Susanna M Badalyan
email: s.badalyan@ysu.am

Anush Barkhudaryan

Sylvie Rapior

Medicinal Macrofungi as Cosmeceuticals: A Review

Keywords: anti-aging, bioactive compounds, cosmeceuticals, enzyme inhibitors, medicinal mushrooms and macrofungi, nutracosmetics, nutricosmetics ABA, antibacterial activity, ABTS, 2,2'-azino-bis dermatitis, AIA, anti-inflammatory activity, AMA, antimicrobial activity, AOA, antioxidant activity, CWE, cold-water extract, EPS, exopolysaccharide, FIPs, fungal immunomodulatory proteins, FMG, fucomannogalactan, HaCaT, human keratinocytes, HWE, hot-water extracts, MRSA, methicillin-resistant Staphylococcus aureus, NF-κB, nuclear factor-κB, NO, nitric oxid, PUFA, polyunsaturated fatty acids, ROS, reactive oxygen species, SIRT1, Sirtuin 1, SPF, sun protection factor, TCM, traditional Chinese medicine, UV, ultraviolet I

come

INTRODUCTION

The natural process of aging, ultraviolet (UV) radiation and hormonal alterations affect the skin and other body parts, which result in the generation of reactive oxygen species (ROS) causing oxidative stress. [START_REF] Usman | Nutritional, medicinal, and cosmetic value of bioactive compounds in Button Mushroom (Agaricus bisporus): A review[END_REF] Cosmetic skin and hair care formulations based on naturally derived active ingredients, including plants and fungi, are currently preferred by consumers for their antioxidant effects.

Currently, the cosmetic industry is in a constant search for bioactive ingredients to design novel cosmeceuticals possessing lower toxicity. [START_REF] Kaur | Nanotechnology: A new paradigm in cosmeceuticals[END_REF] Cosmeceuticals are products between cosmetics and pharmaceuticals containing bioingredients with anti-aging, anti-inflammatory, antioxidant, and photoprotective effects. They can be extracted from different organisms or obtained biotechnologically (cultivation, fermentation, enzymatic synthesis, etc.).

Recent progress in the biology and biotechnology of macrofungi (commonly referred to as mushrooms), genomics, proteomics and myco-pharmacology has contributed to their usage in medicine and food industries. [START_REF] Niego | Macrofungi as a nutraceutical source: Promising bioactive compounds and market value[END_REF] Macrofungi are also considered a valuable source of multifunctional cosmetic ingredients to be exploited as cosmeceuticals. [START_REF] Wu | Mushroom cosmetics: the present and future[END_REF][START_REF] Badalyan | Biotechnological exploitation of macrofungi for the production of food, pharmaceuticals and cosmeceuticals[END_REF][START_REF] Angelini | Micoterapici per la medicina estetica sistemica: i funghi contro l'invecchiamento cutaneo[END_REF] They are mainly used in skin and hair care products, and the trend for their industrial usage is expected to increase over the years. [START_REF] Pareek | Mushrooming of herbals in new emerging markets of cosmeceuticals[END_REF] Wild or cultivable, edible or inedible, medicinal or poisonous macrofungi are incorporated in many cosmetic products as producers of bioactive compounds (phenolics, glucans and other polysaccharides and terpenoids) with anti-aging, antibacterial, antifungal, anti-inflammatory, antioxidant, hypocholesterolemic, hypoglycemic, immune-modulating as well as anticollagenase, anti-elastase, anti-hyaluronidase, anti-tyrosinase, wound-healing and other activities. [START_REF] Badalyan | Biotechnological exploitation of macrofungi for the production of food, pharmaceuticals and cosmeceuticals[END_REF][START_REF] Taofiq | A mushroom extracts and compounds in cosmetics, cosmeceuticals and nutricosmetics. A review[END_REF][START_REF] Badalyan | Perspectives in the usage of bioactive substances of medicinal mushrooms in pharmaceutical and cosmetic industries[END_REF][START_REF] Badalyan | Recent progress in research on the pharmacological potential of mushrooms and prospects for their clinical application[END_REF][START_REF] Sharifi-Rad | Mushrooms-rich preparations on wound healing: from nutritional to medicinal attributes[END_REF] However, there are numerous undescribed or not yet cultivated mushroom species with a potential cosmetic application. [START_REF] Angelini | Micoterapici per la medicina estetica sistemica: i funghi contro l'invecchiamento cutaneo[END_REF][START_REF] Hyde | Fungi -an unusual source for cosmetics[END_REF][START_REF] Lindequist | Macrofungi in pharmacy, medicine, cosmetics and nutrition -An appraisal[END_REF] Edible mushrooms are considered well-known untapped source of valuable food and nutraceuticals which can improve human health and quality of life. However, inedible or poisonous species (i.e., Amanita muscaria, Ganoderma lucidum, Inonotus obliquus, Phellinus igniarius, Ramaria formosa, etc.) represent a source of bioactive secondary metabolites to develop myco-pharmaceuticals (dietary supplements or nutriceuticals) with health-enhancing and therapeutic properties. [START_REF] Badalyan | Biotechnological exploitation of macrofungi for the production of food, pharmaceuticals and cosmeceuticals[END_REF][START_REF] Badalyan | Perspectives in the usage of bioactive substances of medicinal mushrooms in pharmaceutical and cosmetic industries[END_REF] Macrofungi-derived nutraceuticals and nutriceuticals are obtained from mycelia and/or fruiting bodies and administered per os in the form of capsules, tablets, or extracts. [START_REF] Chang | Current and future research trends in agricultural and biomedical applications of medicinal mushrooms and mushroom products (Review)[END_REF][START_REF] Wong | Mushroom nutriceuticals in Malaysia[END_REF] Currently, macrofungi-derived numerous commercial products, such as cosmeceuticals (applied topically, i.e., creams, lotions, and ointments) and nutracosmetics (administered per os) are available in the market. Their usage is significantly high due to minimal regulation and safety compared to traditional drugs. [START_REF] Brandt | Cosmeceuticals: current trends and market analysis[END_REF][START_REF] Elkhateeb | What medicinal mushroom can do?[END_REF] The current review reports the main genera and species of macrofungi presently used, or patented to be used, in formulations of cosmeceuticals and nutracosmetics, such as Agaricus subrufescens, A. bisporus, Choiromyces maeandriformis, Ophiocordyceps sinensis, Ganoderma lingzhi, G. lucidum, Grifola frondosa, Hypsizygus ulmarium, Inonotus obliquus, Lentinula edodes, Polyporus spp., Phellinus spp., Schizophyllum commune, Trametes versicolor, Tremella fuciformis, and Tuber spp. (Table 1).

We also describe the main ranges of human cosmetic and cosmeceutical products incorporating macrofungi or their extracts as anti-aging, antioxidant, hair and skin care, anti-acne or perfumers. Cosmetic brands include Bliss (Hut.com Ltd., Cheshire, UK), La Roche (F.

Hoffmann-La Roche Ltd., Basel, Switzerland), Nu-Derm (Obagi Medical Products Inc., Irvine, CA, USA), SensiClear (Mission Scientific Skincare Inc., Gold River, CA, USA) and others (Table 2).

II. MACROFUNGI-DERIVED BIOACTIVE COMPOUNDS AS POTENTIAL

COSMECEUTICALS

Over the past years, growing consumers' demand for natural low toxicity organic products with effective skin and hair repairing effects have caused cosmetic industry to search for alternative ingredients. Macrofungi are considered sources of bioactive compounds increasingly used as cosmeceutical ingredients. Scientific data showed that medicinal, edible and inedible mushrooms are resources of raw extracts, fractions of extracts, bioactive molecules, i.e. fatty acids, phenolics, peptides, polysaccharides, terpenoids, vitamins and volatile organic compounds, some of them being also enzyme-inhibitory compounds; that large spectrum of bioactive natural sources possess significant anti-aging, anti-inflammatory, antioxidant, antipigmentation, antimicrobial, anti-wrinkle, moisturizing, and skin-whitening effects, may act as inhibitors of collagenase, elastase, hyaluronidase and be used to regulate tyrosinase activity; consequently, they may be potentially used as natural cosmeceuticals or nutracosmetics.

However, the mechanisms behind the mentioned bioactivities are still unknown.

