STEIN S METHOD AND THE LAPLACE DISTRIBUTION. John Pike Department of Mathematics Cornell University

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1 STEIN S METHOD AND THE LAPLACE DISTRIBUTION John Pike Department of Mathematics Cornell University jpike@cornell.edu Haining Ren Department of Mathematics University of Southern California hren@usc.edu Astract. Using Stein s method techniques, we develop a framework which allows one to ound the error terms arising from approximation y the Laplace distriution and apply it to the study of random sums of mean zero random variales. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distriutional transformation which is related to zero-iasing. 1. Background and Introduction Beginning with the pulication of Charles Stein s seminal paper [16], proailists and statisticians have developed a wide range of techniques ased on characterizing equations for ounding the distance etween the distriution of a random variale X and that of a random variale Z having some specified target distriution. The metrics for which these techniques are applicale are of the form d H L X, L Z = sup h H E[hX] E[hZ] for some suitale class of functions H, and include as special cases the Wasserstein, Kolmogorov, and total variation distances. The Kolmogorov distance gives the L distance etween the associated distriution functions, so H = {1,a] x : a R}. The total variation and Wasserstein distances correspond to letting H consist of indicators of Borel sets and 1-Lipschitz functions, respectively. The asic idea is to find an operator A such that E[AfX] = for all f elonging to some sufficiently large class of functions F if and only if L X = L Z. For example, Stein showed that Z N, σ if and only if E[A N fz] = E[ZfZ σ f Z] = for all asolutely continuous functions f with f < [16], and shortly thereafter Louis Chen proved that Z Poissonλ if and only if E[A P fz] = E[ZfZ λfz +1] = for all functions f for which the expectations exist []. Similar characterizing operators have since een worked out for several other common distriutions e.g. [6, 11, 1, 13]. 1 Mathematics Suject Classification. 6F5. Key words and phrases. Stein s Method, Laplace distriution, geometric summation. 1

2 STEIN S METHOD AND THE LAPLACE DISTRIBUTION Given a Stein operator A for L Z, one then shows that for every h H, the equation Afx = hx E[hZ] has solution f h F. Taking expectations, asolute values, and suprema yields d H L X, L Z = sup E[hX] E[hZ] = sup E[Af h X]. h H h H In practice, this is usually how one proves that E[AfX] = for f F is sufficient for L X = L Z. The intuition is that since E[AfZ] = for f F, the distriution of X should e close to that of Z when E[AfX] is close to zero. Remarkaly, it is often easier to work with the right-hand side of the aove equation, and the tools for analyzing distances etween distriutions in this manner are collectively known as Stein s method. For more on this rich and fascinating suject, the authors recommend [17, 15, 3, 4]. In this paper, we apply the aove ideas to the Laplace distriution. For a R, >, a random variale W Laplacea, has distriution function { 1 F W w; a, = e w a, w a 1 1 w a e, w a and density f W w; a, = 1 w a e, w R. If W Laplace,, then its moments are given y { E[W k, k is odd ] = k k!, k is even, and its characteristic function is 1 ϕ W t = 1 + t. This distriution was introduced y P.S. Laplace in 1774, four years prior to his proposal of the second law of errors, now known as the normal distriution. Though nowhere near as uiquitous as its younger siling, the Laplace distriution appears in numerous applications, including image and speech compression, options pricing, and modeling sizes of sand particles, diamonds, and eans. For more properties and applications of the Laplace distriution, the reader is referred to the text [1]. Our interest in the Laplace distriution was sparked y the fact that if X 1, X,... is a sequence of random variales satisfying certain technical assumptions and Np X i converges N p Geometricp is independent of the X i s, then the sum p 1 weakly to the Laplace distriution as p [1]. Such geometric sums arise in a variety of settings [9], and the general setup distriutional convergence of sums of random variales is exactly the type of prolem for which one expects Stein s method computations to yield useful results. Indeed, Erol Peköz and Adrian Röllin have applied Stein s method arguments to generalize a theorem due to Rényi concerning the convergence of sums of a random numer of positive random variales to the exponential distriution [1]. By an analogous line of reasoning, we are ale to carry out a similar program for convergence of random sums of certain mean zero random variales to the Laplace distriution.

