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Orthogonal Polynomials in Stein's Method

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Orthogonal Polynomials in Stein's Method Orthogonal Polynomials in Steins Metho d Wim Schoutens KatholiekeUniversiteit Leuven Department of Mathematics Celestijnenlaan B B Leuven Belgium Abstract We discuss Steins metho d for Pearsons and Ords family of distributions We giveasys tematic treatment including the Stein equation its solution and smo othness conditions Akey role in the analysis is played by the classical orthogonal p olynomials AMS Subject Classication J H Keywords Steins Method Orthogonal Polynomials Pearsons Class Ords Class Approximation Distributions Markov Processes Postdo ctoral Researcher of the Fund for Scientic Research Flanders Belgium Running Head Steins Method Orthogonal Polynomials Mailing Address Wim Schoutens Katholieke Universiteit Leuven Department of Math ematics Celestijnenlaan B B Leuven Belgium Email WimSchoutenswiskuleuvenacbe Intro duction Steins Metho d provides a way of nding approximations to the distribution say of a random variable which at the same time gives estimates of the approximation error involved The strenghts of the metho d are that it can b e applied in many circumstances in which dep endence plays a part In essence the metho d is based on a dening equation or equivalently an op erator of the distribution and a related Stein equation Up to now is was not clear which equation to take One could think at a lot of equations We show how for a broad class of distributions there is one equation who has a sp ecial role We give a systematic treatment including the Stein equation its solution and smo othness conditions A key to ol in Steins theory is the generator metho d develop ed by Barb our Barb our suggested employing for the op erator of the Stein equation the generator of a Markov pro cess In us lo oks for a dening equation for which is related to the generator this generator metho d one th of a Markov pro cess For a given distribution there may be various Markov pro cesses who t in Barb ours metho d However up to now it was still not clear which Markov pro cess to take to obtain good results We show how for a broad class of distributions there is a sp ecial Markov pro cess a birth and death pro cess or a diusion which takes a leading role in the analysis Furthermore a key role is played by the classical orthogonal p olynomials P Diaconis and S Zab ell already mentioned this connection It turns out that the dening op erator is based on a hyp ergeometric dierence or dierential equation which lies at the heart of the classical orthogonal p olynomials Furthermore the sp ectral representation of the transition probabilities of the Markov pro cess involved will b e in terms of orthogonal p olynomials closely related to the distribution to b e approximated This systematic treatment together with the intro duction of orthogonal p olynomials in the analysis seems to b e new Furthermore some earlier uncovered examples like the Beta the Students t and the Hyp ergeometric distribution are nowworked out Prelimaries Birth and Death Pro cesses and Diusions A birth and death process fX t g is a Markov pro cess on the state space S f g with t stationary transition probabilities ie P t PrX j jX i i j S is not dep ending on ij ts s s and with innitesimal generator A given by Af i f i f i f i i S i i i i for all b ounded realvalued functions f B S and where we take for i not on the i i b oundary of S On the b oundary of S we must have We will always work with i i S N f g in which case we set or take S f Ng with N a p ositive in teger and in which case Furthermore we do not allow the existence of an N absorbing state ie a b oundary state with birth parameter and death parameter equal to zero The parameters and are called resp ectivelythebirth and death rates It can b e shown that i i the limits lim P tp j S exist and are indep endentof the initial state i It turns out t ij j P j S where j S nfg that the p are given by p j j j j j j k k S P and and In order that the sequence fp g denes a distribution we must have j k k P P p We say that fp g is the limiting stationary distribution If then clearly j k k k k then all p are zero and we do not have a limiting stationary distribution j In the analysis of birth and death pro cesses a prominent role is played by a sequence of p olynomials fQ xn S g called birthdeath polynomials They are determined uniquely by n the recurrence relation xQ x Q x Q x Q x n S n n n n n n n n together with Q xandQ x Karlin and McGregor proved that the transition function P can b e represented as Z xt e Q xQ xdx i j S t P t i j ij j where is a p ositive Borel measure with total mass and