Antioxidants against harmful free radicals and cellular damage are the most demanded cosmeceuticals since their topical administration is essential for protection of aging skin. Skin hyperpigmentation is caused by several factors that upregulate melanogenesis and controlling this unwanted process is a major challenge in dermatology and cosmetology to develop safe and effective anti-melanogenic products. Macrofungi-derived compounds, such as p-coumaric acid, can be used to reduce oxidative stress and inflammation, as anti-aging, and skin whitening agents. [START_REF] Alvarez | The suitability of lipases for the synthesis of bioactive compounds with cosmeceutical applications[END_REF] Previous studies have shown that p-coumaric acid inhibited melanin synthesis in murine melanoma cells, human epidermal melanocytes, showed efficient transdermal delivery and functional efficacy in reducing erythema development and skin UV-induced pigmentation, therefore has a potential to be used as a skin-lightening ingredient. [START_REF] Boo | p-Coumaric acid as an active ingredient in cosmetics: A review focusing on its antimelanogenic effects[END_REF] The bioactive peptides with plant and animal origins can have a beneficial effect on multiple physiological pathways in the skin which makes them ideal candidates for development of cosmeceuticals. Significant antioxidant and dermatoprotective effects of these peptides after topical application or per os administration were revealed which suggests that they can be used in skin care formulations as functional ingredients. However, the mechanisms underlying the skin-protective effect have not been completely elucidated. [START_REF] Aguilar-Toalá | Potential role of natural bioactive peptides for development of cosmeceutical skin products[END_REF] Phenolic compounds and tocopherols derived from edible mushrooms Boletus edulis, Lentinula edodes and Xerocomus badius are the most responsible for their AOA. [START_REF] Heleno | Nutritional value, bioactive compounds and antioxidant properties of three edible mushrooms from Poland[END_REF] The exopolysaccharide (EPS-1) isolated from fermentation broths of rare, valuable edible medicinal polypore fungus Abortiporus biennis also possesses prominent radical-scavenging and macrophage-stimulatory activities in vitro and could be explored as a functional ingredient with potential applications in medical and cosmetic industries. [START_REF] Long | Preliminary characterization of exopolysaccharides produced by Abortiporus biennis in submerged fermentation[END_REF] Sclerotia of medicinal polyporoid species Polyporus umbellatus have been used in traditional Chinese medicine (TCM) to treat oedema due to its diuretic effect. It contained bioactive compounds responsible for AOA and AIA and was used as an ingredient in several biotech products. [START_REF] Bandara | Polyporus umbellatus, an edible-medicinal cultivated mushroom with multiple developed health-care products as food, medicine and cosmetics: a review[END_REF] Polysaccharides, triterpenes, and fungal immunomodulatory proteins (FIPs) found in Ganoderma species have been exploited for potential ingredients in the cosmetic industry to develop new cosmeceuticals and nutracosmetics with antioxidant, antibacterial, antiinflammatory, and melanin-inhibitory effects. However, numerous questions need to be addressed before FIPs can be accepted and used in cosmetic products. [START_REF] Li | Ganoderma proteins and their potential applications in cosmetics[END_REF] Tyrosinase is an enzyme which has a key role in the formation of melanin. Ganoderma lucidum extract was found to have potent anti-tyrosinase activity and is part of several cosmetic, skin-lightening beauty products in China. [START_REF] Jiang | Ganoderma lucidum (Reishi Mushroom): potential application as health supplement and cosmeceutical ingredient[END_REF][START_REF] Chien | Effects of tyrosinase activity by the extracts of Ganoderma lucidum and related mushrooms[END_REF] G. lucidum also contains α/β-D-glucans, alkaloids, triterpenoids (ganoderic acids, ganoderenic acids, ganoderol, ganoderiol, lucidenic acids), sterols (ergosterol), proteins (LZ-8, LZ-9), nucleosides (adenosine, inosine, uridine), and nucleotides (adenine, guanine) with different pharmacological potential. [START_REF] Badalyan | Perspectives in the usage of bioactive substances of medicinal mushrooms in pharmaceutical and cosmetic industries[END_REF] Various marketed products of G.

lucidum with nutritional and cosmeceutical value are already available in the market. [START_REF] Ahmad | Ganoderma lucidum (reishi) an edible mushroom; a comprehensive and critical review of its nutritional, cosmeceutical, mycochemical, pharmacological, clinical, and toxicological properties[END_REF] AOA and ABA of secondary metabolites of Pleurotus species have also been reported. [START_REF] Cardoso | Antioxidant and antimicrobial influence on oyster mushrooms (Pleurotus ostreatus) from substrate supplementation of calcium silicate[END_REF] In vitro safety of ethanolic extracts derived from G. lucidum and P. ostreatus incorporated into base cosmetic creams was evaluated. The absence of toxicity for keratinocytes and fibroblasts in a concentration-dependent manner was revealed. Protocatechuic and syringic acids were the only compounds permeating from G. lucidum extract. These results have shown the safety of studied extracts as skin care ingredients. [START_REF] Taofiq | Mushroom ethanolic extracts as cosmeceuticals ingredients: Safety and ex vivo skin permeation studies[END_REF] Extracellular polysaccharides derived from medicinal mushroom Grifola frondosa (GF-EPS) were applicable in the development of functional cosmeceuticals. The GF-EPS, together with mycelial extract of G. frondosa (GF-MPS), showed AOA, stimulatory effect on collagen biosynthesis and anti-melanogenesis activities, without significant cytotoxicity. GF-EPS and GF-MPS were shown to be promising cosmetic ingredients. [START_REF] Kim | Grifola frondosa polysaccharides as cosmeceuticals[END_REF] Chemical constituents isolated from medicinal bracket fungus Phellinus igniarius could also be used as ingredients to develop antiinflammatory cosmetic products. [START_REF] Jiang | Anti-inflammatory activity of chemical constituents isolated from the willow bracket medicinal mushroom Phellinus igniarius (Agaricomycetes)[END_REF] Edible medicinal cultivated mushroom Schizophyllum commune contains phenolics and β-glucans beneficial to skin health. However, data on the development of Sch. commune as a cosmeceutical ingredient is lacking. The optimal water extraction conditions at 30°C for 1 h were suggested for effective antioxidant and anti-tyirosinase properties of β-glucans in Sch.

commune. [START_REF] Razak | Effects of different extracting conditions on anti-tyrosinase and antioxidant activities of Schizophyllum commune fruit bodies[END_REF] The β-glucans isolated from ethanol extracts of Sch. commune also showed potent AOA and may be used as cosmeceuticals. [START_REF] Phuket | Biological activity of ß-glucans from edible mushroom, Schizophyllum commune in Thailand[END_REF] Edible mushroom Tremella fuciformis has attracted increasing attention due to its various bioactivities. The polysaccharides extracted from T. fuciformis, mainly consisting of mannose and uronic acid, could efficiently reduce water and collagen losses of the skin, and inhibit the increase of glycosaminoglycans. Moreover, after oral treatment with polysaccharides a histopathological study showed that, UV-induced structural alterations of the skin were alleviated, as well as repairing endogenous collagen breakdown and increasing AOA compared to the irradiated control group (without treatment). Thus, polysaccharides from T. fuciformis can be used as a skin-protective cosmetic ingridient. [START_REF] Wen | Effect of polysaccharides from Tremella fuciformis on UV-induced photoaging[END_REF] Atopic dermatitis (AD) is an inflammatory skin disease usually accompanied by Staphylococcus aureus (MRSA) infection due to cutaneous barrier-function damage.

Benzenoid aromatic compound, coenzyme Q 0 isolated from Antrodia cinnamomea, possesses ABA and AIA. The strongest anti-MRSA activity was achieved by inhibiting DNA polymerase, topoisomerases, reducing the ribosomal proteins, as well as downregulating the expression of interleukin (IL)-6, chemokine (C-C motif) ligand (CCL)5, and CCL17 in HaCaT cells, respectively. CoQ 0 at 0.5 μg/mL could recover the filaggrin decreased by HaCaT activation to the normal control and is effective in mitigating AD symptoms associated with bacterial infection. [START_REF] Chou | Coenzyme Q0 from Antrodia cinnamomea exhibits drug-resistant bacteria eradication and keratinocyte inflammation mitigation to ameliorate infected atopic dermatitis in mouse[END_REF] Yellow strain Flammulina velutipes (FVY) has become popular among customers due to its distinct texture different from F. velutipes white strain (FVW). The physicochemical properties and AOA of FVY and FVW strains were evaluated. The results showed that FVY-80 possesses the highest AOA based on 2,2-diphenyl-1-picrylhydrazyl (DPPH) (IC 50 = 2.22 mg/mL) and 2,2'-azino-bis-3-ethylbenzthiazoline-6-sulphonic acid (ABTS) radical assay (IC 50 = 2.04 mg/mL). FVY-80 significantly reduced the ROS content in L929 cell by 55.96% and was suggested as the best source for cosmeceutical ingredient. [START_REF] Hu | Characterization and antioxidant activities of yellow strain Flammulina velutipes (Jinhua Mushroom) polysaccharides and their effects on ROS content in L929 cell[END_REF] The study of ABA, AOA and sun protection factor (SPF) of ethyl acetate extracts of mycelia of selected edible mushrooms revealed ABA against Staphylococcus epidermidis, Escherichia coli, Micrococcus luteus and Bacillus megaterium, high SPF activity (34.02 at 200 μg/mL) and AOA, particularly of cultural broth extracted from F. velutipes, P. florida, and P.

ostreatus. The results showed the presence of bioactive ingredients in these fungi with promising application in cosmeceuticals. [START_REF] Saraswat | Assessing cosmeceuticals properties of some macrofungi for improved skin care[END_REF] Anti-melanogenetic activity and AIA of hot water extracts (HWE) of culinary-medicinal mushrooms Agaricus bisporus (white and brown), G. lucidum, Hypsizygus marmoreus, Pleurotus floridanus, and P. pulmonarius were evaluated. Brown A. bisporus reduced intracellular melanin content to the largest extent (up to 57.05 ± 3.90 %) without a cytotoxic effect on B16F10 melanoma cells, as well as showed anti-tyrosinase activity to 17.93 ± 2.65 %.

