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[ [ section ] ]    large deviations of empirical measures have been widely studied in the literature since the celebrated sanov s theorem , which gives the large deviations principle in the scale of @xmath0 of the empirical measures of i.  i.  d. random variables with the relative entropy @xmath1 as the rate function . another entropy , voiculescu s non - commutative entropy @xmath2 , arises in the study of fluctuations of eigenvalues of random matrices , see hiai & petz@xcite and the references therein . chan @xcite interprets empirical measures of eigenvalues of random matrices as a system of interacting diffusions with singular interactions . [ [ section-1 ] ]    in this paper we study empirical measures which can be thought of as a decoupled version of the empirical measures generated by random matrices . we are interested in empirical measures on @xmath3 generated by pairs of random variables that are tied together by a totally symmetric , and hence non - local , interaction , see formula ( [ f ] ) for the ( unnormalized ) joint density . under certain assumptions , we prove that the large deviation principle in the scale @xmath4 holds for the joint empirical measures , and the rate function is non - convex . as a corollary , we derive a large deviations principle for the univariate average empirical measures with a rate function that superficially resembles the rate function of random matrices , see corollary [ c2 ] ; an interesting feature here is the emergence of concave rate functions , see remark [ rc2 ] . ( eigenvalues of random matrices are exchangeable and the large deviation rate function for their empirical measures is convex ; infinite exchangeable sequences often lead to non - convex rate functions , see dinwoodie & zabell @xcite and ( * ? ? ? * example 3 ) . ) [ [ section-2 ] ]    let @xmath5 be a continuous function which satisfies the following conditions .    [ b0 ] @xmath6 for all @xmath7 . [ p - int ] for every @xmath8 , @xmath9 . [ b1 ] @xmath10 is bounded , @xmath11 .    in the following statements we use the convention that @xmath12 . [ b2 ] the function @xmath13 has compact level sets : for every @xmath14 the set @xmath15 is compact . the purpose of the next assumption is to allow singular interactions , where @xmath16 ; this assumption is automatically satisfied with @xmath17 if @xmath18 for all @xmath19 . [ b3 ] there is a @xmath20 such that @xmath21 extends from @xmath22 to the continuous function on @xmath3 . examples of functions that satisfy these assumptions are : the gaussian kernel @xmath23 for @xmath24 , see the proof of proposition [ iig2 ] ; and a singular kernel @xmath25 for @xmath20 , see the proof of proposition [ log ] . define @xmath26 clearly , @xmath27 depends on @xmath0 ; we will suppress this dependence in our notation and we will further write @xmath28 as a convenient shorthand for @xmath29 .    assumptions [ b0 ] , [ p - int ] , and [ b1 ] imply that @xmath27 is integrable . indeed , since @xmath30 , @xmath31 @xmath32    we are interested in joint empirical measures @xmath33 considered as random variables with values in the polish space of probability measures @xmath34 ( equipped with the topology of weak convergence ) , with the distribution induced on @xmath34 by the probability measure @xmath35 defined by @xmath36    [ t1 ] if @xmath10 satisfies assumptions [ b0 ] , [ p - int ] , [ b1 ] , [ b2 ] , and [ b3 ] , then the joint empirical measures @xmath37 satisfy the large deviation principle in the scale @xmath4 with the rate function @xmath38 $ ] given by @xmath39 where @xmath40 and @xmath41 . we say that @xmath42 is a negative definite kernel if @xmath43 and @xmath44 for all @xmath45 such that @xmath46 .    condition ( [ neg - def ] ) is satisfied for @xmath47 , where @xmath48 is positive - definite . consider the average empirical measures @xmath49    [ c2 ] suppose that the assumptions of theorem [ t1 ] hold true , and in addition @xmath50 is continuous and negative - definite . then the average empirical measures @xmath51 satisfy the large deviation principle in the scale @xmath4 with the rate function @xmath52 and @xmath53 . this follows from the contraction principle . the mapping @xmath54 is continuous in the weak topology . the rate function is @xmath55 . write @xmath56 . if @xmath57 then ( * ? ? ? * theorem 3 ) implies that @xmath58 . thus @xmath59 . another form of the cited inequality is that for any two probability measures @xmath60 we have @xmath61 in particular , @xmath62 , which implies that @xmath63 . [ rc1 ] inequality ( [ 2k ] ) implies that the rate function satisfies @xmath64 [ [ section-3 ] ]    let @xmath65 . then @xmath66 and @xmath67 .    