3 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 3 We egin in Section y introducing a Stein operator which we show completely characterizes the mean zero Laplace distriution. Specifically, we prove Theorem 1.1. Let W Laplace, and define the operator A y Afx = fx f f x. Then E[AgW ] = for every function g such that g and g are locally asolutely continuous and E g W, E g W <. Conversely, if X is any random variale such that E[AgX] = for every twicedifferentiale function g with g, g, g <, then X Laplace,. In Section 3, we use this characterization to ound the distance to the Laplace distriution. For technical reasons, we work in the ounded Lipschitz metric, d BL, which is defined in terms of 1-Lipschitz test functions with sup norm 1. We egin y defining the centered equilirium transformation X X L in terms of the functional equation E[fX] f = 1 E[X ]E[f X L ] for all twice-differentiale functions f such that f, f, and f are ounded. After estalishing that X L exists whenever X has mean zero and finite nonzero variance, we derive the following coupling ound. Theorem 1.. Suppose that X is a random variale with E[X] =, E[X ] =,, and let X L have the centered equilirium distriution for X. Then d BL L X, Laplace, + E X X L. Finally, in Section 4 we apply these tools to the study of random sums of mean zero random variales. As a special case, we show Theorem 1.3. Let X 1, X,... e a [ sequence of independent random variales with E[X i ] =, E[Xi ] =, sup i 1 E X i 3] = ρ <, and let N Geometricp with strictly positive support e independent of the X i s. Then N d BL L p 1 X i, Laplace, p 1 for all p, ρ 6. Characterizing the Laplace Distriution Our first order of usiness is to estalish a characterizing operator for the Laplace distriution. As is typical in Stein s method constructions, we split the proof of Theorem 1.1 into two parts. We egin with Lemma.1. Suppose that W Laplace,. If g and g are locally asolutely continuous with E g W, E g W <, then E[gW ] g = E[g W ].

4 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 4 Proof. Applying Fuini s theorem twice shows that g xe x dx = g 1 y x dy dx = 1 = 1 = 1 ˆ y = 1 = 1 = 1 x e g xe y dxdy y g ye y g dy e y dy g 1 y e z dz dy g ˆ z g ye z dydz g gze z dz g gze z dz g g. e z dz g Setting hy = g y, it follows from the previous calculation that ˆ g xe x dx = g ye y dy = h ye y dy = 1 = 1 = 1 ˆ Summing the aove expressions gives hze z h dz h g ze z g dz + g gze z dz g + g. ˆ E[g W ] = 1 g xe x 1 dx + g xe x dx = 1 [ ˆ 1 gze z g dz + g ˆ 1 + gze z g dz = 1 ˆ 1 gze z g dz ] g = 1 E[gW ] g. Note that since the density of a Laplace, random variale is given y f W w = w 1 e, the density method [3] suggests the following characterizing equation for the Laplace distriution: g w 1 sgnwgw = g w + f W w gw =, f W w

5 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 5 and indeed one can verify that if W Laplace,, then E[g W ] = 1 E[sgnW gw ] for all asolutely continuous g for which these expectations exist. Thus if g is such a function as well, setting Gw = sgnw gw g gives E[g W ] = 1 E[sgnW g W ] = 1 E[G W ] = 1 E[sgnW GW ] = 1 E[gW ] g, so the general form of the equation in Theorem 1.1 can e ascertained y iterating the density method. Alternatively, it is known [15] that if Z Exponential1, then E[g Z] = E[gZ] g for all asolutely continuous g with E g Z <. Thus if g is also asolutely continuous and E g Z <, then E[g Z] = E[g Z] g = E[gZ] g g. Using this oservation, one can derive the equation in Lemma.1 y noting that if J is independent of Z with PJ = 1 = PJ = 1 = 1, then W = JZ has the Laplace, distriution. We include each of these approaches ecause their analogues may e variously applicale in different situations involving the construction of characterizing equations. Oserve that there is an iterative step involved in each case. By manipulating the integral defining E[g W ] or using the usual Stein equation for the exponential along with the representation W = JZ, one arrives at the first order equation E[g W ] = 1 E[sgnW gw ] suggested y one application of the density method. However, we were not ale to get much mileage out of the operator Ãgx = g x 1 sgnxgx, while the second-order operator Agx = gx g g x turned out to e quite effective. This idea of iterating more traditional procedures to otain higher order characterizing equations which are simpler to work with is one of the key insights of this paper. Now, in order to estalish the second part of Theorem 1.1, we will show that any X satisfying the hypotheses has d BL L X, Laplace, = where d BL denotes the ounded Lipschitz distance given y d BL L X, L Y = sup E[hX] E[hY ], h H BL H BL = {h : h 1 and hx hy x y for all x, y R}. The claim will follow since d BL is a metric on the space of Borel proaility measures on R [19]. In keeping with the general strategy laid out in the introduction, we consider the initial value prolem gx g x = hx W h, g = where h H BL and W h := E[hW ], W Laplace,.

