converges to zero as s → 0+, which yields [START_REF] Burago | A course in metric geometry[END_REF]. Condition (9) follows from the symmetry of

X (d)
, which is inherited from the symmetry of ξ . For a proof (standard but technical) of part (b), we refer the reader to Lemma 5.2 in the Appendix.

d H (A, B) = inf{r > 0 : A ⊂ U r (B), B ⊂ U r (A)}.
Here

, U r (A) = {m ∈ M : ρ(A, m) < r} is the r-neighborhood of A in M. It is well known that (K (M), d H ) is a metric space. If M is complete, then (K (M), d H ) is also complete.
By an isometry between two sets E 1 and E 2 living in possibly different metric spaces (M 1 , ρ 1 ) and (M 2 , ρ 2 ), we understand a bijective mapping J : E 1 → E 2 such that ρ 2 (J(x), J(y)) = ρ 1 (x, y), for all x, y ∈ M 1 .

The Gromov-Hausdorff distance d GH (E 1 , E 2 ) between two compact metric spaces E 1 and E 2 is defined as the infimum of d H (ϕ 1 (E 1 ), ϕ 2 (E 2 )), where the infimum is taken over all metric spaces (M, ρ) and all isometric embeddings (injective isometries) ϕ 1 : E 1 → M and ϕ 2 : E 2 → M. It is known that the set of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff distance, becomes a complete separable metric space, called the Gromov-Hausdorff space.

2.5.2.

Hausdorff distance up to isometry in ℓ 2 . Compact metric spaces we are interested in live in the same Hilbert space ℓ 2 . This suggests that the full power of the general notion of Gromov-Hausdorff distance might not be needed. Addressing this question, the following notion of closeness between compact subsets of ℓ 2 , called Hausdorff distance up to isometry, has been proposed in [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF]. Note that a very close concept can be found in Exercise 5.26 in [START_REF] Petrunin | Pure metric geometry: introductory lectures[END_REF].

Introduce the following equivalence relation ∼ on K (ℓ 2 ), the collection of non-empty compact subsets of ℓ 2 . Two subsets K 1 ⊂ ℓ 2 and K 2 ⊂ ℓ 2 are considered equivalent if there is an isometry J : K 1 → K 2 , that is a bijection between K 1 and K 2 that preserves distances. The equivalence class of a compact set K is denoted by

[K] := {K ′ ∈ K (ℓ 2 ) : K ∼ K ′ }.
The set of all such equivalence classes is denoted by

H := K (ℓ 2 )/ ∼. Now we introduce a metric on H. For K 1 , K 2 ∈ K (ℓ 2 ), the Hausdorff distance up to isometry between [K 1 ] and [K 2 ] is defined by d ∼ ([K 1 ], [K 2 ]) = inf K ′ 1 ∈[K 1 ],K ′ 2 ∈[K 2 ] d H (K ′ 1 , K ′ 2 ). (18)
Proposition 2.7 (Proposition 2.10 in [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF]). The function d ∼ :

H × H → [0, ∞) is a metric on H.
The following result establishes equivalence of Gromov-Hausdorff convergence and convergence in (H, d ∼ ).

Theorem 2.8 (Theorem 2.12 in [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF]). Let K 1 , K 2 , . . . and K be compact subsets of ℓ 2 . Then,

[K n ] → [K]
in (H, d ∼ ) if and only if K n → K in the Gromov-Hausdorff sense (where K n and K are regarded as metric spaces with the induced ℓ 2 -metric). Remark 2.9. The notion used in [START_REF] Petrunin | Pure metric geometry: introductory lectures[END_REF] differs from our definition (18) by two aspects. Firstly, the space ℓ 2 is replaced by a universal homogeneous metric space (Urysohn space) U ∞ . Secondly, the infimum used in the definition of d ∼ is taken over global isometries of U ∞ . This defines a pseudometric on the family of compact subsets on U ∞ , for which the corresponding metric space is isometric to the Gromov-Hausdorff space.

In view of Theorem 2.8, we immediately obtain the following: Corollary 2.10. Under the assumptions of Theorem 2.3, the following holds true:

[M (d) ⌊nT ⌋ ] =⇒ [R ν (T )], d → ∞, (19)
weakly on the space of probability measures on (H, d ∼ ).

Let conv (respectively, conv) denote the operation of taking convex (respectively, closed convex) hull.

Lemma 4.3 in [START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF] and the continuous mapping theorem yield the following result.