The extract obtained from brown A. bisporus has AIA decreasing the levels of NO and TNFαand can be used to develop skin-whitening and anti-inflammatory products. Other mushroom extracts showed moderate activity. [START_REF] Saad | Antimelanogenesis and anti-inflammatory activity of selected culinary-medicinal mushrooms[END_REF] The ethanolic extract from Agaricus blazei (=A. The results suggest that ABM residues can be used as an inexpensive and sustainable source of nutraceutical and cosmeceutical ingredients. [START_REF] Jiang | Ganoderma lucidum (Reishi Mushroom): potential application as health supplement and cosmeceutical ingredient[END_REF] Bioactive compounds extracted from macrofungi showed that the extract of Trametes versicolor suppressed the UV-induced cellular senescence in HaCaT cells possibly through augmenting Sirtuin 1 (SIRT1) protein expression. [START_REF] Chong | Mushroom extract inhibits ultraviolet B-induced cellular senescence in human keratinocytes[END_REF] Phenolic compounds (cinnamic acid, pcoumaric acid, p-hydroxybenzoic acid, pyrogallol) with strong AOA and ABA against Bacillus subtilis, E. coli, Listeria monocytogenes, Salmonella typhimurium, S. aureus were detected in Macrocybe lobayensis. As a natural antioxidant and antibacterial agent, this fungus was suggested to be used in pharmaceutical, cosmetic or food industries. [START_REF] Khatua | Functional ingredients and medicinal prospects of ethanol extract from Macrocybe lobayensis[END_REF] Macrofungi are also producers of bioactive compounds with anti-aging tyrosinase, hyaluronidase, collagenase and elastase enzymes inhibitory effects which are incorporated in many cosmetic products (Tables 1 and2). [START_REF] Badalyan | Biotechnological exploitation of macrofungi for the production of food, pharmaceuticals and cosmeceuticals[END_REF][START_REF] Taofiq | A mushroom extracts and compounds in cosmetics, cosmeceuticals and nutricosmetics. A review[END_REF][START_REF] Badalyan | Perspectives in the usage of bioactive substances of medicinal mushrooms in pharmaceutical and cosmetic industries[END_REF][START_REF] Sharifi-Rad | Mushrooms-rich preparations on wound healing: from nutritional to medicinal attributes[END_REF] Ascomycetous Cordyceps (= Ophiocordyceps) species is a unique source of bioactive cordycepin with anti-aging property and represents a potential ingredient for cosmeceutical applications. [START_REF] Kunhorm | Enrichment of cordycepin for cosmeceutical applications: culture systems and strategies[END_REF] Evaluation of skin care benefits of ethyl acetate, ethanol and hot water mycelial extracts from valuable medicinal fungus O. sinensis exhibited in vitro moderate or significant anti-collagenase and photoprotective activity with a SPF up to 25. The ethyl acetate extracts showed the highest anti-tyrosinase and anti-elastase activity at 0.14 -0.47 mg/mL, comparable to arbutin and epigallocatechin gallate. These results revealed the potential of O. sinensis as a source of cosmetic bioingredients for skin-care products. [START_REF] Cheng | Cosmetic and skincare benefits of cultivated mycelia from the Chinese caterpillar mushroom, Ophiocordyceps sinensis (Ascomycetes)[END_REF] The tyrosinase-inhibitory effect, AIA, AOA and ABA of ethanolic extracts obtained from A. bisporus, P. ostreatus, and L. edodes were reported and their cosmeceutical potential has been evaluated. The phenolic acids and ergosterol were suggested as basic ingredients for cosmetic creams. The formulations presented 85 -100% of these compounds can further be exploited as cosmeceuticals. [START_REF] Taofiq | A mushroom extracts and compounds in cosmetics, cosmeceuticals and nutricosmetics. A review[END_REF] AOA of mushroom-derived biomolecules could provide a skin-lightening effect by inhibition of tyrosinase, while anti-collagenase and anti-elastase activities can help to restore skin elasticity. Three different polysaccharide extracts obtained from G. lucidum were screened for their antioxidant, anti-collagenase and anti-elastase activities. In this study, smaller quantities of proteins, phenols and flavonoids were detected, as well. None of the analysed extracts showed overt toxicity to HaCaT cells at concentrations up to 2 mg/mL and displayed superior scavenging ability on ABTS and DPPH radicals. The hot water crude polysaccharide (HWCP) extract and partially purified (HWPP) fraction were found to be effective inhibitors of lipid peroxidation (LPx) with a two-fold higher inhibition of LPx compared with ascorbic acid (EC 50 = 1.65 ± 0.08 mg/mL), a common supplement in cosmeceutical formulations. HWCP showed the strongest anti-tyrosinase activity and inhibition of skin extracellular matrix degradation enzymes. Thus, G.

lucidum may be considered as a promising source of natural cosmeceutical ingredients. The ethanolic extracts obtained from A. bisporus and P. ostreatus can be used as a source of cosmeceutical ingredients since they possess antioxidant, anti-tyrosinase and antimicrobial effects. All extracts may be used in free or encapsulated forms in cosmeceutical formulations.

The encapsulated forms will provide a controlled release of the cosmeceutical ingredients resulting bioactivity control. [START_REF] Taofiq | Mushroom-based cosmeceutical ingredients: Microencapsulation and in vitro release profile[END_REF][START_REF] Kozarski | Ganoderma lucidum as a cosmeceutical: antiradical potential and inhibitory effect on hyperpigmentation and skin extracellular matrix degradation enzymes[END_REF] The chemical content and cosmetic properties, such as inhibition of tyrosinase and hyaluronidase activity, SPF values of mycelial extracts derived from Ganoderma applanatum, Laetiporus sulphureus, and T. versicolor have been reported. [START_REF] Sułkowska-Ziaja | Mycelial culture extracts of selected wood-decay mushrooms as a source of skin-protecting factors[END_REF] The total amount of phenolic acids in the extracts ranged from 2.69 mg/100 g (G. applanatum) to 10.30 mg/100 g (T.

versicolor) dry weight, the total amount of sterols at 48.40 mg/100 g (T. versicolor) to 201.04 mg/100 g (L. sulphureus) dry weight, and indoles at 2.90 mg/100 g (G. applanatum) to 16.74 mg/100 g (L. sulphureus) dry weight. Kojic acid was revealed in the extracts of L. sulphureus and G. applanatum. L. sulphureus extract caused a dose-dependent anti-hyaluronidase activity, while all other extracts inhibited tyrosinase activity. The extract of G. applanatum exhibited an SPF value of ~ 9. The screened species may be used as bioingredients in the formulation of cosmeceuticals. [START_REF] Sułkowska-Ziaja | Mycelial culture extracts of selected wood-decay mushrooms as a source of skin-protecting factors[END_REF] The assessment of HWE and cold-water extract (CWE) from fruiting bodies of cultivated macrofungi Auricularia polytricha, P. ostreatus, G. lucidum, and Sch. commune revealed their huge cosmeceutical potential. Total phenolic, polysaccharide and glucan contents were determined, the AOA was estimated. Anti-hyaluronidase activity was used as an indicator of anti-aging and anti-inflammatory properties, while anti-tyrosinase activity was evaluated to assess the whitening property of these extracts. The total polysaccharide content of P. ostreatus extracts was the highest, while extracts from G. lucidum contained the lowest level of glucan (10.12 -10.67%). CWE from Sch. commune exhibited 98.15% anti-tyrosinase activity and 94.82% AOA. G. lucidum HWE revealed the strongest anti-hyaluronidase activity (72.78%). The cosmeceutical properties of these mushrooms can be mainly attributed to the combination of polysaccharides and phenolics. Overall, CWE of Sch. commune and HWE of G. lucidum revealed the best outcome and suggested them for further study as potential cosmeceuticals. [START_REF] Razak | Assessment of cosmeceutical potentials of selected mushroom fruit body extracts through evaluation of antioxidant, anti-hyaluronidase and anti-tyrosinase activity[END_REF] An culinary-medicinal mushroom A. bisporus is rich in nutrients and minerals, and possesses potential anticancer, antioxidant, anti-obesity, and anti-inflammatory properties. The revealed anti-tyrosinase activity and ergothioneine with skin-whitening properties have shown that this fungus is not only a source of nutraceuticals and pharmaceuticals, but also cosmeceutical products. [START_REF] Usman | Nutritional, medicinal, and cosmetic value of bioactive compounds in Button Mushroom (Agaricus bisporus): A review[END_REF] Many commercially available macrofungi-derived cosmetic products have replaced synthetic compounds that have long-term adverse effects. [START_REF] Sujarit | Mushrooms: Splendid gifts for the cosmetic industry[END_REF] However, the use of macrofungi and their extracts obtained from fruiting bodies or mycelia as cosmeceuticals need to undergo several steps of biotechnological cultivation and production (extraction optimization, assessment of efficacy and safety) to be produced as a cosmetic product. Previous reports regarding stability, compatibility, and toxicological studies of macrofungi also need to be considered. 48

III. CONCLUSIONS

Macrofungi (mushrooms) have been regarded as a traditional source of natural low-and high-molecular-weight secondary metabolites and used for centuries to prevent and treat many diseases; however, over the past decade, they are being exploited for potential ingredients in the cosmetic industry. Several species and their extracts derived from fruiting bodies and/or mycelia are either used or patented to be used as cosmeceuticals and nutracosmetics for their anti-aging, anti-wrinkle, skin-whitening and moisturizing, wound-healing, anti-collagenase, anti-elastase, anti-hyaluronidase, anti-tyrosinase, anti-hyperpigmentation, anti-inflammatory, and antioxidant effects. Among known biodiversity, only a few taxonomically identified and cultivated species are utilized as cosmeceuticals. Further evaluation of natural resources of macrofungi will promote their exploitation in cosmetic industry. Advances in fungal biology and biotechnology, genomics, proteomics, metabolomics, as well as systems pharmacology and molecular cosmetology, study of the molecular mechanisms of medicinal and cosmetic effects of macrofungi will support the discovery of new species to be used in the formulation of organic cosmetic products and cosmeceuticals.