6 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 6 Lemma.. For h H BL, W Laplace,, hx = hx W h, a ounded, twice-differentiale solution to the initial value prolem is given y g h x = 1 gx g x = hx, g = e x x e y hydy + e x ˆ x e y hydy. This solution satisfies g h, g h, g h 4 and g h + 3. Proof. The general solution to the homogeneous equation g x gx = is given y g x = C 1 e x + C e x, so, since the associated Wronskian is nonzero, the variation of parameters method suggests that a solution to the inhomogeneous equation g x gx = hx is where u h x = 1 Differentiation gives x g h x = u h xe x + vh xe x e y hydy, vh x = 1 ˆ x g hx = hx u hxe x 1 + hx 1 v hxe x 1 = and thus g hx = 1 hx + u h xe x + vh xe x e y hydy. uh xe x vh xe x, = 1 g h x hx, so g h is indeed a solution. To see that the initial condition is satisfied, we oserve that g h = u h + v h = 1 = hyf W ydy W h e y hydy = f W y hy W h dy f W ydy = W h W h =. Moreover, since h 1, ˆ W h = 1 hxe x dx 1 e x dx = 1, and thus hx hx + W h. Consequently, and u h xe x 1 e x vh xe x 1 e x x ˆ x e y dy = 1 e y dy = 1, for all x R, and the ounds on g h and g h follow. Noting that g h x = 1 g h x hx and thus g h x = 1 g 3 h x 1 h x completes the proof since h and h 1.

7 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 7 With the preceding result in hand, we can finish of the proof of Theorem 1.1 via Lemma.3. If X is a random variale such E[gX] g = E[g X] for every twice-differentiale function g with g, g, g <, then X Laplace,. Proof. Let W Laplace, and, for h H BL, let g h e as in Lemma.. Because g h = and g h, g h, g h are ounded, it follows from the aove assumptions that E[hX] E[hW ] = E[g h X g hx] =. Taking the supremum over h H BL shows that d BL L X, L W =. Before moving on, we oserve that the reason we are working with the ounded Lipschitz distance is that the ounds on g h and its derivatives depended on oth h and h having finite sup norm. As d BL is not especially common at least explicitly in the Stein s method literature, we conclude this section with a proposition relating it to the more familiar Kolmogorov distance d K L X, L Y = sup P{X x} P{Y x}. x R Proposition.4. If Z is an asolutely continuous random variale whose density, f Z, is uniformly ounded y a constant C <, then for any random variale X, d K L X, L Z C + dbl L X, L Z. Proof. We first note that the inequality holds trivially if d BL L X, L Z = as d BL and d K are metrics. Also, since d K P, Q 1 for all proaility measures P and Q, C+ 1, and d BL L X, L Z 1 implies d BL L X, L Z 1, we have d K L X, L Z 1 C + dbl L X, L Z whenever d BL L X, L Z 1. d BL L X, L Z, 1. Now, for x R, ε >, write h x z = 1,x] z = Then for all x R, Thus it suffices to consider the case where { 1, z x, z > x and h x,εz = 1, z x 1 z x ε, z x, x + ε]., z > x + ε E[h x X h x Z] = E[h x X] E[h x,ε Z] + E[h x,ε Z] E[h x Z] ˆ x+ε E[h x,ε X] E[h x,ε Z] + 1 z x f Z zdz x ε E[h x,ε X] E[h x,ε Z] + Cε.