with supp ort on the nonnegative real axis is called the spectral measure of P Taking t in one easily sees that the p olynomials fQ xn S g are orthogonal with resp ect to In the examples we will encounter a variety of n birth and death pro cesses Other examples can be found in the litterature see for example and Another class of Markov pro cesses will app ear also in the analysis Diusions with state space S a b a b We refer to for a general intro duction Supp ose A is the generator of the diusion In a clear pro of is given of the fact that A is of the form Af xxf x xf x where x is called the drift co ecient and x the diusion co ecient We will highlight the sp ectral representation for some of diusion pro cesses in the examples see also Steins Metho d Normal Approximation and Poisson Approximation In Stein published a metho d to prove Normal approximation It is based on the fact that a random variable Z has a Standard Normal distribution N if and only if for all dieren tiable functions f suchthat E jf X j whereX has a Standard Normal distribution N Z E f Z Zf Hence it seems reasonable that if E f W WfW is small for a large class of functions f then the distribution of W is close to the Standard Normal distribution Supp ose we wish to estimate the dierence b etween the exp ectation of a smo oth function h with resp ect to the random variable W and E hZ where Z has a Standard Normal distribution Stein showed that for any smo oth realvalued b ounded function h there is a function f f solving the now called Stein h equation for the Standard Normal f x xf xhx E hZ with Z a Standard Normal random variable The unique b ounded solution of the ab ove equation is given by Z x f xexpx hy E hZ exp y dy h Then we estimate E f W Wf W h h and hence E h W E hZ The next step is to show that the quantity is small In order to do this we will use the structure of W For instance it mightbethat W is a sum of indep endent random variables In addition we will use some smo othness conditions on f Stein showed the h following inequalities r jjf jj jjh E hZ jj h jjf jj suph inf h h jjf jj jjh jj h where jj jj denotes the supremum norm In this way we can bound the distance of W from the Normal in terms of a test function h the immediate b ound of the distance is one of the key advantages of Steins metho d compared to moment generating functions or characteristic functions Chen applied Steins idea in the context of Poisson approximation The Stein equation for the Poisson distribution Pisnow a dierence equation x xf x hx E hZ f where h is a b ounded realvalued function dened on the set of the nonnegativeintegers and Z has a Poisson distribution P The choice of the left hand side of equation is based on the fact that a random variable W on the set of the nonnegativeintegers has a Poisson distribution with parameter if and only if for all b ounded realvalued functions f on the integers E f W WfW The solution of the Stein equation for the Poisson distribution Pisgiven by x k X x x hk E hZ f xx h k k This solution is the unique except at x b ounded solution the value f x for negative x do es h not enter into consideration and is conventionally taken to be zero In one nds the following estimates of the smo othness for f by an analytic argument h jjf jj jjhjj min andjjf jj jjhjj min h h where f xf x f x General Approach For an arbitrary distribution the general pro cedure is Find a go o d characterization of the desired distribution in terms of an equation that is of the t yp e Z is a rv with distribution if and only if E Af Z for all smo oth functions f where A is an op erator asso ciated with the distribution Thus in the standard normal case Af xf x xf x x R We will call such an op erator a Stein operator Next assume Z to have distribution and consider the Stein equation hx E hZ Af x or anyrandomvariable W For every smo oth h nd a corresp onding solution f of this equation F h E hW E hZ E Af W Hence to estimate the proximityof W and Z it is sucientto h estimate E Af W for all p ossible solutions of h However in this pro cedure it is not completely clear which characterizing equation for the distribution to cho ose one could think of a whole set of p ossible equations The aim is to b e able to solve for a suciently large class of functions htoobtainconvergence in a known top ology Barb ours Generator Metho d Akey to ol in Steins theory is the generator metho d develop ed by Barb our Replacing f by f in the Stein equation for the Standard Normal gives f x xf x hx E hZ If we equation can be rewritten as A f hx E hZ The key set A f xf x xf x this advantage is that A is also the generator of a Markov pro cess the OrnsteinUhlenbeck pro cess with Standard Normal stationary
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