Currently, different mushroom-derived cosmeceuticals and nutracosmetics with various properties are available in the multi-billion-dollar cosmetic market. Despite the availability of a number of products, the industry is largely unregulated, and consumers should consult a doctor prior to the use of these products. Recent developments in cosmeceutical biotechnology, including nano-systems and nano-delivery technologies, have been introduced to improve the efficacy and safety of usage of organic products in commercial production of anti-aging cosmeceuticals. Further realization of clinical studies, manufacturing of high-quality standardized products, and sustainable production of cosmeceuticals under controlled conditions are warranted.

  brasiliensis) (ABM) was incorporated into a semi-solid base cosmetic cream (safety tests for nutraceutical applications) and cell viability effects of the extract, and of the final cream formulation, on a keratinocyte cell line (HaCaT) (safety tests for cosmeceutical applications) were studied. The study detected proteins, carbohydrates, and a low-fat content. Overall, 22 fatty acids including 53% of polyunsaturated fatty acids (PUFA)were estimated the most abundant fraction. The cell viabilities of Caco-2 and HT29 cell lines were maintained up to 100 μg/mL. After incorporation into the base cream, a formulation with a pale-yellow colour and favourable pH was obtained. The viability of HaCaT cells in the presence of the extract and the final cream formulation was maintained in a concentration-dependent manner, which indicates the safety of this extract for use in cosmetic industry.

TABLE 1 :

 1 66. Lima ATM, Santos MN, de Souza LAR, Pinheiro TS, Paiva AAO, Dore CMPG, Costa MSSP, Santos ND, Baseia YG, Araújo RM, Leite EL. Chemical characteristics of a heteropolysaccharide from Tylopilus ballouii mushroom and its antioxidant and anti-Medicinal macrofungi-derived bioactive compounds and their cosmetic effects AIA: anti-inflammatory activity; AMA, antimicrobial activity; AOA, antioxidant activity; FIPs, fungal immunomodulatory proteins; FMG, fucomannogalactan; NF-κB, nuclear factor-κB; SIRT1, Sirtuin 1; SPF, sun protection factor;

	inflammatory activities. Carbohydr Polym. 2016;144:400-9.
	Mushroom species	Bioactive compounds	Cosmetic effects	Refs.
	Abortiporus biennis (Bull.) Singer	Exopolysaccharides	AOA	22
	Agaricus bisporus (J.E. Lange)	Caffeic, chlorogenic, cinnamic,	AOA, AIA, ABA, anti-	1,8,3
	Imbach	ferulic, gallic, p-	tyrosinase	8,43,
		hydroxybenzoic, p-coumaric		54
		and protocatechuic acids,
		alkaloids, polyphenols
	Agaricus subrufescens Peck	Polysaccharides, benzenoid	AOA, anti-pigmentation	29
		aromatic compound
	Amanita muscaria (L.) Lam.	β-D-Glucan, FMG	AIA	49
	Antrodia cinnamomea T.T. Chang &	Benzenoid aromatic compound	ABA, AIA	35
	W.N. Chou.	coenzyme Q 0
	Auricularia polytricha (Mont.)Sacc.	Phenolics, glucans	Anti-hyaluronidase, anti-	46
			tyrosinase, AIA
	Boletus edulis Bull.	Ergothioneine, glutathione	AOA	50
	Calocybe gambosa (Fr.) Donk	Ascorbic acid (vitamin C)	Anti-radiation, stimulating	51
			collagen synthesis
	Clitocybe aurantiaca (Wulfen)	Clitocybin A, D	Anti-elastase, stimulation	52,53
	Studer-Steinhäuslin [=		procollagen synthesis,
	Hygrophoropsis aurantiaca (Wulfen)		AOA

TABLE 2 :

 2 Medicinal macrofungi incorporated in different cosmetic products

	Mushroom species	Cosmetic Product	Cosmetic Effect
	Agaricus subrufescens	Vitamega Facial Moisturizing Mask	Renews and
			revitalizes skin
	Albatrellus confluens (Alb.	REN Clean Skincare Evercalm Ultra	Skin feels smooth
	& Schw. ex Fr.) Kotl.	Comforting Rescue Mask	with even tone
	Ophiocordyceps sinensis	Kosé Sekkisei Cream	Moisturizer,
			suppress
			melanin production
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ACKNOWLEDGMENT

The work was supported by the Science Committee of Republic of Armenia, in the frames of the thematic research project № 21T-1F228.

The authors declare to have no conflict of interest


Benaych-Georges Florent

Thierry Cabanal-Duvillard
email: thierry.cabanal-duvillard@mi.parisdescartes.fr

Florent Benaych
email: florent.benaych@gmail.com

Thierry Cabanal

A MATRIX INTERPOLATION BETWEEN CLASSICAL AND FREE MAX OPERATIONS. I. THE UNIVARIATE CASE

Keywords: MSC 2000 subject classifications. 15A52, 46L54 random matrices, free probability, max-stable laws 1

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.

of concepts in both theories: many fundamental notions or results of classical probability theory, such as Law of Large Numbers, Central Limit Theorem, Gaussian distribution, convolution, cumulants, infinite divisibility have their precise counterpart in free probability theory.

Recently, Ben Arous and Voiculescu [START_REF] Ben Arous | Free extreme values[END_REF] have added a new item in this dictionary, bringing to light the fact that the operation called the classical upper extremal convolution, which associates the distribution of the supremum X ∨ Y to the distributions of two independent random variables X and Y , has an analogue in free probability theory, called the free upper extremal convolution, which associates the distribution of the supremum X ∨ Y to the distributions of two free random variables X and Y . From the point of view of cumulative distribution functions 1 , both of these operations have a concise interpretation: the classical (resp. free) upper extremal convolution of two probability measures with distribution functions F, G is the probability measure with distribution function F G (resp. max(0, F + G -1)). The purpose of this paper is to construct random matrix models providing a better understanding of the free upper extremal convolution and a new insight on the relation between classical and free upper extremal convolutions.

Let us first notice the following fact (which is an immediate consequence of remark 2.1): for µ, ν probability measures on the real line, for (X i ) i≥1 and (Y i ) i≥1 two independent families of independent identically distributed random variables, with respective distributions µ, ν, if for each N ≥ 1, one denotes by Z N,1 ≥ Z N,2 ≥ • • • ≥ Z N,N the N largest elements of the multiset (X 1 , . . . , X N , Y 1 , . . . , Y N ), then the empirical probability measure 1 N N i=1 δ ZN,i converges almost surely to the free upper extremal convolution of µ and ν as N tends to infinity. This result may appear as a coincidence. But, if we consider (X 1 , . . . , X N ) and (Y 1 , . . . , Y N ) as the eigenvalues of two independent random matrices X (N ) and Y (N ) which are invariant in law under conjugation by any unitary matrix, then (Z N,1 , . . . , Z N,N ) are the eigenvalues of the supremum 2 of X (N ) and Y (N ) with respect to the spectral order introduced by Olson [START_REF] Olson | The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice[END_REF] and used by Ben Arous and Voiculescu in their paper. Since X (N ) and Y (N ) are known to be asymptotically free, this gives actually a first interpretation of the free upper extremal convolution.

We would like to give a deeper insight on the analogy between the classical and free settings. There is a morphism Λ ∨ from the set of probability measures endowed with the classical upper convolution to the same set endowed with the free upper convolution. It maps any probability measure with distribution function F to the one with distribution function max(0, 1 + log F ). Moreover, Ben Arous and Voiculescu have established that Λ ∨ provides a remarkable correspondence between classical max-stable laws and free max-stable laws, which preserves the domains of attraction. In this paper, we shall give a concrete realization of this morphism via a random matrix model. More precisely, for each probability measure µ on the real line and each positive integer N , we shall define a law Λ ∨ N (µ) on the set of N by N Hermitian matrices such that: a) a Λ ∨ N (µ)-distributed random matrix is invariant under conjugation by any unitary matrix, b) for any pair µ 1 , µ 2 of probability measures on the real line and any pair M 1 , M 2 of independent random matrices distributed respectively with respect to Λ ∨ N (µ 1 ) and Λ ∨ N (µ 2 ), the law of the supremum

M 1 ∨ M 2 of M 1 and M 2 is Λ ∨ N (µ)
, where µ is the classical upper extremal convolution of µ 1 and µ 2 , c) if for all N , M N is a random matrix with law Λ ∨ N (µ), then the empirical spectral law of M N tends almost surely to Λ ∨ (µ) as N tends to infinity. Such a model produces a new clue with regards to the relevance of the analogy between the two maxoperations, and it may be used to provide more intuitive proofs of some of Ben Arous and Voiculescu's results. For instance, we have already mentioned that unitarily-invariant random matrices behave asymptotically as free random variables. Therefore one can re-prove immediately, as a consequence of a), b) and c), the fact that Λ ∨ is a morphism between the classical and the free upper extremal convolutions (and thus the formula F X∨Y = max(0, F X + F Y -1) for free random variables X, Y ). We shall see later 1 The cumulative distribution function of a random variable X with distribution µ is the function, denoted either by F X or Fµ, is defined by Fµ(x) = µ((-∞, x]), for all x ∈ R. 2 The supremum of two Hermitian matrices is defined in section 2.3.

that Ben Arous and Voiculescu's results about max-stable laws and domains of attraction can also be proved with our random matrix model.

In a forthcoming paper, we will show how this approach is also appropriate to study the multivariate counterpart of the free upper extremal convolution.