8 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 8 Since d BL L X, L Z, 1, if we take ε = d BL L X, L Z, 1, then εh x,ε H BL and thus E[h x X h x Z] 1 ε E[εh x,εx] E[εh x,ε Z] + Cε 1 ε d BL L X, L Z + Cε = C + dbl L X, L Z. A similar argument using E[h x Z h x X] = E[h x Z] E[h x ε,ε Z] + E[h x ε,ε Z] E[h x X] Cε + E[h x ε,εz] E[h x ε,ε X] shows that E[h x X] E[h x Z] C + dbl L X, L Z for all x R, and the proposition follows y taking suprema. 1 Remark. When C 1, we can take ε = C d BL L X, L Z in the aove argument to otain an improved ound of d K L X, L Z 3 CdBL L X, L Z. To the est of the authors knowledge, the aove proposition is original, though the proof follows the same asic line of reasoning as the well-known ound on the Kolmogorov distance y the Wasserstein distance see Proposition 1. in [15]. It seems that the primary reason for using the Wasserstein metric, d W, is that it enales one to work with smoother test functions while still implying convergence in the more natural Kolmogorov distance. Proposition.4 shows that d BL also upper-ounds d K while enjoying all of the resulting smoothness of Wasserstein test functions and with additional oundedness properties to oot. Moreover, the Wasserstein distance is not always well-defined e.g. if one of the distriutions does not have a first moment, whereas d BL always exists. Finally, d BL is a fairly natural measure of distance since it metrizes weak convergence [19]. However, we always have d W L X, L Z d BL L X, L Z, and it is possile for a sequence to converge in d BL ut not in d W or d K. Furthermore, the ounded Lipschitz metric does not scale as nicely as the Kolmogorov or Wasserstein distances when the associated random variales are multiplied y a positive constant. For the remainder of this paper, we will state our results in terms of d BL with the corresponding Kolmogorov ound eing implicit therein, though one should note that, as with the Wasserstein ound on d K, Kolmogorov ounds otained in this fashion are not necessarily optimal, often giving the root of the true rate. 3. The Centered Equilirium Transformation Our next task is to use the characterization in Theorem 1.1 to otain ounds on the error terms resulting from approximation y the Laplace distriution. To this end, we introduce the following definition.

9 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 9 Definition 3.1. For any nondegenerate random variale X with mean zero and finite variance, we say that the random variale X L has the centered equilirium distriution with respect to X if E[fX] f = 1 E[X ]E[f X L ] for all twice-differentiale functions f such that f, f, and f are ounded. We call the map X X L the centered equilirium transformation. Note that the centered equilirium distriution is uniquely defined ecause E[f X] = E[f Y ] for all twice continuously differentiale functions with compact support implies X = d Y. The nomenclature is in reference to the equilirium distriution from renewal theory, which was used in a similar manner in [1] for a related prolem involving the exponential distriution. Since the characterizing equation for the Laplace distriution involves second derivatives and a variance term, one expects some kind of relation etween X L and the zero ias distriution for X defined y E[XfX] = E[X ]E[f X z ] for all asolutely continuous f for which E XfX < [7] in much the same way as the equilirium distriution is related to the size ias distriution [1]. The following theorem shows that this is indeed the case. Moreover, since the zero ias distriution is defined for any X with E[X] = and VarX,, it will follow that every such random variale has a centered equilirium distriution. Theorem 3.. Suppose that X has mean zero and variance σ,. Let X z have the zero ias distriution with respect to X and let B Beta, 1 e independent of X z. Then X L := BX z satisfies E[fX] f = σ E[f X L ] for all twice-differentiale f with f, f, f <. Proof. Applying the fundamental theorem of calculus, Fuini s theorem, the definition of X z, and the fact that B has density px = x1 [,1] x gives [ˆ 1 ] ˆ 1 E[fX] f = E Xf uxdu = E [Xf ux] du ˆ 1 [ˆ 1 = σ ue[f ux z ]du = σ E = σ E [f BX z ] The assumptions ensure that all of the functions are integrale. ] uf ux z du Corollary 3.3. If X has variance E[X ] = and X L has the centered equilirium distriution with respect to X, then X L is asolutely continuous with density f X Lx = 1 ˆ 1 [ E X; X > x ] dv. v Proof. X L = d BX z with B and X z as in Theorem 3., so the claim follows from the fact [7] that X z is asolutely continuous with density f X zx = 1 E[X; X > x] y the usual method of computing the density of a product.