denote by @xmath68 the @xmath69-th moment of a measure @xmath70 . [ iig2 ]    * if @xmath24 then the empirical measures @xmath71 satisfy the large deviation principle in the scale @xmath4 with the rate function @xmath72 * if @xmath73 then the average empirical measures @xmath49 satisfy the large deviation principle in the scale @xmath4 with the rate function @xmath74    ( in the formulas above , use @xmath75 if @xmath76 . )    the marginal density relevant in proposition [ iig2](i ) is @xmath77    [ rc2 ] both rate functions in proposition [ iig2 ] are concave . \(i ) it is easy to see that the assumptions of theorem [ t1 ] are satisfied . indeed , @xmath78 is continuous , bounded from below . furthermore @xmath79 so @xmath50 has compact level sets . finally , for @xmath80 by a change of variables we see that @xmath81 so assumption [ p - int ] is satisfied , too . the result follows by the contraction principle : taking a marginal of a measure in @xmath34 is a continuous mapping . the rate function is @xmath82 . but since @xmath83 is infinite on non - product measures , this is the same as @xmath84 . since @xmath85 here , it remains to notice that @xmath86 . \(ii ) this follows from corollary [ c2 ] : if @xmath87 then @xmath88 is positive - definite . thus @xmath67 is a negative definite kernel . [ [ section-4 ] ]    next , we consider a model which can be interpreted as a  decoupled " version of a model studied in relation to eigenvalue fluctuations of random matrices , where one encounters @xmath89 instead of our @xmath90 , compare ( * ? ? ? * section 5 ) , ( * ? ? ? * formula ( 1.9 ) ) . we consider here a slightly more general situation when @xmath91 then @xmath92 and @xmath93 . we assume that functions @xmath94 are continuous , @xmath20 , and that @xmath95    [ log ] the bivariate empirical measures @xmath96 defined by ( [ mu_n ] ) satisfy the large deviation principle in the scale @xmath4 with the rate function @xmath83 given by ( [ i ] ) .    in particular , if @xmath97 , then the rate function is @xmath98    we verify that the hypotheses of theorem [ t1 ] are satisfied . assumption [ b0 ] holds trivially . assumption [ b3 ] holds trivially since @xmath99 is continuous .    to verify assumption [ b1 ] notice that @xmath100    since @xmath101 is a continuous function which by ( [ vw ] ) tends to infinity as @xmath102 , it is bounded from below , @xmath103 for some @xmath104 similarly , @xmath105 . we now verify assumption [ b2 ] . the set @xmath106 is closed since @xmath107 is lower semicontinuous . furthermore , ( [ k > v+w ] ) implies that @xmath108 is contained in a level set of the continuous function @xmath109 . the latter set is bounded since @xmath110 for all large enough @xmath111 and similarly @xmath112 for all large enough @xmath113 .    to verify assumption [ p - int ] we use inequality ( [ k > v+w ] ) again . it implies @xmath114 by assumption ( [ vw ] ) , there is @xmath115 such that for @xmath116 we have @xmath117 . by the previous argument the integrand is bounded ; thus @xmath118 .    therefore , by theorem [ t1 ] the empirical measures @xmath96 satisfy the large deviation principle with the rate function @xmath119 . if @xmath120 then @xmath121 by calculus . the proof relies on varadhan s functional method , see ( * ? ? ? * theorem t.1.3 ) , ( * ? ? ? * theorem 4.4.10 ) . it consists of two steps : verification that the varadhan functional @xmath122 is well defined for a large enough class of bounded continuous functions @xmath123 , and the proof of exponential tightness of @xmath37 . denote by @xmath144 the set of absolutely continuous probability measures @xmath145 on @xmath146 with compact support @xmath147 , and continuous density @xmath27 . let us first record the well - known fact . since measures @xmath60 are absolutely continuous and have compact supports , for every integer @xmath155 we can find partitions @xmath156 and @xmath157 of @xmath158 , @xmath159 respectively such that @xmath160 then , denoting @xmath161\times[a_1,a_2]\times\dots\times[a_{n-1},a_n]$ ] and @xmath162\times[b_1,b_2]\times\dots\times[b_{n-1},b_n]$ ] , we have @xmath163    write @xmath164 . by our choice of the partitions , functions @xmath165}\ ] ] and @xmath166}\ ] ] are probability densities . let @xmath167 integrating over a smaller set @xmath168 on the right hand side of ( [ ef ] ) we get @xmath134 @xmath169 using jensen s inequality , applied to the convex exponential function in the last integral , we get @xmath170 where @xmath171        lemma [ h ] says that the entropies @xmath175 are finite . thus the functions @xmath50 , @xmath176 , and @xmath177 are @xmath172-integrable . applying claim [ int ] , we get @xmath178 , @xmath179 , and @xmath180 . therefore , @xmath181        combining inequalities ( [ * ] ) and ( [ mmm ] ) , we get @xmath185 since functions @xmath186 are continuous and @xmath60 have compact support and continuous densities @xmath27 , @xmath187 , therefore @xmath154-almost surely @xmath188 ( to see this , notice that for fixed @xmath189 , the sequence convergences for all @xmath190 such that @xmath191 . ) , and the functions are bounded . since @xmath192 ranges over a finite set of values only we have @xmath193 @xmath194 letting @xmath195 in ( [ lu ] ) we obtain ( [ ll - low ] ) . clearly the left - hand side of ( [ comp * ] ) can not exceed the right hand side . to show the converse inequality , fix @xmath214 and @xmath215 such that @xmath216 since the supremum is finite , see ( [ finite ] ) , and @xmath50 is bounded from below , therefore @xmath217 . for @xmath218 large enough , define probability measures @xmath219 by @xmath220)}{\nu_j([-l , l ] ) } , \ , j=1,2.