The paper is organized as follows. First, we shall focus on the upper extremal convolution for classical, free and Hermitian random variables. Then, we shall introduce our models, via a strategy which is close to the one of our previous papers [START_REF] Benaych-Georges | Classical and free infinitely divisible distributions and random matrices[END_REF] and [START_REF] Cabanal-Duvillard | A matrix representation of the Bercovici-Pata bijection[END_REF], and set the main theorems. The last part will be dedicated to the proofs.

The upper extremal convolutions

2.1. For independent real random variables. Let X and Y be real independent random variables, with cumulative distribution functions F X and F Y , and let us denote

X ∨Y = max(X, Y ). The cumulative distribution function of X ∨ Y is then equal to F X∨Y = F X F Y .
2.2. For free random variables. Let (M, τ ) be a tracial W * -probability space, that is a von Neumann algebra M endowed with an ultraweakly continuous faithful trace-state τ . The maximum X ∨ Y of two self-adjoints elements X and Y in M is defined with respect to the spectral order introduced by Olson in 1971 (cf [START_REF] Olson | The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice[END_REF]; see also [START_REF] Ando | Majorization, doubly stochastic matrices and comparison of eigenvalues[END_REF][START_REF] Ben Arous | Free extreme values[END_REF]):

• If p and q are two self-adjoint projectors, then q ∨ q is the Hermitian projector on ℑ(p) + ℑ(q). • If a and b are two self-adjoint elements, with resolutions of identity E a and E b , then a ∨ b is the selfadjoint element with resolution of identity given by E a ∨ E b : a ∨ b is the only self-adjoint element h such that for all real number t,

χ (t,∞) (h) = χ (t,∞) (a) ∨ χ (t,∞) (b),
where χ (t,∞) denotes the indicator function of the interval (t, ∞) and is applied via the functional calculus.

In 2006, in [START_REF] Ben Arous | Free extreme values[END_REF], Ben Arous and Voiculescu have determined the cumulative distribution function of X ∨ Y whenever X, Y are free:

(1) ∀t ∈ R, F X∨Y (t) = max(0, F X (t) + F Y (t) -1) =: F X 2 ∨ F Y (t).
It is easy to note that the function Λ ∨ (u) = max(0, 1 + log u) is a kind of morphism between classical and free upper extremal convolutions:

Λ ∨ (F X F Y ) = Λ ∨ (F X )2 ∨ Λ ∨ (F Y ).
Notations: • In order to avoid the use of too many notations, we shall define the binary operation 2 ∨ on the set of probability measures on the real line with the same symbol as the operation 2 ∨ on the set of cumulative distribution functions. For all pair µ, ν of probability measures on the real line, µ2 ∨ ν is defined by:

F µ2 ∨ ν = F µ 2 ∨ F ν ( = max(0, F µ (t) + F ν (t) -1) ).
• We shall also define the operator Λ ∨ on the set of probability measures on the real line with the same symbol as the function Λ ∨ (x) = max(0, 1 + log x) on [0, +∞): for all probability measure µ on the real line, Λ ∨ (µ) is the probability measure with cumulative distribution function:

F Λ ∨ (µ) (t) = Λ ∨ (F µ (t)) ( = max(0, 1 + log F µ (t)) ).
Remark 2.1. The free upper extremal convolution also appears in a much simpler situation than the maximum of two free operators. Let x 1 , . . . , x N , y 1 , . . . , y N ∈ R, and z 1 ≥ . . . ≥ z N the N largest elements of the multiset (x 1 , . . . , x N , y 1 , . . . , y N ). Then 2.3. For independent Hermitian random matrices. Let N be a positive integer. The maximum operation introduced by Olson is also defined on the space of N × N Hermitian matrices, as a particular case of the definition given in section 2.2.

Concretely, for A, B Hermitian matrices with the same size, A ∨ B can be defined in the following way. Let {λ 1 > λ 2 > • • • > λ p } be the union of the spectrums of A and of B. For i = 0, . . . , p, let us define E i = i j=1 (ker(Aλ j I) + ker(Bλ j I)). Then A ∨ B is the Hermitian matrix with eigenvalues the λ j 's such that E j = E j-1 , with associated eigenspace E j ∩ (E ⊥ j-1 ). In other words, if, for any subspace F , P F designs the orthogonal projector onto F , we have

A ∨ B = λ 1 P E1 + λ 2 P E2∩(E ⊥ 1 ) + λ 3 P E3∩(E ⊥ 2 ) + • • • + λ p P Ep∩(E ⊥ p-1 )
.

The following proposition resumes the characteristics of the operation ∨ on unitarily invariant random matrices. Recall that the empirical spectral law of a matrix is the uniform measure on the multiset of its eigenvalues counted by multiplicity.

F A∨B = max(0, F A + F B -1). (2)
Remark 2.3. It is noteworthy that formulas (1) and (2) are identical, showing that the matricial and free upper extremal convolutions are somehow the same operation. As free random variables can be approximated by independent unitarily invariant random matrices, the rather trivial result (2) provides an intuitive proof of formula (1).

The matricial interpolation

The aim of this section is to introduce random matrices giving quite a natural interpretation of the mapping Λ ∨ . We simply follow the same approach as we did for the Bercovici-Pata bijection in [START_REF] Benaych-Georges | Classical and free infinitely divisible distributions and random matrices[END_REF] and [START_REF] Cabanal-Duvillard | A matrix representation of the Bercovici-Pata bijection[END_REF].

Let F µ denote its cumulative distribution function for µ probability measure on the real line. For

all integer k ≥ 1, F 1 k
µ is the cumulative distribution function of a probability measure on the real line, which will be denoted by µ

1 k . It is clear that if X 1 , . . . , X k are independent random variables distributed according to µ 1 k , max{X 1 , . . . , X k } should be distributed according to µ.
Let us consider N, k ≥ 1. Let X 1 , . . . , X N be independent random variables distributed according to µ 1 k and U an unitary, Haar distributed, N × N random matrix independent of X 1 , . . . , X N . Define the random matrix M by

(3) M = U    X 1 . . . X N    U * Let M 1 , . . . , M k be independent replicas of M . Theorem 3.1. As k tends to infinity, the distribution of M 1 ∨• • •∨M k converges to a probability measure Λ ∨ N (µ)
on the space of N × N Hermitian matrices. This measure is invariant under the action of the unitary group by conjugation and in the case where µ has no atom, under Λ ∨ N (µ), the joint distribution of the ranked eigenvalues is absolutely continuous with respect to µ ⊗N , with density

(4) N N 1 t1≥•••≥tN N i=1 F µ (t N ) F µ (t i )
at any point (t 1 , . . . , t N ) ∈ R N such that for all i = 1, . . . , N , F µ (t i ) = 0 (the density being set to zero anywhere else).

Moreover, if ν is another probability measure on the real line and A, B are independent random matrices respectively distributed according to

Λ ∨ N (µ), Λ ∨ N (ν), then A ∨ B is distributed according to Λ ∨ N (ρ)
, where ρ is the probability measure on the real line such that

F ρ = F µ F ν .
Corollary 3.2. With the preceding notations, we have

(i) Λ ∨ N (µ) is max-infinitely divisible, (ii) if µ is max-stable, then so is Λ ∨ N (µ).
In the following theorem, we are interested in the limit of the spectral measure of a Λ ∨ N (µ)-distributed random matrix, when its dimension N tends to infinity. Theorem 3.3. Let µ be a probability measure on the real line. For all N ≥ 1, let M N a random matrix with law Λ ∨ N (µ), λ 1 , . . . , λ N its eigenvalues and μN = 1 N N i=1 δ λi its empirical spectral law. Then, when N goes to infinity, μN converges weakly almost surely to Λ ∨ (µ).

The previous theorem may be used to provide new, more intuitive proofs of some of the Ben Arous and Voiculescu's results in [START_REF] Ben Arous | Free extreme values[END_REF].

A first example has been given in the introduction: it is a new derivation of the formula (1) defining the free upper extremal convolution in terms of cumulative distribution functions.

For second example, the fact that Λ ∨ maps any classical max-stable law to a free one is a direct consequence of (ii) of Corollary 3.2 and of formulas (1) and (2).

Finally, let us consider the preservation of the domains of attraction. Ben Arous and Voiculescu have proved that for any classical max-stable law µ, Λ ∨ (µ) is freely max-stable and that for any cumulative distribution function F , for any sequences

a k > 0, b k ∈ R, we have (5) F (a k • +b k ) k -→ k→∞ F µ (•) =⇒ F (a k • +b k ) 2 ∨k -→ k→∞ Λ ∨ (F µ (•))
(to state the reciprocal implication, one needs to choose a right inverse to Λ ∨ , which we shall not do here). This result can be generalized as follows: for all sequence (F k ) of cumulative distribution functions, if F is a cumulative distribution function, we have

(6) F k (•) k -→ k→∞ F (•) =⇒ F k (•) 2 ∨k -→ k→∞ Λ ∨ (F (•))
(the proof of ( 6) is given in the last section of the paper).

Let us now explain how our random matrix model gives an heuristic explanation of [START_REF] Collins | Product of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF]. For sake of simplicity, let F k be equal to F 1 k , with F the cumulative distribution function of µ (for more general F k , a slight modification of our random matrix model is enough to apply what follows).

For each N ≥ 1 and for each k ≥ 1, let us consider a family M 1 (N, k), . . . , M k (N, k) of independent replicas of the random matrix M of (3), i.e.

M = U    X 1 . . . X N    U * ,
where X 1 , . . . , X N are independent random variables with cumulative distribution function F

1 k and U an unitary, Haar distributed, N × N random matrix independent of X 1 , . . . , X N .