10 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 1 Remark. In an earlier version of this paper, we estalished the existence of the centered equilirium distriution y showing that for certain random variales X, X L can e otained y iterating the X P ias transformation from [8] with P x = sgnx. Though there may e some merit to such a strategy and it provides another example of how results for higher order Stein operators may e otained y iterating more traditional techniques, in our case it required the rather artificial assumption that the variates in the domain of the transformation have median zero. Those interested in the iterated X P ias approach are referred to the article [5] y Christian Döler, which contains the essential technical details of our original argument and suggests certain improvements. Lemma.3 shows that, up to scaling, the mean zero Laplace distriution is the unique fixed point of the centered equilirium transformation. Thus one expects that if a random variale is close to its centered equilirium transform, then its distriution is close to the Laplace law. Theorem 1. formalizes this intuition. Proof of Theorem 1.. If X is a random variale with E[X ] = < and X L has the centered equilirium distriution for X, then for all h H BL, taking g h as in Lemma., we see that W h E[hX] = E[g h X g hx] = E[ g hx L g hx] E g hx L g hx g h E X L X = + E X X L. For the example in Section 4, we will also need the following complementary result. Proposition 3.4. If Y L has the centered equilirium distriution for Y and E[Y ] =, then E Y Y L E Y E[ Y 3 ]. Proof. We may assume that E[ Y 3 ] < as the inequality is trivial otherwise. The result will follow immediately from the triangle inequality if we can show that E Y L = 1 6 E Y 3, which is what one would otain y formally plugging fy = y 3 into the definition of the transformation Y Y L. Of course, neither f, f, nor f is ounded, so we must proceed y approximation. To this end, define x 3, x n n 3 + 3n x n 3 x n, n < x n + n f n x = n 3 + 3n n + n x 3 n + n x, n + n < x n + n n + n x 3, n + n < x n + n, x > n + n By construction, f n is smooth with compact support and satisfies f n i f i for all n N, i =, 1,, thus fulfilling the conditions in the definition of the centered

11 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 11 equilirium transformation when E[ Y 3 ] <. Moreover, f n i f i pointwise for i =, 1,, so it follows from the dominated convergence theorem that E[fY ] = lim n E[f ny ] = lim n E[f ny L ]. Fatou s lemma shows that f Y L is integrale since E [ f Y L ] [ ] = E lim inf f ny L lim inf E [ f ny L ] = 1 E[fY ] <, n n so another application of dominated convergence gives E[fY ] = lim n E[f ny L ] = E[f Y L ]. The same general argument shows that if Y L has the centered equilirium distriution for Y and E [ Y n ] <, then E[qY ] q = 1 E[Y ]E[q Y L ] for any polynomial q of degree at most n. As is often the case with distriutional transformations defined in terms of functional equations, we find it more convenient to define X X L in terms of a relatively small class of test functions and then argue y approximation when we want to apply the relation more generally. 4. Random Sums The p-geometric summation of a sequence of random variales X 1, X,... is defined as S p = X 1 + X X Np where N p is geometric with success proaility p - that is, P{N p = n} = p1 p n 1, n N - and is independent of all else. A result due to Rényi [14] states that if X 1, X,... are i.i.d., positive, nondegenerate random variales with E[X i ] = 1, then L ps p Exponential1 as p. In fact, just as the normal law is the only nondegenerate distriution with finite variance that is stale under ordinary summation in the sense that if X, X 1, X,... are i.i.d. nondegenerate random variales with finite variance, then for every n N, there exist a n >, n R such that X = d a n X X n + n, it can e shown that if X, X 1, X,... are i.i.d., positive, and nondegenerate with finite variance, then there exists a p > such that a p X X Np =d X for all p, 1 if and only if X has an exponential distriution. Similarly, if we assume that Y, Y 1, Y,... are i.i.d., symmetric, and nondegenerate with finite variance, then there exists a p > such that a p Y Y Np =d Y for all p, 1 if and only if Y has a Laplace distriution. Moreover, it must e the case that a p = p 1. In addition, we have an analog of Rényi s theorem [1]: Theorem 4.1. Suppose that X 1, X,... are i.i.d., symmetric, and nondegenerate random variales with finite variance σ, and let N p Geometricp e independent of the X i s. If X i d X as p, a p N p then there exists γ > such that a p = p 1 γ + op 1 and X has the Laplace distriution with mean and variance σ γ.