Theorem 2.11. Under the assumptions of Theorem 2.3, the following holds true:

[convM (d) ⌊nT ⌋ ] =⇒ [conv R ν (T )], d → ∞,
weakly on the space of probability measures on (H, d ∼ ).

The limiting closed convex hull can be characterized as follows. Let G ↓ denote the set of nonincreasing

functions g : [0, T ] → [0, 1]. Then convR ν (T ) = ∑ k:x k ≤T e k √ y k g(x k ) : g ∈ G ↓ .

PROOF OF THEOREM 2.3

Fix s > 0, define the truncated variables

X (d) k (s) := X (d) k ½ { X (d) k 2 2 ≥sa(n)} , k ∈ N,
the corresponding random walk

S (d) 0 (s) := 0, S (d) k (s) := X (d) 1 (s) + X (d) 2 (s) + • • • + X (d) k (s), k ∈ N,
and the sets

M (d) k (s) := S (d) 0 (s) a(n) , S (d) 1 (s) a(n) , . . . , S (d) k (s) a(n) , k ∈ N,
which we regard as a.s. finite metric spaces endowed with the induced ρ 2 metric.

Define the random set

R (s) ν (T ) := ∑ k: x k ≤t e k √ y k ½ {y k >s} : 0 ≤ t ≤ T , (20)
c.f. Definition 2.2, and note that it is a.s. finite for every fixed s > 0, since there are a.s. finitely many points

(x k , y k ) of P in [0, T ] × (s, ∞).
Since the Gromov-Hausdorff space is complete and separable, according to Theorem 3.2 on p. 28 in [START_REF] Billingsley | Convergence of probability measures[END_REF] it suffices to check that the following three relations hold true:

(M (d) ⌊nT ⌋ (s), ρ 2 ) =⇒ (R (s) ν (T ), ρ 2 ), d → ∞, (21)
for every fixed s > 0, weakly on the Gromov-Hausdorff space;

(R (s)

ν (T ), ρ 2 ) -→ (R ν (T ), ρ 2 ), s → 0+, (22)
a.s. on the Gromov-Hausdorff space; and lim s→0+ lim sup

d→∞ P{d GH (M (d) ⌊nT ⌋ (s), M (d) ⌊nT ⌋ ) > ε} = 0, (23)
for every fixed ε > 0. The easiest relation to prove among (21), ( 22) and ( 23) is the second one: it follows from Lemma 4.2 below by using the obvious inequality d GH ≤ d H .

PROOF OF (21). The convergence stated in (21) is the weak convergence of probability measures on the Gromov-Hausdorff space. A natural way to deal with it could be working with the Gromov-Hausdorff-Prohorov metric; see [START_REF] Abraham | A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces[END_REF][START_REF] Greven | Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)[END_REF][START_REF] Miermont | Tessellations of random maps of arbitrary genus[END_REF]. However, in our setting we are able to avoid this heavy machinery by an appeal to a version of the Skorokhod representation theorem.

For k ∈ N, put R (d) k := X (d) k 2 . Let M p := M p ([0, ∞)× (0, ∞]
) be the space of locally finite point measures on [0, ∞) × (0, ∞], endowed with the vague topology. This space is known to be complete and separable; see Proposition 3.17 in [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF]. Furthermore, under the assumption ( 6), the following convergence in distribution on M p holds true:

∑ k≥1 δ (k/n,(R (d) k ) 2 /a(n)) =⇒ P, d → ∞;
see Proposition 3.21 in the same reference. Now we want to apply a version of Skorokhod's representation theorem, which will allow us to pass to a new probability space on which the distributional convergence above can be replaced by the a.s. convergence. Note that the left-hand side of the latter formula can be viewed as an image of a measurable map φ n from (ℓ 2 ) N to M p , defined by

φ n (X (d) 1 , X (d) 2 , . . .) = ∑ k≥1 δ (k/n, X (d) k 2 2 /a(n))
.

