LOCAL FIELD U{STATISTICS STEVEN N. EVANS. sequence of independent, identically distributed random variables taking values. 1.

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1 LOCAL FIELD U{STATISTICS STEVEN N. EVANS Abstract. Using the classical theory of syetric functions, a general distributional liit theore is established for U{statistics constructed fro a sequence of independent, identically distributed rando variables taking values in a local eld with zero characteristic.. Introduction Since the work of Hoeding and Halos in the 940s, U{statistics constructed fro sequences of independent, identically distributed real rando variables have played a central role in theoretical and applied statistics. They have also attracted considerable attention fro probabilists because they exhibit a rich liit theory that parallels that of i.i.d. sequences (for exaple, strong laws, central liit theores, large deviation results, and Berry{Esseen{type theores have been established for the). Surveys with extensive bibliographies ay be found in [Ser80, KB94, Lee90]. Our ai in this paper is to initiate an investigation into the properties of U{ statistics on algebraic structures other than the reals: naely, local elds. A local eld K is any locally copact, non-discrete eld other than the eld of real nubers or the eld of coplex nubers. All local elds are totally disconnected, and are either nite algebraic extensions of the eld of p-adic nubers{in which case the characteristic is zero { or nite algebraic extensions of the the less failiar p-series eld (the eld of foral Laurent series with coecients drawn fro the nite eld with p eleents) { in which case the characteristic is non{zero. We give an overview of soe of the basic properties of local elds in x2. Probability on local elds has a substantial, if soewhat scattered, literature, and a coprehensive book{length treatent has yet to be written. For the convenience and interest of the reader we have included a representative (but by no eans coplete) bibliography in the references. The natural denition of the notion of U{statistic on the local eld K is the following direct translation of the failiar Euclidean denition. Denition.. Let f k g k2nbe an innite sequence of independent, identically distributed rando variables taking values in the local eld K. Fix a syetric Borel function : K! K for soe 2 N (that is, the value of the function is unchanged by perutations of its arguents). The sequence of U {statistics corresponding to f k g k2nand is the sequence of K{valued rando variables Date: March 4, Matheatics Subject Classication. 60B5, 60F05, 05E05. Key words and phrases. local eld, p{adic, U{statistic, syetric function, rando walk. Research supported in part by NSF grants DMS and DMS

2 2 STEVEN N. EVANS fz k g k given by Z k := i <i 2<<i k ( i ; i2 ;:::; i ): The following result is proved in x3. Soe rearks on the hypotheses are given after the proof. Theore.2. Suppose that the local eld K has characteristic zero, the support of the coon distribution of the rando variables k is copact, and the function is continuous. Let fk(h)g h2nbe a sequence of positive integers such that k(h) converges to innity as h!and also k(h) thought of as an eleent of K converges to soe k 2 K as h!. Then the sequence fz k(h) g h2nof U {statistics converges in distribution as h!. 2. Local fields This section is essentially a suary of selected results fro [Tai75, Sch84]. We refer the reader to these works for a fuller account. Before giving the general denition of a local eld, we begin with the prototypical exaple. Exaple 2.. Fix a positive priep. We can write any non-zero rational nuber r 2 Qnf0g uniquely as r = p s (a=b) where a and b are not divisible by p. Set jrj = p,s.ifwesetj0j = 0, then the ap jj has the properties: (2.) jxj =0, x =0; jxyj = jxjjyj; jx + yj jxj_jyj: The ap (x; y) 7! jx, yj denes a etric on Q and we denote the copletion of Q in this etric by Q p. The eld operations on Q extend continuously to ake Q p a topological eld called the p-adic nubers. The ap jjalso extends continuously and the extension continues to have properties (2.). The closed unit ball around 0, Z p := fx 2 Q p : jxj g, is the closure in Q p of the integers Z, and is thus a ring (this is also apparent fro the properties (2.)) called the p{adic integers. As Z p = fx 2 Q p : jxj <pg, the set Z p is also open. Any other ball around 0 is of the for fx 2 Q p : jxj p,k g = p k Z p for soe integer k. Such a ball is the closure of the rational nubers divisible by p k, and is thus a Z p {odule (this is again also apparent fro the properties (2.)). In particular, such a ball is an additive subgroup of Q p. Arbitrary balls are translates (= cosets) of these closed and open subgroups. In particular, the topology of Q p has a base of closed and open sets, and hence Q p is totally disconnected. Further, each of these balls is copact, and hence Q p is also locally copact. A local eld is a locally copact, non{discrete, totally disconnected, topological eld. (As an aside, a locally copact, non-discrete, topological eld that is not totally disconnected is necessarily either the real or the coplex nubers. A local eld with characteristic zero is a nite algebraic extension of the p{adic nuber eld for soe prie p. A local eld with non{zero characteristic is a nite algebraic extension of the p{series eld; that is, the eld of foral Laurent series with coecients drawn fro the nite eld with p eleents for soe prie p.) Fro now on, let K be a xed local eld. There is a real{valued apping on K which we denote by x 7! jxj. This ap has the properties (2.) and it takes the