12 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 1 A recent theorem due to Alexis Toda [18] gives the following Lindeerg-type conditions for the existence of the distriutional limit in Theorem 4.1. Theorem 4. Toda. Let X 1, X,... e a sequence of independent ut not necessarily identically distriuted random variales such that E[X i ] = and VarX i = σi, and let N p Geometricp independent of the X i s. Suppose that lim n n α σn = for some < α < 1, and for all ɛ >, lim p σ 1 := lim n n n σi > exists, 1 p i 1 pe[xi ; X i ɛp 1 ] =. Then as p, the sum p 1 Np X i converges weakly to the Laplace distriution with mean and variance σ. Remark. The original statement of Toda s theorem is slightly more general, allowing for convergence to a possily asymmetric Laplace distriution. In 11, Peköz and Röllin were ale to generalize Rényi s theorem y using a distriutional transformation inspired y Stein s method considerations [1]. Specifically, for a nonnegative random variale X with E[X] <, they say that X e has the equilirium distriution with respect to X if E[fX] f = E[X]E[f X e ] for all Lipschitz f and use this to ound the Wasserstein and Kolmogorov distances to the Exponential1 distriution. The equilirium distriution arises in renewal theory, ut its utility in analyzing convergence to the exponential distriution comes from the fact that a Stein operator for the exponential distriution with mean one is given y A E fx = f x fx+f, so X Exponential1 is the unique fixed point of the equilirium transformation [15]. This transformation and the similarity etween our characterization of the Laplace and the aove characterization of the exponential inspired our construction of the centered equilirium transformation, and the fact that oth distriutions are stale under geometric summation led us to parallel their argument for ounding the distance etween p-geometric sums of positive random variales and the exponential distriution in order to otain corresponding results for the Laplace case. Our results are summarized in the following theorem. Theorem 4.3. Let N e any N-valued random variale with µ = E[N] < and let X 1, X,... e a sequence of independent random variales, [ independent of N, with N ] [ E[X i ] =, and E[Xi ] = N ] σ i,. Set σ = E = E X i σ i and let M e any N-valued random variale, independent of the X i s and defined on the same space as N, satisfying P{M = m} = σ m P{N m}. σ

13 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 13 Then d BL L µ 1 N X i m=1, Laplace, µ σ σ µ E XM X L Proof. We first note that m σ = P{N = m} σi = M + sup i 1 P{N m}σm, m=1 [ ] σ i E N M 1. so M is well-defined. Now, taking V = µ 1 N X i, we claim that V L = µ 1 M 1 X i + XM L has the centered equilirium distriution with respect to V. Throughout, Xm L is taken to e independent of M, N, and X k for k m. To see that this is so, let f e any function satisfying the assumptions in Definition 3.1. Then, using the notation m X = {X i } i 1, g m X = f µ 1 X i, f s x = f µ 1 s + µ 1 x, letting ν m denote the distriution of S m 1 = and oserving that, y independence, m 1 E[h Xm S L m 1 = s] = E[h Xm] L = σm E[hX m h] = σm E[hX m h S m 1 = s] for all s R and all twice differentiale h with h, h, h ounded, we see that E [f µ 1 m 1 ] ˆ X i + µ 1 X L m = X i, [ ] E f µ 1 s + µ 1 X L m S m 1 = s dν m s ˆ = E [ µf s Xm L S m 1 = s ] dν m s = µ ˆ σm E [f s X m f s S m 1 = s] dν m s = µ ˆ [ σm E fµ 1 s + µ 1 Xm [ = µ σm E f µ 1 fµ 1 s Sm 1 = s m X i f = µ σm E [g m X g m 1 X] µ 1 ] dν m s m 1 X i ]