Thus, applying an extended version of the Skorokhod representation theorem, stated in Lemma 5.1, we can pass to a new probability space (Ω, F , P) which accommodates the following objects:

• for every d ∈ N, a distributional copy (X (d) k ) k∈N of the sequence (X (d) k ) k∈N ; • a distributional copy P := ∑ k δ (x k ,y k ) of the Poisson point process P; such that with R (d) k := X (d) k 2 , k ∈ N, it holds P n := ∑ k≥1 δ (k/n,(R (d) k ) 2 /a(n)) -→ ∑ k δ (x k ,y k ) , P -a.s. as d → ∞. ( 24
) Define M (d) k (s) and R (s) ν (T ) in the obvious manner via (X (d)
k ) k∈N and P, respectively. More precisely, put

M (d) k (s) :=      ℓ ∑ j=1 X (d) j ½ {(R (d) j ) 2 ≥sa(n)} a(n) : ℓ = 0, . . . , k      , k ∈ N, and
R (s) ν (T ) := ∑ k: x k ≤t e k y k ½ {y k >s} : 0 ≤ t ≤ T .
We shall prove that (24) yields

d GH (M (d) ⌊nT ⌋ (s), R (s)
ν (T ))

P -→ 0, d → ∞, (25)
in the Gromov-Hausdorff space, for every fixed s > 0. The latter is clearly sufficient for (21), since

E f (M (d) ⌊nT ⌋ (s)) = E f (M (d) ⌊nT ⌋ (s)) -→ E f (R (s) ν (T )) = E f (R (s) ν (T )), d → ∞,
for every bounded continuous f , with E denoting the expectation with respect to P.

Let Ω ′ be an event of probability one on the new probability space such that (24) holds for all ω ∈ Ω ′ , and fix any ω ∈ Ω ′ . For notational simplicity, we suppress the dependence on ω below. According to Proposition 3.13 in [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF], there exist an integer P = P(ω) ∈ N and an enumeration of the atoms of P and

P n in [0, T ] × [s, ∞)
such that for all sufficiently large n ∈ N,

P n (• ∩ ([0, T ] × [s, ∞))) = P ∑ j=1 δ (k j (n)/n,(R (d) k j (n) ) 2 /a(n))
and

P(• ∩ ([0, T ] × [s, ∞))) = P ∑ j=1 δ (x k j ,y k j ) ,
and, moreover,

lim d→∞   k j (n) n , (R (d) k j (n) ) 2 a(n)   = (x k j , y k j ), j = 1, . . . , P. ( 26
)
Without loss of generality, we assume that the enumeration is chosen such that

x k 1 < x k 2 < • • • < x k P . Then it is clear that M (d) ⌊nT ⌋ (s) =    ℓ ∑ j=1 X (d) k j (n) a(n) : ℓ = 0, . . . , P    . Also, R (s)
ν (T ) = ℓ ∑ j=1
e k j y k j : ℓ = 0, . . . , P .

Define the bijective mapping

I n : R (s) ν (T ) -→ M (d) ⌊nT ⌋ (s) by I n ℓ ∑ j=1 e k j y k j = ℓ ∑ j=1 X (d) k j (n) a(n) , ℓ = 0, . . . , P.
By Corollary 7.3.28 on p. 258 of [START_REF] Burago | A course in metric geometry[END_REF], the Gromov-Hausdorff distance between M (d) ⌊nT ⌋ (s) and R (s) ν (T ) is bounded above by twice the distortion of the map I n , that is

d GH (M (d) ⌊nT ⌋ (s), R (s) ν (T )) ≤ 2 sup 0≤ℓ≤m≤P I n m ∑ j=1 e k j y k j -I n ℓ ∑ j=1 e k j y k j 2 - m ∑ j=1 e k j y k j - ℓ ∑ j=1 e k j y k j 2 = 2 sup 0≤ℓ≤m≤P m ∑ j=ℓ+1 X (d) k j (n) a(n) 2 - m ∑ j=ℓ+1 e k j y k j 2 . Note that sup 0≤ℓ≤m≤P m ∑ j=ℓ+1 X (d) k j (n) a(n) 2 - m ∑ j=ℓ+1 e k j y k j 2 2 ≤ sup 0≤ℓ≤m≤P m ∑ j=ℓ+1 X (d) k j (n) a(n) 2 2 - m ∑ j=ℓ+1 e k j y k j 2 2 = sup 0≤ℓ≤m≤P m ∑ j=ℓ+1   (R (d) k j (n) ) 2 a(n) -y k j   + 1 a(n) ∑ i, j∈{ℓ+1,...,m},i = j X (d) k i (n) , X (d) k j (n) 2 .
In the first step we used the inequality |x -y| 2 ≤ |x -y||x + y| = |x 2y 2 | for x, y ≥ 0. In view of (26), the first sum on the right-hand side converges to 0 for all ω ∈ Ω ′ . Therefore, it remains to check that