3 LOCAL FIELD U{STATISTICS 3 values fq k : k 2 Zg[f0g, where q = p c for soe prie p and positive integer c (so that for K = Q p wehave c =). A ap with properties (2.) is called a non-archiedean valuation. The property jx + yj jxj _jyj is known as the ultraetric inequality or the strong triangle inequality. The apping (x; y) 7! jx, yj on K K is a etric for K that gives the topology of K. A consequence of (2.) is that if jxj 6= jyj, then jx + yj = jxj_jyj. This latter result iplies that for every \triangle" fx; y; zg K we have thatat least two of the lengths jx, yj, jx, zj, jy, zj ust be equal and is therefore called the isosceles triangle property. 3. Proof of Theore.2 Write E for the support of the coon distribution of the k. By the ultraetric Stone{Weierstrass theore (see, for exaple, xa.4 of [Sch84]), polynoials are uniforly dense in the space of continuous functions fro the copact set E into K. Therefore, for each >0 there exists a polynoial Q such that sup jq(x ;:::;x ), (x ;:::;x )j <: (x ;:::;x )2E Dene a syetric polynoial Q : E! K by Q(x ;:::;x )=! 2S Q(x () ;:::;x () ); where S denotes the syetric group on letters and we have used the assuption that K has zero characteristic to conclude that! 6= 0. By the strong triangle inequality and the syetry of, sup (x ;:::;x )2E = sup (x ;:::;x )2E j!j, : Q(x ;:::;x ), (x ;:::;x ) Q(x() ;:::;x () ), (x () ;:::;x () )! 2S Thus, again by the strong triangle inequality, Q( i ; i2 ;:::; i ), Z k i <i 2<<i k j!j, : It thus clearly suces to consider the special case of the theore when is a syetric polynoial. By replacing k by k, c and (x ;:::;x )by (x + c;:::;x + c), we ay further suppose that 0 2 E. Moreover, because li h i <i 2<<i k(h) = li h k(h) = k (k, ) :::(k, +)! by assuption, we ay suppose that has no constant ter. Given a positive integer n and integers 2 ::: 0with0= n+ = n+2 = :::, dene the corresponding onoial syetric function M n; in the variables (x ;:::;x n )by M n; (x ;:::;x n ):= x ;

4 4 STEVEN N. EVANS where the su is over all distinct perutations =( ;:::; n )of( ;:::; n ) and x := x P :::xn n : The syetric polynoials M n; with i i d for a basis for the vector space of syetric polynoials in(x ;:::;x n ) of total degree at ost d (cf. Ch I of [Mac95] orch7of[sta99]). Consequently, wehave (x ;:::;x )= c M ; (x ;:::;x ); for suitable constants c, where the su is over all with 0 = + = +2 = :::, only nitely any of the c are non{zero, and c 0 =0(by our added assuption that has no constant ter). Observe for such that if ` := axfr : r > 0g, then k, ` M ; ( i ;:::; i )= M k; ( ;:::; k ):, ` i <i 2<<i k Note that li h, k(h),`,` exists. It therefore suces to show that the rando vectors (M k; ( ;:::; k )); where ranges over those non{zero partitions such that 0 = + = +2 = ::: and P i i is at ost the total degree of, converge in distribution as k!. Given a non-negative integer j, dene the power su syetric function P n;j in the variables (x ;:::;x n )by P n;j (x ;:::;x n ):=x j + + xj n ; and given integers 2 ::: 0 with 0 = n+ = n+2 = :::, set We have P n; (x ;:::;x n ):= Y i M n; (x ;:::;x n )= P n;i (x ;:::;x n ): c P n; (x ;:::;x n ) where the su is over all with P i i = P i i and, iportantly, the constants c do not depend on n (cf. Ch I of [Mac95] or Ch 7 of [Sta99], and note that we are again using the assuption that K has characteristic 0). It thus suces to show for any positive integer J that the rando vectors k i= i ; k i= 2 i ;:::; converge in distribution as k!. However, this process is just a rando walk on the copact subgroup of the additive group of K J generated by f(x; x 2 ;:::;x J ): x 2 Eg, and the added assuption that 0 2 E ensures that the rando walk converges in distribution to Haar easure on this subgroup. Reark 3.. The hypothesis that K has zero characteristic was used several ties in the above proof. We do not know whether the result has a counterpart for non{zero characteristics. k i= J i!