14 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 14 for all m N, hence E[f V L ] = P{M = m}e [f m=1 = µ σ σ σ m=1 m = E[V ] E µ 1 m 1 X i + µ 1 X L m ] P{M = m}e [g m X g m 1 X] [ ] P{N m} g m X g m 1 X m=1 = E[V ] E[g N X] g X = E[V E[fV ] f. ] Having shown that V L does in fact have the centered equilirium distriution for V, we can apply Theorem 1. with B = E[V ] = σ µ to otain d BL L V, Laplace, B 1 + E V V L B = µ σ N M E XM XM L + sgnn M µ E X M XM L + E σ i=n M+1 N M X i i=n M+1 X i. Finally, since the X i s are independent with mean zero, the Cauchy-Schwarz inequality gives E N M i=n M+1 X i = P{N M = j, N M = k}e j=1 k=1 P{N M = j, N M = k}e j=1 k=1 j=1 k=1 σ i = sup i 1 and the proof is complete. j+k i=j+1 j+k i=j+1 P{N M = j, N M = k}k 1 sup σ i i 1 k=1 X i X i [ ] k 1 P{ N M = k} = sup σ i E i 1 N M 1, Remark. [ When the summands in Theorem 4.3 have common variance σ1, we have N ] σ = E = µσ1, so M is defined y σ 1 P{M = m} = σ m σ P{N m} = 1 P{N m}. µ In other words, M is the discrete equilirium transformation of N defined in [13] for use in geometric approximation. Thus the E[ N M 1 ] term in the ound 1

15 STEIN S METHOD AND THE LAPLACE DISTRIBUTION 15 from Theorem 4.3 may e regarded as measuring how close L N is to a geometric distriution. If the summands are i.i.d., then X M = d X 1 as M is assumed to e independent of all else, so Theorem 1. shows that the E XM XM L term measures how close the distriution of the summands is to the Laplace. To put this into context, recall that if the X i s are i.i.d. Laplace, and N Geometricp, then p 1 N X i Laplace,. We conclude our discussion with a proof of Theorem 1.3, which gives sufficient conditions for weak convergence in the setting of Theorem 4.1. Though it requires that the X i s have uniformly ounded third asolute moments, the condition of symmetry is dropped and the identical distriution assumption is reduced to the requirement that the X i s have common variance. This result is not quite as general as Theorem 4., ut it does provide ounds on the error terms. Proof of Theorem 1.3. We are trying to ound the distance etween the Laplace, distriution and that of p 1 N X i where X 1, X,... are independent mean zero random variales with common [ variance E[Xi ] = and uniformly ounded third asolute moments sup i 1 E X i 3] = ρ <, and N Geometricp is independent of the X i s. In the language of Theorem 4.3, we have µ = 1 p, σ = µ, and σm P{N m} = pp{n m} = p σ 1 p m 1 = p1 p m 1 = P{N = m}, i=m so we can take M = N to otain N d BL L p 1 X i, Laplace, p 1 p 1 + E XN XN L. Applying Proposition 3.4 gives E XN XN L = P{N = n}e Xn Xn L and the result follows n=1 n=1 n=1 P{N = n} E X n + 1 [ 6 E X n 3] P{N = n} E [ Xn ] 1 + ρ 6 = + ρ 6, Acknowledgements The authors would like to thank Larry Goldstein for suggesting the use of Stein s method to get convergence rates in the general setting of Theorem 4. and for many helpful comments throughout the preparation of this paper. Thanks also to Alex Rozinov for several illuminating conversations concerning the material in Section 4 and to the anonymous referees whose careful notes were of great help to us.

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