1 a(n) sup 0≤ℓ≤m≤P ∑ i, j∈{ℓ+1,...,m},i = j X (d) k i (n) , X (d) k j (n) 2 P -→ 0, d → ∞. (27)
Using that

∑ i, j∈{ℓ+1,...,m},i = j X (d) k i (n) , X (d) k j (n) 2 ≤ P 2 sup i, j∈{ℓ+1,...,m},i = j X (d) k i (n) , X (d) k j (n) 2 ≤ P 2 sup i, j∈{1,...,⌊nT ⌋},i = j X (d) i ½ { X (d) i 2 2 ≥sa(n)} , X (d) j ½ { X (d) j 2 2 ≥sa(n)} 2 ,
and recalling that (X

(d) k ) k∈N is a distributional copy of (X (d) k ) k∈N , we see that (27) is a consequence of 1 a(n) sup i, j∈{1,...,⌊nT ⌋},i = j X (d) i ½ { X (d) i 2 2 ≥sa(n)} , X (d) j ½ { X (d) j 2 2 ≥sa(n)} 2 P -→ 0, d → ∞. (28)
To check the latter we note that, for every fixed ϑ > 0,

P sup i, j∈{1,...,⌊nT ⌋},i = j X (d) i ½ { X (d) i 2 2 ≥sa(n)} , X (d) j ½ { X (d) j 2 2 ≥sa(n)} 2 > ϑ a(n) ≤ T 2 n 2 P{ X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), | X (d) 1 , X (d) 2 2 | > ϑ a(n)} = T 2 n 2 P{ X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), X (d) 1 2 X (d) 2 2 ≥ Sa(n), | X (d) 1 , X (d) 2 2 | > ϑ a(n)} + T 2 n 2 P{ X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), X (d) 1 2 X (d) 2 2 < Sa(n), | X (d) 1 , X (d) 2 2 | > ϑ a(n)},
where S > s is fixed. Note that by ( 6)

n 2 P{(a(n)) -1 ( X (d) 1 2 2 , X (d) 2 2 2 ) ∈ •} v -→ d→∞ (ν ⊗ ν)(•), (29)
on (0, ∞] × (0, ∞] and, therefore,

lim d→∞ n 2 P{ X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), X (d) 1 2 X (d) 2 2 ≥ Sa(n)} = (ν ⊗ ν) [s, ∞) × [s, ∞) ∩ {(x, y) ∈ (0, ∞] 2 : xy ≥ S 2 } ,
for all but countably many S > s. The right-hand side converges to zero as S → ∞. On the other hand,

n 2 P X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), X (d) 1 2 X (d) 2 2 < Sa(n), | X (d) 1 , X (d) 2 2 | > ϑ a(n) ≤ n 2 P X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), X (d) 1 2 X (d) 2 2 < Sa(n), Θ (d) 1 , Θ (d) 2 2
> ϑ S -1

≤ n 2 P X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n), Θ (d) 1 , Θ (d) 2 2 > ϑ S -1 ≤ C(s)P Θ (d) 1 , Θ (d) 2 2 > ϑ S -1 X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n) ,
where C(s) is a positive constant which depends on s. The last passage is again a consequence of [START_REF] Blumenthal | Some theorems on stable processes[END_REF]. As d → ∞, the conditional probability on the right-hand side of the last display converges to zero by [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. This completes the proof of (28) as well as of (21).

PROOF OF (23). It is clear that

d GH (M (d) ⌊nT ⌋ (s), M (d) ⌊nT ⌋ ) ≤ d H (M (d) ⌊nT ⌋ (s), M (d) ⌊nT ⌋ ) ≤ 1 a(n) max k=1,...,⌊nT ⌋ k ∑ j=1 X (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 . Further, max k=1,...,⌊nT ⌋ k ∑ j=1 X (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 ≤ max k=1,...,⌊nT ⌋ k ∑ j=1 X (d) j ½ { X (d) j 2 2 ≤sa(n)} -EX (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 + max k=1,...,⌊nT ⌋ k ∑ j=1 E X (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 =: Z (d) 1 (s) + Z (d) 2 (s).
In order to estimate Z (d)