5 LOCAL FIELD U{STATISTICS 5 Reark 3.2. The role played by the hypothesis that k(h) converges in K is apparent fro the proof. However, because the assuption initially looks rather unusual, we ephasise its iportance with the following siple exaple. Suppose that K is the eld of p{adic nubers Q p, =2,and (x ;x 2 )=x + x 2. Then Z k =(k, ) As we observed in the proof, if 0 is in the support of the coon distribution of k, then P k i= i converges in distribution as k!to Haar easure on the subgroup of Q p generated by the support. Note that if k is of the for p s +, then jk, j = p,s, whereas if k is of the for p s +2,s, then p does not divide k, and hence jk, j =. Consequently, weust take k!along a subsequence in order for Z k to have a liit in distribution. Reark 3.3. In principle, the steps in the proof can be reversed to describe the liiting distribution as the push{forward by an appropriate function of Haar easure on the copact additive subgroup of KN generated by (x; x 2 ;x 3 ;:::)forx in suitable xed translate of the support of the distribution of the k. It does not appear that one can give a ore eective characterisation of the liit. In the next section we exaine soe particularly siple exaples where it possible to say soething concrete about the liit. k i= i : 4. Soe exaples Suppose that K is the eld of p{adic nubers Q p for soe prie p and k takes only the values 0 and with positive probability. Then P k i= i = P k i= 2 i = :::, and these sus converge in distribution to Haar easure on the ring of p{adic integers Z p. Write E n;r for the r th eleentary syetric function of n variables; that is, E n;r (x ;:::;x n ):= i <:::<i rn x i :::x ir ; n r:, k,r If we put = E ;r, then Z k =,r Ek;r ( ;:::; k ). Write U for a rando variable with Haar distribution on Z p. Then, by a classical deterinantal identity (see Exaple 8 in xi.2 of [Mac95]), E k;r ( ;:::; k )converges in distribution as k!to r! det 0 U 0 ::: 0 0 U U 2 ::: U U U ::: U r, U U U ::: U U C A = U(U, ) :::(U, r +) U = r! r (this is also clear fro eleentary considerations: if fx ;:::;x n gf0; g, then E n;r (x ;:::;x n ) counts how any subsets of size r can be drawn fro a set of (x + + x n ) objects). We note in passing that the rando variable U is the siplest exaple of the natural analogue on Q p of a Gaussian rando variable. Moreover, the rando variables, U r are, in a very natural sense, orthogonal and appear in a theory of

6 6 STEVEN N. EVANS stochastic integration and Wiener chaos on Q p. We refer the reader to [Eva89a, Eva9, Eva93, Eva95] for details. Write H n;r for the r th coplete syetric function of n variables; that is, H n;r (x ;:::;x n ) is the su of all onoials of total degree r in the variables x ;:::;x n. By another classical deterinantal identity (see Exaple 8 in xi.2 of [Mac95]), H k;r ( ;:::; k ) converges in distribution as k!to r! det 0 U, 0 ::: 0 0 U U,2 ::: U U U ::: U,(r, ) U U U ::: U U,U C A =(,)r r Note that fh k;r ( ;:::; k )g is not a sequence of U{statistics for soe function. Finally, given a partition of soe integer r, let S n; (x ;:::;x n )betheschur function in the variables x ;:::;x n associated with (see xi.3 of [Mac95]). Fro a classical deterinantal forula expressing Schur functions in ters of the coplete syetric functions (see Equation (3.4) of [Mac95]) S k; ( ;:::; k )=det(h k;i,i+j ( ;:::; k )) i;jn for any N that is at least the length of the partition (that is, the nuber of non{zero parts in ). By the above, the right{hand side converges in distribution as k!to,u det (,) i,i+j,u i, i + j i;jn =(,) r det i, i + j Fro Exaple 4 in xi.3 of [Mac95], we see that the right{hand side is Y x2 0 U, c 0(x) ; h 0(x) : : i;jn where 0 is the partition dual to and c 0(x) (resp. h 0(x)) denotes the content (resp. hook length) of 0 at x (that is, if x is the boxinthei th row andj th colun of the Young diagra of 0, then c 0(x) :=j, i and h 0(x) is the nuber of boxes in the hook with corner at x). There is a natural identication of boxes in the Young diagra of a partition with boxes in the Young diagra of the dual partition. Under this identication, c 0(x) =,c (x) and h 0(x) = h (x). Consequently, S k; ( ;:::; k )converges in distribution as k!to [AK9] [AK94] Y x2 U + c (x) : h (x) References S. Albeverio and W. Karwowski, Diusion on p-adic nubers, Gaussian rando elds (Nagoya, 990), Ser. Probab. Statist., no., World Sci. Publishing, River Edge, NJ, 99, pp. 86{99., A rando walk on p-adics the generator and its spectru, Stochastic Process. Appl. 53 (994), {22. [AKZ99] S. Albeverio, W. Karwowski, and. Zhao, Asyptotics and spectral results for rando walks on p-adics, Stochastic Process. Appl. 83 (999), 39{59.