• Let k tend to infinity. Following the definition of Λ ∨ N (µ) given in theorem 3.1, we get:

(7) M 1 (N, k) ∨ • • • ∨ M k (N, k) -→ k→∞ M N in distribution,
where for all N , M N is a Λ ∨ N (µ)-distributed random matrix. Let now N tend to infinity. From theorem 3.3, we deduce:

(8) Empirical Spectral Law(M N ) -→ N →∞ Λ ∨ (µ).
• Now, let N tend to infinity before k does. According to the law of large numbers, the empirical spectral law of M i (N, k) converges almost surely, for each k and i = 1, . . . , k, to the law with distribution function F 1 k , which shall be denoted by µ 1 k . Therefore, we deduce from the asymptotic freeness of independent randomly rotated random matrices (more precisely by the equality between formulas (1) and ( 2)), the following almost sure convergence:

(9) Empirical Spectral Law(M 1 (N, k) ∨ • • • ∨ M k (N, k)) -→ N →∞ µ 1 k 2 ∨ • • • 2 ∨ µ 1 k .
Joining ( 7), ( 8), ( 9), we get the following diagram

M 1 (d, k) ∨ • • • ∨ M k (d, k) k→∞ ------→ Λ ∨ N (µ) | | N → ∞ N → ∞ ↓ ↓ Empirical Spectral Law: µ 1 k 2 ∨ • • • 2 ∨ µ 1 k Empirical Spectral Law: Λ ∨ (µ)
The right-hand side of (6), i.e. the fact that (µ

1 k ) 2 ∨k -→ k→∞ Λ ∨ (µ)
, only means that one can add an edge -→ k→∞ between both bottom vertices of the diagram, i.e. that the operations k → ∞ and N → ∞ are commutative, which is quite expected.

Proofs

4.1. Preliminary results. Both of the theorems will be proved first for measures which are absolutely continuous with respect to the Lebesgue measure, and then extended to all measures by approximation. In this context, the appropriate approximation tool is given by the following lemma. Lemma 4.1. Let us define, for µ probability measure on the real line, the function F <-1> µ on (0, 1) by

F <-1> µ (u) = min{x ∈ R ; F µ (x) ≥ u}.
(i) For µ, ν probability measures on the real line, we have the following equalities (between quantities which can be infinite)

inf{ε > 0 ; F ν (• -ε) ≤ F µ (•) ≤ F ν (• + ε)} = ||F <-1> µ -F <-1> ν || ∞ = inf ||X -Y || ∞ ,
where the infimum in the third term is taken on pairs (X, Y ) of random variables defined on a same probability space with respective distributions µ, ν.

(ii) For any probability measure µ on the real line, for any ε > 0, there exists a probability measure µ ε on the real line such that F µε is smooth and

F µε (• -ε) ≤ F µ (•) ≤ F µε (• + ε).
Proof. (i) We prove these equalities by cyclic majorizations.

-Let us prove first

inf{ε > 0 ; ∀x ∈ R, F ν (x -ε) ≤ F µ (x) ≤ F ν (x + ε)} ≥ ||F <-1> µ -F <-1> ν || ∞ . Let us consider ε > 0 such that F ν (• -ε) ≤ F µ (•) ≤ F ν (• + ε). Since this is equivalent to F µ (• - ε) ≤ F ν (•) ≤ F µ (• + ε),
this inequation is symmetric in µ and ν. Hence, it suffices to prove that F <-1> ν (u) -F <-1> µ (u) ≤ ε for all u ∈ (0, 1). We have:

F ν (F <-1> µ (u) + ε) ≥ F µ (F <-1> µ (u)) ≥ u, hence, F <-1> ν (u) ≤ F <-1> µ (u) + ε. -The inequality ||F <-1> µ -F <-1> ν || ∞ ≥ inf ||X -Y || ∞
is due to the fact that for U random variable with uniform distribution on (0, 1),

X := F <-1> µ (U ), Y := F <-1> ν (U ) are respectively distributed according to µ, ν. -To conclude, it suffices to prove inf ||X -Y || ∞ ≥ inf{ε > 0 ; F ν (• -ε) ≤ F µ (•) ≤ F ν (• + ε)}.
So let us consider a pair (X, Y ) of random variables defined on the same probability space with respective distributions µ, ν. Consider ε > 0 such that |X -Y | ≤ ε uniformly on the probability space. Then, for all real number x,

P(Y ≤ x -ε) ≤ P(X ≤ x) ≤ P(Y ≤ x + ε).
(ii) Let ε > 0, and F a smooth non decreasing function on the real line such that for all k ∈ Z, F (kε) = F µ (kε). Such a function exists obviously, as an example, one can consider

F (x) = x -∞ f (t) dt, for f (t) = k∈Z [F (kε) -F ((k -1)ε)]ϕ(t -kε)
with ϕ smooth non negative function on the real line, with support contained in [-ε, 0] satisfying

0 -ε ϕ(t) dt = 1.
Hence F is a càdlàg non decreasing function on the real line such that F (x) tends to zero (resp. one) as x tends to -∞ (resp. +∞). It follows that F = F µε for a certain probability measure µ ε on the real line. Moreover, for all real number x, if k ∈ Z is such that kε ≤ x < (k + 1)ε, then, one has:

F µε (x -ε) ≤ F µε (kε) = F µ (kε) ≤ F µ (x) ≤ F µ ((k + 1)ε) = F µε ((k + 1)ε) ≤ F µε (x + ε).
For any positive integer N , let us define max N to be the function from ∪ n≥N R n to R N which maps any vector x of R n , n ≥ N , to the vector of the N largest coordinates of x ranked in decreasing order. The following property is a basic result, see for instance [START_REF] David | Order statistics[END_REF]. Proposition 4.2. Let F be the cumulative distribution function of a probability measure µ with no atom on the real line. Let, for n ≥ N , X 1 , . . . , X n be independent random variables with law µ. The distribution of max N (X 1 , . . . , X n ) has density

(t 1 , . . . , t N ) → n! (n -N )! 1 t1≥•••≥tN F (t N ) n-N
with respect to µ ⊗N .

Before stating the next proposition, we shall recall that for µ be a probability measure on the real line and k positive integer, µ 1 k is the probability measure on the real line with cumulative distribution function F 1 k µ , i.e. the law m such that for X 1 , . . . , X k independent random variables with law m, max(X 1 , . . . , X k ) has law µ. Proposition 4.3. (i) Let µ be a probability measure on the real line. For all integer N ≥ 1, the push forward of the probability measure (µ 1 k ) ⊗kN on R kN by the function max N converges weakly, as the integer k tends to infinity, to a probability measure on R N denoted by µ N .

(ii) When µ has no atom, µ N is absolutely continuous with respect to µ ⊗N , with density

N N 1 t1≥•••≥tN N i=1 F µ (t N ) F µ (t i )
at any point (t 1 , . . . , t N ) ∈ R N such that for all i = 1, . . . , N , F µ (t i ) = 0.

(iii) Let us endow R N with the norm ||x|| = max i |x i |. Then, for any probability measures µ, ν on the real line, inf{||V -W || ∞ ; V, W random vectors defined on the same space with respective laws µ N , ν N } ≤ inf{||X -Y || ∞ ; X, Y random variables defined on the same space with respective laws µ, ν}.

(iv) For any pair µ, ν of probability measures on the real line, if ρ is the probability measure on the real line such that F ρ = F µ F ν , for all N ≥ 1, ρ N is the push-forward, by the function max N , of the probability measure µ N ⊗ ν N on R 2N .

Proof. Let k, N ≥ 1, and µ a probability measure with no atom. From property 4.2, we infer first that µ

1 k = 1 k F 1 k -1 µ
dµ, and then that the distribution max N ((µ

1 k ) ⊗kN ) has a density with respect to µ ⊗N equal to (t 1 , . . . , t N ) → (dk)! k N (dk -N )! 1 t1≥•••≥tN N i=1 F µ (t N ) 1-1 k F µ (t i ) 1-1 k
at any point (t 1 , . . . , t N ) ∈ R N such that for all i = 1, . . . , N , F (t i ) = 0 (the density can obviously be set to zero anywhere else). As k tends to infinity, this density stays uniformly bounded and converges pointwise to

N N 1 t1≥•••≥tN N i=1 F µ (t N ) F µ (t i ) ,
hence (i) for probability measures with no atom and (ii) are proved. Now, let us complete the proof of (i). Let us consider a probability measure µ and, for all positive integer k, a family X(k, 1), . . . , X(k, kN ) of independent random variables with law µ 1 k . To prove (i), by Theorem 1.12.4 of [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF], it suffices to prove that for any real bounded Lipschitz function f on R N , the sequence E(f (max N (X(k, 1), . . . , X(k, kN )))) converges as k tends to infinity. So let us fix such a function. We shall prove that the previous sequence is a Cauchy sequence. Let us fix ε > 0. Let us consider µ ε as in (ii) of lemma 4.1. Note that for all positive integers k, we also have:

F 1 k µε (• -ε) ≤ F 1 k µ (•) ≤ F 1 k µε (• + ε).
So, based on (i) of lemma 4.1, we may suppose that for all k ≥ 1, on the probability space where the X(k, i)'s are defined, there is a family

Y (k, 1), . . . , Y (k, kN )
of random variables such that (a) the Y (k, i)'s are independent and distributed according to µ