1 (s), we note that k ∑ j=1 X (d) j ½ { X (d) j 2 2 ≤sa(n)} -EX (d) j ½ { X (d) j 2 2 ≤sa(n)} , k ∈ N,
is an R d -valued martingale with respect to the natural filtration of the sequence (X (d) k ) k∈N . Thus, by Jensen's inequality,

k ∑ j=1 X (d) j ½ { X (d) j 2 2 ≤sa(n)} -EX (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 2
, k ∈ N, is a submartingale. By Doob's maximal inequality

P Z (d) 1 (s) ≥ 2 -1 ε a(n) = P (Z (d) 1 (s)) 2 ≥ 4 -1 ε 2 a(n) ≤ 4 ε 2 a(n) E ⌊nT ⌋ ∑ j=1 X (d) j ½ { X (d) j 2 2 ≤sa(n)} -EX (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 2 = 4⌊nT ⌋ ε 2 a(n) E X (d) ½ { X (d) 2 2 ≤sa(n)} -EX (d) ½ { X (d) 2 2 ≤sa(n)} 2 2 ≤ 8⌊nT ⌋ ε 2 a(n) E X (d) 2 2 ½ { X (d) 2 2 ≤sa(n)} .
Condition [START_REF] Burago | A course in metric geometry[END_REF] implies that lim

s→0+ lim sup d→∞ P Z (d) 1 (s) ≥ 2 -1 ε a(n) = 0.
It remains to show that lim

s→0+ lim sup d→∞ Z (d) 2 (s) a(n) = 0. ( 30
)
Note that

Z (d) 2 (s) = max k=1,...,⌊nT ⌋ k ∑ j=1 E X (d) j ½ { X (d) j 2 2 ≤sa(n)} 2 ≤ T n EX (d) ½ { X (d) 2 2 ≤sa(n)} 2 .
Thus, (30) follows from [START_REF] Glynn | Extensions of the queueing relations L = λW and H = λ G[END_REF].

In Section 2.1, we remarked that if the distribution of X (d) 2 2 does not depend on d, then (10) implies [START_REF] Burago | A course in metric geometry[END_REF]. Here is the proof of this fact. Using p. 579 in [START_REF] Feller | An introduction to probability theory and its applications[END_REF], Condition [START_REF] Greven | Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)[END_REF] implies

E X (d) 2 2 ½ { X (d) 2 2 ≤sa(n)} ∼ α 1 -α sa(n)P{ X (d) 2 2 ≥ sa(n)}, d → ∞. Therefore, lim d→∞ n a(n) E X (d) 2 2 ½ { X (d) 2 2 ≤sa(n)} = αs 1-α (1 -α) , (31)
and the right-hand side converges to zero, as s → 0+.

PROPERTIES OF THE CRINKLED SUBORDINATOR

Lemma 4.1. For every T > 0, the set R ν (T ) in Definition 2.2 is a.s. compact in ℓ 2 .

Proof. It suffices to show that the set {C ν (t) : 0 ≤ t ≤ T } is a.s. totally bounded in ℓ 2 , that is, for every ε > 0 there exists an a.s. finite ε-net for {C ν (t) : 0 ≤ t ≤ T }.

By Lebesgue dominated convergence theorem,

lim s→0+ ∑ k: x k ≤T y k ½ {y k ≤s} = 0, a.s. (32)
In particular, for every ε > 0, there exists a (random) δ > 0 such that

∑ k: x k ≤T y k ½ {y k ≤δ } ≤ ε 2 .
Recall that the set in definition (20) is a.s. finite, and let us show that R (δ ) ν (T ) is the sought a.s. finite ε-net for {C ν (t) : 0 ≤ t ≤ T }. For every t ∈ [0, T ], we have

C ν (t) -∑ k: x k ≤t e k √ y k ½ {y k >δ } 2 2 = ∑ k: x k ≤t e k √ y k ½ {y k ≤δ } 2 2 = ∑ k: x k ≤t y k ½ {y k ≤δ } ≤ ∑ k: x k ≤T y k ½ {y k ≤δ } ≤ ε 2 ,
and the proof is complete. Proof. The proof follows from the inequalities

d H (R (s) ν (T )), R ν (T )) ≤ sup t∈[0,T ] C ν (t) -∑ k: x k ≤t e k √ y k ½ {y k >s} 2 = sup t∈[0,T ] ∑ k: x k ≤t y k ½ {y k ≤s} = ∑ k: x k ≤T y k ½ {y k ≤s} .
In view of (32), the right-hand side converges to 0 a.s. as s → 0+. Proof. This follows from the fact that the Hausdorff dimension of R ν (T ) is equal to α; see Theorem 3.2 in [START_REF] Blumenthal | Some theorems on stable processes[END_REF]. [START_REF] Hu | Strong representation of weak convergence[END_REF]; see also [START_REF] Bai | Strong representation of week convergence[END_REF], where this result appeared for the first time.