\ ] ] by definition , measures @xmath221 have compact support . since @xmath222 and @xmath107 is @xmath172-integrable , by lebesgue s dominated convergence theorem @xmath223)\nu_2([-l , l])}= { { \mathbb k}}(\nu_1\otimes \nu_2).\ ] ] similarly , @xmath224 thus ( [ comp * ] ) follows . trivially , @xmath226 to show the converse inequality , fix @xmath214 and compactly supported @xmath227 such that @xmath228 see lemma [ comp ] . as previously , @xmath50 is @xmath172-integrable , see ( [ finite ] ) . consider the convolution @xmath229 , where @xmath230 and @xmath231 . measures @xmath232 have continuous densities , and since @xmath60 have compact supports , @xmath232 also have compact support . thus @xmath233 and @xmath234      assumption [ b3 ] asserts that @xmath239 is a continuous function . thus @xmath240 is bounded on the compact set @xmath241 . since the supports of @xmath242 are contained in @xmath241 , and @xmath243 , we get @xmath244 this concludes the proof if @xmath17 . if @xmath245 , then @xmath246 is @xmath172-integrable as a linear combination of integrable functions , @xmath247 . therefore we have @xmath248 @xmath249 @xmath250 taking the lim sup as @xmath235 , from ( [ * 00 ] ) , ( [ * * ] ) , ( [ * * * ] ) , and ( [ * 1 ] ) we get @xmath251 @xmath252 since @xmath214 is arbitrary , to end the proof we use the following . proof of claim [ log - est ] claim [ log - est ] is established by the argument in @xcite . for completeness , we repeat it here . let @xmath254 be independent random variables with distributions @xmath60 respectively and let @xmath255 . since @xmath256 is integrable , @xmath257 . let @xmath258 $ ] be a r.  v. independent of @xmath259 with the density @xmath260 . it is easy to see that @xmath261 , and the inequality to prove reads @xmath262 for fixed @xmath263 we have @xmath264 indeed , since @xmath265 we get @xmath266 therefore , @xmath267 if @xmath268 we get @xmath269 . if @xmath270 , then @xmath271 . thus in both cases , ( [ joh ] ) holds true . proof of theorem combining lemmas [ upper bound ] and [ lower bound ] we have @xmath277 @xmath278 @xmath279 @xmath280 by ( [ reduct ] ) , all of the above inequalities are in fact equalities . thus ( [ ll ] ) holds true . recall that @xmath37 is exponentially tight if for every @xmath281 there is a compact subset @xmath282 such that @xmath283 our proof of exponential tightness is a concrete implementation of de acosta @xcite . assumption [ b1 ] implies that @xmath284 . let @xmath285 $ ] be given by @xmath286        assumption [ b2 ] says that for every @xmath189 the set @xmath293 is a compact subset of @xmath3 . for every @xmath294 by chebyshev s inequality we have @xmath295 thus @xmath292 is pre - compact , and its weak closure @xmath296 is compact . notice that by ( [ ll ] ) used with @xmath301 we have @xmath302 . since @xmath303 is finite , see ( [ finite ] ) , therefore by lemma [ exp(q ) ] we have @xmath304 fix @xmath281 . let @xmath305 be the pre - compact set from lemma [ pre - comp ] corresponding to @xmath306 . proof of theorem [ t1 ] recall that the space @xmath310 of probability measures on @xmath3 with the topology of weak convergence is a polish space . by theorem [ exp - t ] , @xmath37 is exponentially tight . theorem [ v - f ] says that the varadhan functional @xmath311 is defined on all functions @xmath131 given by ( [ phi ] ) , and @xmath312 @xmath313 thus @xmath314    functions @xmath131 defined by ( [ phi ] ) form a subset of @xmath315 which separates points of @xmath34 and is closed under the operation of taking pointwise minima . thus by ( * ? ? ? * theorem t.1.3 ) or ( * ? ? ? * theorem 4.4.10 ) , the empirical measures @xmath37 satisfy the large deviation principle with the rate function @xmath316 here , the supremum is taken over all @xmath317 and @xmath318 is defined by ( [ phi ] ) .        now consider @xmath323 . assumptions [ b1 ] and [ b3 ] imply that @xmath324 is a bounded continuous function for every real @xmath325 . thus @xmath326 is given by ( [ phi ] ) . since @xmath327 , from ( [ @ ] ) we get @xmath328 . thus @xmath329 this together with ( [ i+ ] ) proves ( [ i ] ) for product measures .      clearly , @xmath336 is a bounded continuous function , which vanishes on product measures . by the upper bound ( [ ll - u ] ) we therefore have @xmath337 . so @xmath338 . since @xmath339 can be arbitrarily large , @xmath331 .
we prove the large deviation principle for the joint empirical measure of pairs of random variables which are coupled by a  totally symmetric " interaction . 
 the rate function is given by an explicit bilinear expression , which is finite only on product measures and hence is non - convex . 
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