1 k ε , (b) for all i = 1, . . . , kN , |X(k, i) -Y (k, i)| ≤ 2ε almost surely.
Note that µ ε has no atom, hence by (a) and what we just proved, the sequence

E(f (max N (Y (k, 1), . . . , Y (k, kN ))))
is a Cauchy sequence. Note also that by (b), if C is a Lipschitz constant for f with respect to the norm ||x|| = max i |x i |, then for all k,

|E(f (max N (X(k, 1), . . . , X(k, kN )))) -E(f (max N (Y (k, 1), . . . , Y (k, kN ))))| ≤ 2Cε. Hence there is k 0 ≥ 1 such that for all k, k ′ ≥ k 0 , |E(f (max N (X(k, 1), . . . , X(k, kN )))) -E(f (max N (X(k ′ , 1), . . . , X(k ′ , k ′ d))))| ≤ (4C + 1)ε.
Thus we have proved (i) for any probability measure µ. Now, let us prove (iii). Consider µ, ν probability measures on the real line. Note that by part (i) of lemma 4.1, it suffices to prove that for all positive ε such that

F ν (• -ε) ≤ F µ (•) ≤ F ν (• + ε),
for all α > ε, there exists a pair V, W of random vectors defined on the same space with respective laws µ N , ν N such that ||V -W || ∞ ≤ α. Let us consider such a positive ε and α > ε. For all k ≥ 1, we also have:

F 1 k ν (• -ε) ≤ F 1 k µ (•) ≤ F 1 k ν (• + ε).
So, according to part (i) of lemma 4.1, we shall consider, for all k ≥ 1, a family

X(k, 1), . . . , X(k, kN ), Y (k, 1), . . . , Y (k, kN )
of random variables defined on the same space such that (a) the X(k, i)'s are independent and distributed according to µ 1 k , (b) the Y (k, i)'s are independent and distributed according to ν

1 k , (c) for all i = 1, . . . , kN , |X(k, i) -Y (k, i)| ≤ α almost surely.
Let, for all k ≥ 1, τ k be the joint law, on R 2N , of the random vector

(max N (X(k, 1), . . . , X(k, kN )), max N (Y (k, 1), . . . , Y (k, kN )).
The law of max N (X(k, 1), . . . , X(k, kN )) (resp. of max N (Y (k, 1), . . . , Y (k, kN ))) converges weakly to µ N (resp. to ν N ) as k tends to infinity. It follows that the sequence τ k is tight, i.e. relatively compact for the topology of weak convergence (Theorem 6.1 of [START_REF] Billingsley | Convergence of probability measures[END_REF]). Let τ be the weak limit of a subsequence of τ k . Let (V, W ) be a τ -distributed random vector of R N × R N . Then the law of V (resp. of W ) is µ N (resp. ν N ). Moreover, it is easy to notice that for all k, τ k is supported by {(x, y)

; x ∈ R N , y ∈ R N , ||x -y|| ≤ α}. Indeed, let σ, τ ∈ S n such that X(k, σ(1)) ≥ • • • ≥ X(k, σ(kN )) and Y (k, τ (1)) ≥ • • • ≥ Y (k, τ (kN )); then for any i = 1, . . . , kN , X(k, σ(i)) = max V ⊂{1,...,kN },#V =i min(X(k, j), j ∈ V ) ≥ max V ⊂{1,...,kN },#V =i min(Y (k, j) -α, j ∈ V ) = Y (k, τ (i)) -α a.s.
By symmetry, this proves |X(k, σ(i)) -Y (k, τ (i))| ≤ α a.s., and this implies max N (X(k, 1), . . . , X(k, kN ))

-max N (Y (k, 1), . . . , Y (k, kN )) = max i=1,...,N |X(k, σ(i)) -Y (k, τ (i))| ≤ α a.s.
Letting k go to infinity, this establishes the inequality V -W ≤ α almost surely.

To prove (iv), consider for any k ≥ 1,

X(k, 1), . . . , X(k, kN ), Y (k, 1), . . . , Y (k, kN )
is a family of independent random variables such that for all i = 1, . . . , kN , X(k, i)

(resp. Y (k, i)) is distributed according to µ 1 k (resp. ν 1 k ). Then X(k, 1) ∨ Y (k, 1), . . . , X(k, kN ) ∨ Y (k, kN ) are i.i.d. with law ρ 1 k
, and consequently µ N , ν N and ρ N are the respective weak limit distributions, as k tends to infinity,

of max N [X(k, 1), . . . , X(k, kN )], max N [Y (k, 1), . . . , Y (k, kN )] (10) and max N [X(k, 1) ∨ Y (k, 1), . . . , X(k, kN ) ∨ Y (k, kN )].
Hence, by continuity and commutativity of the maximum operations, max N (µ N ⊗ ν N ) is the weak limit, as k tends to infinity, of the distribution of (11) max N [X(k, 1), . . . , X(k, kN ), Y (k, 1), . . . , Y (k, kN )].

To conclude, we shall establish that the distributions of the vectors of ( 10) and ( 11) have the same weak limits as k tends to infinity. Obviously, it reduces to prove that the probability of the event

{max N [X(k, 1) ∨ Y (k, 1), . . . , X(k, kN ) ∨ Y (k, kN )] (12) = max N [X(k, 1), . . . , X(k, kN ), Y (k,
(A)) + Im(χ (t,∞) (B))] = min{N, dim[Im(χ (t,∞) (A))] + dim[Im(χ (t,∞) (B)]} a.s. Let p = dim[Im(χ (t,∞) (A))], q = dim[Im(χ (t,∞) (B))].
Due to the unitarily invariance of A (resp. B), the law of Im(χ (t,∞) (A)) (resp. Im(χ (t,∞) (B))) is also invariant under the action of any unitary matrix, hence is uniform on the set of all subspaces of dimension p (resp. q). Moreover, Im(χ (t,∞) (A)) and Im(χ (t,∞) (B)) are independent.

Let us consider a family (g i,j , i ∈ {1, . . . , N } , j ≥ 1) of independent complex standard Gaussian random variables. The matrix (g i,j , i, j ∈ {1, . . . , N }) is distributed w.r.t. the Circular Unitary Ensemble. Hence, it is unitarily invariant and a.s. inversible. Therefore, the subspace generated by vectors (g i,j , i ∈ {1, . . . , N }) p j=1 (resp. (g i,j , i ∈ {1, . . . , N }) p+q j=p+1 , (g i,j , i ∈ {1, . . . , N }) p+q j=1 ) has dimension p (resp. q, min(N, p + q)) a.s., and its law is uniform on the set of all subspaces of same dimension. This proves (13).

M (k) = M 1 ∨ • • • ∨ M k , then M (k)
is an Hermitian random matrix, whose law is invariant under conjugation with any unitary matrix and whose eigenvalues λ

(k) 1 ≥ . . . ≥ λ (k)
N are equal to the N largest of the N k eigenvalues of M 1 , . . . , M k which are all independent with same distribution µ 1/k . Therefore, by theorem 4.3.5 of [START_REF] Hiai | The semicircle law, free random variables and entropy[END_REF], there exists U unitary, Haar distributed and independent of the vector (λ

(k) 1 , . . . , λ (k)
N ), such that:

M (k) = U     λ (k) 1 . . . λ (k) N     U * .
Thus, using the fact that

N -1 p=0 x p p! = e x +∞ x 1 (N -1)! u N -1 e -u du, we get: E e λμN ((-∞,t]) = e λ F µ (t) N N -1 p=0 (-N e -λ N log F µ (t)) p p! + -log Fµ(t) 0 N N (N -1)! u N -1 e -N u du = e λ F µ (t) N e -N e -λ N log Fµ(t) +∞ -N e -λ N log Fµ(t) 1 (N -1)! u N -1 e -u du + -log Fµ(t) 0 N N (N -1)! u N -1 e -N u du = e λ+N " 1-e -λ N « log Fµ(t) +∞ -e -λ N log Fµ(t) N N (N -1)! u N -1 e -N u du + -log Fµ(t) 0 N N (N -1)! u N -1 e -N u du.
The following inequalities will be useful, and they can be established from the Chernov inequality or proved directly:

∀x ∈ [0, 1),

x 0 N N (N -1)! u N -1 e -N u du ≤ x N e N (1-x) , ∀x > 1, +∞ x N N (N -1)! u N -1 e -N u du ≤ x N e N (1-x) .
We have three cases to consider:

Case 1: -log F µ (t) > 1. If moreover λ ∈ [0, N log (-log F µ (t))), then -e -λ N log F µ (t) > 1,
and by ( 14),

e λ+N " 1-e -λ N « log Fµ(t) +∞ -e -λ N log Fµ(t) N N (N -1)! u N -1 e -N u du ≤ e λ+N " 1-e -λ N « log Fµ(t) -e -λ N log F µ (t) N e N " 1+e -λ N log Fµ(t) « = e N (1+log Fµ(t)+log(-log Fµ(t))) < 1 since 1 + log F µ (t) + log(-log F µ (t)) < 0.
Hence for all N ≥ 1, E e λμN ((-∞,t]) ≤ 2, and from the Chernov inequality, we get then: for all ε > 0,

P {μ N ((-∞, t]) > ε} ≤ inf λ∈[0,N log(-log Fµ(t)))
2e -ελ = 2e -εN log(-log Fµ(t)))

This is enough, with Borel-Cantelli lemma, to prove the almost-sure convergence of μN ((-∞, t]) to 0 = max(0, 1 + log F (t)).