Lemma 5.1. Let (M, ρ) and (M 1 , ρ 1 ) be two complete separable metric spaces, and φ n : M → M 1 Borel- measurable mappings, n ∈ N. Suppose that (µ n ) n∈N is a sequence of probability measures on (M, ρ) and µ 0 is a probability measure on (M 1 , ρ 1 ) such that µ n • φ -1 n converges weakly to µ 0 as n → ∞. Then there exist a sequence of M-valued random variables (X n ) n∈N , an M 1 -valued random variable X 0 , all defined on a common probability space (Ω, F , P), such that X n has distribution µ n for all n ∈ N 0 , and φ n (X n ) → X 0 a.s. as n → ∞.

A calculation for Example 3.

Lemma 5.2. In the setting of Example 3, formula (8) holds true for every fixed s > 0 and ε > 0.

Proof. We start by noting that [START_REF] Resnick | Heavy-tail phenomena: Probabilistic and statistical modeling[END_REF] implies

P{|ξ | > t} ∼ t -2α , t → ∞,
and, in particular, E|ξ | α < ∞. Recall that a(n) = n 1/α . By conditional Markov's inequality,

P d ∑ k=1 ξ 1,k ξ 2,k > εd 1/α X (d) 1 2 X (d) 2 2 X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n) ≤ P d ∑ k=1 ξ 1,k ξ 2,k > d 1/α εsa(n) X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n) ≤ 1 d(εsa(n)) α E d ∑ k=1 ξ 1,k ξ 2,k α X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n) = 1 d(εsa(n)) α E ∑ d k=1 ξ 1,k ξ 2,k α ½ { X (d) 1 2 2 ≥sa(n), X (d) 2 2 2 ≥sa(n)} P{ X (d) 1 2 2 ≥ sa(n), X (d) 2 2 2 ≥ sa(n)} ∼ s -2α n 2 d(εsa(n)) α E d ∑ k=1 ξ 1,k ξ 2,k α ½ { X (d) 1 2 2 ≥sa(n), X (d) 2 2 2 ≥sa(n)} = n d • 1 s 3α ε α E d ∑ k=1 ξ 1,k ξ 2,k α ½ { X (d) 1 2 2 ≥sa(n), X (d) 2 2 2 ≥sa(n)} , d → ∞,
where for the asymptotic equivalence we used [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF]. Using the subadditivity of x → x α , α ∈ (0, 1), we conclude that

E d ∑ k=1 ξ 1,k ξ 2,k α ½ { X (d) 1 2 2 ≥sa(n), X (d) 2 2 2 ≥sa(n)} ≤ d ∑ k=1 E|ξ 1,k | α E|ξ 2,k | α ½ { X (d)

n

  for each d ∈ N. Let n = n(d) be an arbitrary sequence of positive integers such that n(d) → ∞ as d → ∞. By default, the notation d → ∞ implies that also n = n(d) → ∞. Denote by S (d) n the piecewise-linear interpolation obtained by joining the consecutive points S by line segments. By construction, every S (d) n

  ) (c) With the sequence (a(k)) k∈N defined in part (

2 . 3 .

 23 dimension of R ν (T ); see Proposition 4.3 below. A formula for the Hausdorff dimension of R ν (T ) is available; see Section 5.1.2 in [3]. Main results. For the sequence of random walks given by (1) and satisfying assumptions (a), (b) and (c) above, define a sequence of finite random metric subspaces of ℓ 2 via M (d)

Theorem 2 . 3 .

 23 Assume that conditions (a), (b) and (c) are fulfilled. Fix T > 0. Then, weakly on the Gromov-Hausdorff space of compact metric spaces, it holds

2. 4 .

 4 Examples. Below we consider three families of random walks satisfying the assumptions (a), (b) and (c).

2. 5 . 2 . 5 . 1 .

 5251 Compact subsets of ℓ 2 and their convergence. In this subsection, we recall the definitions of the Hausdorff and Gromov-Hausdorff metrics, and the notion of Hausdorff distance up to isometry in ℓ 2 . Hausdorff and Gromov-Hausdorff metrics. Let (M, ρ) be an arbitrary metric space. Denote by K (M) the set of non-empty compact subsets of M. Let d H denote the Hausdorff distance between elements of K (M), defined by

Lemma 4 . 2 .