7 LOCAL FIELD U{STATISTICS 7 [Tai75] [Bik99] A. Kh. Bikulov, Stochastic equations of atheatical physics over the eld of p-adic nubers, Theoret. and Math. Phys. 9 (999), 594{604. [BV97] A. Kh. Bikulov and I.V. Volovich, p-adic Brownian otion, Izv. Math. 6 (997), 537{ 552. [Eva89a] S.N. Evans, Local eld Gaussian easures, Seinar on Stochastic Processes, 988 (Gainesville, FL, 988), Progr. Probab., no. 7, Birkhauser Boston, Boston, MA, 989, pp. 2{60. [Eva89b], Local properties of Levy processes on a totally disconnected group, J. Theoret. Probab. 2 (989), 209{259. [Eva9], Equivalence and perpendicularity of local eld Gaussian easures, Seinar on Stochastic Processes, 990 (Vancouver, BC, 990), Progr. Probab., no. 24, Birkhauser Boston, Boston, MA, 99, pp. 73{8. [Eva93], Local eld Brownian otion, J. Theoret. Probab. 6 (993), 87{850. [Eva95], p-adic white noise, chaos expansions, and stochastic integration, Probability easures on groups and related structures, I (Oberwolfach, 994), World Sci. Publishing, River Edge, NJ, 995, pp. 02{5. [Gui89] F. Guiier, Siplicite duspectre de Liapouno d'un produit de atrices aleatoires sur un corps ultraetrique, C. R. Acad. Sci. Paris Sr. I Math. 309 (989), 885{888. [KB94] V.S. Koroljuk and Yu. V. Borovskich, Theory of U-statistics, Matheatics and its Applications, no. 273, Kluwer, Dordrecht, 994. [KM94] W. Karwowski and R. Vilela Mendes, Hierarchical structures and asyetric stochastic processes on p-adics and adeles, J. Math. Phys. 35 (994), 4637{4650. [Koc97] A.N. Kochubei, Stochastic integrals and stochastic dierential equations over the eld of p-adic nubers, Potential Anal. 6 (997), 05{25. [Koc99], Analysis and probability over innite extensions of a local eld, Potential Anal. 0 (999), 305{325. [Lee90] A.J. Lee, U-statistics. Theory and practice, Statistics: Textbooks and Monographs, no. 0, Marcel Dekker, Inc., New York, 990. [Mac95] I.G. Macdonald, Syetric Functions and Hall Polynoials, 2nd ed., Oxford Matheatical Monographs, Oxford University Press, Oxford, 995. [Mad90] A. Madrecki, Minlos' theore in non-archiedean locally convex spaces, Coent. Math. Prace Mat. 30 (990), 0{. [Mad9], Soe negative results on existence of Sazonov topology in l-adic Frechet spaces, Arch. Math. (Basel) 56 (99), 60{60. [Sat94] T. Satoh, Wiener easures on certain Banach spaces over non-archiedean local elds, Copositio Math. 93 (994), 8{08. [Sch84] W.H. Schikhof, Ultraetric Calculus : an Introduction to p-adic Analysis, Cabridge Studies in Advanced Matheatics, no. 4, Cabridge University Press, Cabridge, 984. [Ser80] R.J. Sering, Approxiation Theores of Matheatical Statistics, Wiley Series in Probability and Matheatical Statistics, John Wiley and Sons, Inc., New York, 980. [Sta99] R.P. Stanley, Enuerative Cobinatorics, Cabridge Studies in Advanced Matheatics, no. 62, Cabridge University Press, Cabridge, 999. M.H. Taibleson, Fourier Analysis on Local Fields, Matheatical Notes, no. 5, Princeton University Press, Princeton, NJ, 975. [Yas96] K. Yasuda, Additive processes on local elds, J. Math. Sci. Univ. Tokyo 3 (996), 629{ 654. E-ail address: evans@stat.berkeley.edu Departent of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, CA , U.S.A