Case 2:log F µ (t) < 1. Then by (14),

-log Fµ(t) 0 N N (N -1)! u N -1 e -N u du ≤ (-log F µ (t)) N e N (1+log Fµ(t)) = e N δt with δ t = 1 + log F µ (t) + log(-log F µ (t)) < 0.
• Moreover, we suppose λ ≥ 0. Then

E e λ[μN ((-∞,t])-(1+log Fµ(t))] ≤ e λ F µ (t) N e -N e -λ N log Fµ(t) + e N δt e -λ(1+log Fµ(t)) = e -N log Fµ(t) " λ N -(1-e -λ N ) « + e N (δt-λ N (1+log Fµ(t)))
From the Chernov inequality, we infer then: for all ε > 0

P {μ N ((-∞, t]) -(1 + log F µ (t)) > ε} ≤ inf λ≥0 e -N log Fµ(t) " λ N -(1-e -λ N ) « + e N (δt-λ N (1+log Fµ(t))) e -λε = inf λ≥0 e -N (log Fµ(t)(λ-(1-e -λ ))+λε) + e N (δt-λ(1+log Fµ(t)+ε)) ≤ 2e -N C
for some constant C > 0. This gives, using Borel-Cantelli lemma:

lim sup N -→+∞ μN ((-∞, t]) ≤ 1 + log F µ (t) a.s.
• We suppose now λ ≤ 0. Using the same trick, we get

P {μ N ((-∞, t]) -(1 + log F µ (t)) < -ε} ≤ inf λ≤0 e -N (log Fµ(t)(λ-(1-e -λ ))-λε) + e N (δt-λ(1+log Fµ(t)-ε)) ≤ 2e -N C ′
for some constant C ′ > 0. This gives:

lim inf N -→+∞ μN ((-∞, t]) ≥ 1 + log F µ (t) a.s.
And the result follows:

lim N -→+∞ μN ((-∞, t]) = 1 + log F µ (t) = max(0, 1 + log F µ (t)) a.s.
Case 3: log F µ (t) = -1. We know that F µ (u) tends to 0 as u tends to -∞. Hence, there is u < t such that log F µ (u) < -1. For all N , μN ((-∞, t]) ≥ μN ((-∞, u]), hence, lim inf

N -→+∞ μN ((-∞, t]) ≥ lim inf N -→+∞ μN ((-∞, u]) = 0 = 1 + log F µ (t) a.s.
Moreover, for any positive ε, by continuity of F µ and by the fact that F µ (u) tends to 1 as u tends to +∞, there is u > t such that 1 + log F µ (t) < 1 + log F µ (u) < 1 + log F µ (t) + ε. Hence, lim sup The theorem is proved in the case where F µ is smooth. Now, we consider a probability measure µ on the real line without making any assumption about F µ . By Theorem 1.12.4 of [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF] Equations ( 16), ( 17), (18), together, imply (15). Thus, the theorem is proved.

4.6. Proof of the implication [START_REF] Collins | Product of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF]. Let us end the paper with the proof of the implication [START_REF] Collins | Product of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF]. We consider a sequence F k of cumulative distribution functions and a cumulative distribution function F such that at any x where F is continuous, F k (x) k -→ k→∞ F (x). Let us prove that for any such x, F k (x) 2 ∨k -→ k→∞ Λ ∨ (F (x)). Let us denote, for y ∈ R, y + = max(0, y). Note that since for any a, b ∈ R such that b ≤ 0, we have (a + + b) + = (a + b) + , by induction on k, we prove easily that for all G 1 , . . . , G k distribution functions,

G 1 2 ∨ • • • 2 ∨ G k = (G 1 + • • • + G k -k + 1)
+ . Thus we have to prove

1 NN

 1 i=1 δ zi is the free upper extremal convolution of 1 N N i=1 δ xi and 1 N N i=1 δ yi .

Proposition 2 . 2 .

 22 If A and B are two N × N Hermitian random independent matrices, whose laws are invariant by conjugation by any unitary matrix, then • A ∨ B is an Hermitian random matrix, whose law is invariant under the conjugation by any unitary matrix; • the N eigenvalues of A ∨ B are the N largest of the 2N eigenvalues of A and B (counted with multiplicity); • the cumulative distribution functions F A , F B and F A∨B of the respective empirical spectral laws of A, B and A ∨ B are almost surely linked by the relation

Remark 4 . 4 . 4 . 3 .

 4443 Note that the proof of the second statement could also have been deduced from Theorem 2.2 of[START_REF] Collins | Product of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF]. Proof of theorem 3.1. Following proposition 2.2, if we denote

  N -→+∞ μN ((-∞, t]) ≤ lim sup N -→+∞ μN ((-∞, u]) = 1 + log F µ (u) < 1 + log F µ (t) + ε a.s. Therfore, lim N -→+∞ μN ((-∞, t]) = 1 + log F µ (t) almost surely.

  , the distance d(m, m ′ ) := sup f dmf dm ′ ,where the sup is taken on the set BL 1 bounded Lipschitz functions f on the real line with Lipschitz constant ≤ 1 and such that ||f || ∞ ≤ 1, is a distance which defines the weak topology on the set of probability measures on the real line.Thus we have to prove that almost surely, as N tends to infinity, lim d(μ N , Λ ∨ (µ)) = 0, i.e. that for all ε > 0, almost surely, for N large enough,(15) d(μ N , Λ ∨ (µ)) ≤ ε.So let us fix ε > 0. By part (ii) of lemma 4.1, there exists a probability measure ν on the real line such that F ν is smooth andF ν (•ε/6) ≤ F µ (•) ≤ F ν (• + ε/6).Note that the same obviously holds if one replaces µ by Λ ∨ (µ) and ν by Λ ∨ (ν), since for all probability measure m, F Λ ∨ (m) = max(0, 1 + log F m ). Hence, according to part (i) of lemma 4.1,(16) d(Λ ∨ (µ), Λ ∨ (ν)) ≤ ε/6.Note also that by definition of Λ ∨ N (µ), the vector (X 1 ≥ • • • ≥ X N ) ofranked eigenvalues of M N has law µ N . Following part (iii) of proposition 4.3, one can suppose that for all N , on the same space as M N , there is a random matrix N N with law Λ ∨ N (ν) with ranked eigenvalues (Y 1 ≥ • • • ≥ Y k ) such that for all i = 1, . . . , N , |X i -Y i | ≤ ε/3 almost surely. If one denotes the spectral law of N N by νN , it implies that (17) ∀N ≥ 1, d(μ N , νN ) ≤ ε/3 almost surely. Please note finally that by the first part of the proof, (18) almost surely, for N large enough, d(ν N , Λ ∨ (ν)) ≤ ε/3.

( 1 +

 1 k(F k (x) -1)) + -→ k→∞ (1 + log F (x)) + , i.e. k(F k (x) -1) -→ k→∞ log F (x),which follows directly from the hypothesis F k (x) k -→ k→∞ F (x).

  1), . . . , Y (k, kN )]} tends to zero as k tends to infinity. Note that this event is equivalent to the fact that there is i ∈ {1, . . . , kN } such that X(k, i) and Y (k, i) are both coordinates of max N [X(k, 1), . . . , X(k, kN ), Y (k, 1), . . . , Y (k, kN )] and are not equal to any of the others X(k, j)'s and Y (k, l)'s. Hence, if one denotes byI(k) (resp. J(k)) the set of i's in {1, . . . , kN } such that X(k, i) (resp. Y (k, i)) is one of the coordinates of max N [X(k, 1), . . . , X(k, kN )] (resp. of max N [Y (k, 1), . . . , Y (k, kN )]),the event of (12) is contained in the event I(k) ∩ J(k) = ∅, which probability is equal to kN N -1 , since we may choose for I(k) and J(k) to be independent and to have uniform distribution on the set of subsets of {1, . . . , kN } with cardinality N . Thus, (iv) is proved. 4.2. Proof of proposition 2.2. The first statement is immediate. The third one follows immediately form the second one. To establish the second statement, it suffices to prove that almost surely, for any t ∈ R, Im(χ (t,∞) (A)) + Im(χ (t,∞) (B)) has the maximal dimension conditionally to the dimensions of Im(χ (t,∞) (A) and Im(χ (t,∞) (B), i.e. that (13) dim[Im(χ (t,∞)

Note that the distribution of the vector (λ

N ) is the push forward of the probability measure (µ 1 k ) ⊗kN on R kN by the function max N . Hence, the first part of the theorem follows from (i) and (ii) of proposition 4.3, and the second part follows from (iv) of the same proposition. 4.4. Proof of corollary 3.2. Both assertions are easy consequences of the last part of theorem 3.1.

4.5. Proof of theorem 3.3. Suppose first that F µ is smooth on the real line. It implies that µ is absolutely continuous with respect to the Lebesgue measure and that dµ(x) = F ′ µ (x) dx. Since F µ is continuous on the real line, the almost sure weak convergence of the sequence of random probability measures μN to Λ ∨ (µ) is equivalent to the fact the for all rational number t, μN ((-∞, t]) converges almost surely to max(0, 1 + log F µ (t)). We shall prove that for all t ∈ R, μN ((-∞, t]) converges almost surely to max(0, 1 + log F µ (t)). Let us fix a real number t.

The first step is to compute E e λμN ((-∞,t]) : with the notations λ 0 = +∞, λ N +1 = -∞, we have

Now, note that for all real number T such that F (T ) > 0, for all integer non negative m, (T,∞)

Hence for p = 0, . . . , N -1,

And for p = N :

N -1 dµ(t N ) = log Fµ(t) 0 N N (N -1)! u N -1 e -N u du.