 42 For every T > 0)), R ν (T )) = 0, a.s.

Let d denoteProposition 4 . 3 .

 43 the Hausdorff dimension of the set R ν (T ) (the closed range of the subordinator S ν ; see[START_REF] Kabluchko | Random walks in the high-dimensional limit I: The Wiener spiral[END_REF]) regarded as a subset of [0, ∞) endowed with the Euclidean distance (t, s) → |t -s|. A formula for d can be found in Corollary 5.3 of[START_REF] Bertoin | Subordinators: examples and applications[END_REF]. The Hausdorff dimension of the set R ν (T ) is a.s. equal to 2d, for every fixed T > 0.Proof. By the very definition of the Hausdorff dimension, if a set A ⊂ M has Hausdorff dimension d in a metric space (M, ρ) and β ∈ (0, 1] is a fixed parameter, then A has Hausdorff dimension d/β in the metric space (M, ρ β ). Applying this observation with β = 1/2, we see that as a subset of [0, ∞) endowed with the distance (t, s) → |t -s|, the set R ν (T ) has Hausdorff dimension 2d. It remains to note that isometric sets have the same Hausdorff dimensions and ( R ν (T ), | • -• |) is isometric to (R ν (T ), ρ 2 ); see(14).

Corollary 4 . 4 .

 44 If C ν is a crinkled α-stable subordinator with α ∈ (0, 1), then the Hausdorff dimension of the set R ν (T ) is a.s. equal to 2α, for every fixed T > 0.

Remark 4 . 5 . 1 .

 451 The same arguments in conjunction with the fact that the Wiener spiral is isometric to [0, 1] with the metric (t, s) → |t -s| demonstrate that the Hausdorff dimension of the Wiener spiral is equal to 2An extension of the Skorokhod representation theorem. The following lemma is an extended version of the Skorokhod representation theorem. It is a light version of Theorem 1.1 in the paper

2 = d E|ξ 1 , 1 | α ½ {ξ 2 1, 1

 21121 E|ξ 1,1 | α ½ { X (d)

  + • • • + ξ 2 1,d ≥ d 1/α sa(n)/2} ∼ (d -1) • P{ξ 2 ≥ d 1/α sa(n)/2} ∼ (s/2) -α /n.

	By Eq. (4) in [11],
	P{ξ 2 1,2 It remains to check that
	lim d→∞	n E|ξ | α ½ {ξ 2 ≥d 1/α sa(n)/2}	2	= 0.
	To this end, it is clearly sufficient to show that
	lim A→∞	lim n→∞	n E|ξ | α ½ {ξ 2 ≥Aa(n)}	2	= 0.	(33)

+•••+ξ 2 1,d ≥d 1/α sa(n)} 2 .

Further, since (a + b) 2 ≤ 2a 2 + 2b 2 , we have

d E|ξ 1,1 | α ½ {ξ 2 1,1 +•••+ξ 2 1,d ≥d 1/α sa(n)} 2 ≤ 2d E|ξ 1,1 | α ½ {ξ 2 1,1 ≥d 1/α sa(n)/2} 2 + 2d E|ξ 1,1 | α ½ {ξ 2 1,2 +•••+ξ 2 1,d ≥d 1/α sa(n)/2} 2 = 2d E|ξ 1,1 | α ½ {ξ 2 1,1 ≥d 1/α sa(n)/2} 2 + 2d E|ξ 1,1 | α P{ξ 2 1,2 + • • • + ξ 2 1,d ≥ d 1/α sa(n)/2} 2 .

≤sa(n)} 2 = 0. (9)

ACKNOWLEDGMENTS AM gratefully acknowledges the financial support and hospitality of the University of Angers during his stay in December 2022-March 2023. ZK was supported by the German Research Foundation under Germany's Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics -Geometry -Structure and by the DFG priority program SPP 2265 Random Geometric Systems. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under the Grant Agreement No. 759702 and from Centre Henri Lebesgue, programme ANR-11-LABX-0020-0.

This can be accomplished by an appeal to formula (5.21) on p. 579 in [START_REF] Feller | An introduction to probability theory and its applications[END_REF] applied with β = α/2. According to this formula,

Thus,

and (33) follows.