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Unformatted text preview: Math Camp 1: Functional analysis Sayan Mukherjee + Alessandro Verri About the primer Goal To brie y review concepts in functional analysis that will be used throughout the course.∗ The following concepts will be described 1. Function spaces 2. Metric spaces 3. Convergence 4. Measure 5. Dense subsets ∗ The de nitions and concepts come primarily from \Introductory Real Analysis" by Kolmogorov and Fomin (highly recommended). 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Separable spaces Complete metric spaces Compact metric spaces Linear spaces Linear functionals Norms and semi-norms of linear spaces Convergence revisited Euclidean spaces Orthogonality and bases Hilbert spaces 16. Delta functions 17. Fourier transform 18. Functional derivatives 19. Expectations 20. Law of large numbers Function space A function space is a space made of functions. Each function in the space can be thought of as a point. Examples: 1. 2. 3. , the set of all real-valued continuous functions in the interval a, b] C a, b , the set of all real-valued functions whose absolute value is integrable in the interval a, b] L1 a, b , the set of all real-valued functions square integrable in the interval a, b] L2 a, b Note that the functions in 2 and 3 are not necessarily continuous! Metric space By a metric space is meant a pair (X, ρ) consisting of a space X and a distance ρ, a single-valued, nonnegative, real function ρ(x, y) de ned for all x, y ∈ X which has the following three properties: 1. ρ(x, y) = 0 i x =y 2. ρ(x, y) = ρ(y, x) 3. Triangle inequality: ρ(x, z) ≤ ρ(x, y) + ρ(y, z) Examples 1. The set of all real numbers with distance ρ(x, y ) = |x − y | is the metric space IR1. 2. The set of all ordered n-tuples x = (x1 , ..., xn) of real numbers with distance ( ρ x, y )= is the metric space IRn. n =1 i (xi − yi)2 3. The set of all functions satisfying the criteria f 2 (x)dx < ∞ with distance ( ( ) ( )) = (f1(x) − f2(x))2dx is the metric space L2(IR). ρ f1 x , f2 x 4. The set of all probability densities with Kullback-Leibler divergence p (x) ρ(p1 (x), p2 (x)) = ln 1 p1 (x)dx p2 (x) is not a metric space. The divergence is not symmetric ρ(p1 (x), p2(x)) = ρ(p2 (x), p1 (x)). Convergence An open/closed sphere in a metric space S is the set of points x ∈ IR for which ρ(x0 , x) < r open ρ(x0 , x) ≤ r closed. An open sphere of radius with center x0 will be called an -neighborhood of x0, denoted O (x0). A sequence of points {xn} = x1, x2, ..., xn, ... in a metric space S converges to a point x ∈ S if every neighborhood O (x) of x contains all points xn starting from a certain integer. Given any > 0 there is an integer N such that O (x) contains all points xn with n > N . {xn} converges to x i lim ρ(x, xn) = 0. n→∞ Measure Throughout the course we will see integrals of the form V (f (x), y )dν (x) → V (f (x), y ) p(x)dx ν (x) is the measure. The concept of the measure ν (E ) of a set E is a natural extension of the concept 1) The length l( ) of a line segment 2) The volume V (G) of a space G 3) The integral of a nonnegative function of a region in space. Lebesgue measure Let f be a ν -measurable function (it has nite measure) taking no more than countably many distinct values y1, y2, ..., yn, ... Then by the Lebesgue integral of f over the set A denoted f (x) dν, A we mean the quantity ynν (An) n where An = {x : x ∈ A, f (x) = yn}, provided the series is absolutely convergent. The measure ν is the Lebesgue measure. Lebesgue integral We can compute the integral f (x)dx by adding up the area under the red rectangles. 2 1.8 1.6 1.4 f(x) 1.2 1 0.8 0.6 0.4 0.2 0 −10 −8 −6 −4 −2 0 x 2 4 6 8 10 Riemann integral The more tradition form of the integral is the Riemann integral. The intuition is that of limit of an in nite sum of in nitesimally small rectangles, f (x)dx = f (xn) x. A n Integrals in the Riemann sense require continuous or piecewise continuous functions, the Lebesgue from shown previously relaxes this. Thus, the integral 1 f (x)dx 0 with f : 0, 1] → IR de ned as 1 if t is rational f= 0 otherwise does not exist in the Riemann sense. Lebesgue-Stieltjes integral Let F be a nondecreasing function de ned on a closed interval a, b] and suppose F is continuous from the left at every point a, b). F is called the generating function of the Lebesgue-Stieltjes measure νF . The Lebesgue-Stieltjes integral of a function f is denoted by b f (x) dF (x) a which is the Lebesgue integral f (x) dνF . a,b] An example of dνF is a probability density p(x)dx. Then νF would correspond to the cumulative distribution function. Dense Let A and B be subspaces of a metric space IR. A is said to be dense in B if A ⊂ B. A is the closure of the subset A. In particular A is said to be everywhere dense in IR if A = R. A point x ∈ IR is called a contact point of a set A ∈ IR if every neighborhood of x contains at least on point of A. The set of all contact points of a set A denoted by A is called the closure of A. Examples 1. The set of all rational points is dense in the real line. 2. The set of all polynomials with rational coe cients is dense in C a, b]. 3. Let K be a positive de nite Radial Basis Function then the functions n f (x) = ciK (x − xi) i=1 is dense in L2. Note: A hypothesis space that is dense in L2 is a desired property of any approximation scheme. Separable A metric space is said to be separable if it has a countable everywhere dense subset. Examples: 1. The spaces IR1, IRn, L2 a, b], and C a, b] are all separable. 2. The set of real numbers is separable since the set of rational numbers is a countable subset of the reals and the set of rationals is is everywhere dense. Completeness A sequence of functions fn is fundamental if such that ∀n and m > N , ρ(fn, fm) < . ∀> 0 ∃N A metric space is complete if all fundamental sequences converge to a point in the space. , L1, and L2 are complete. That C2 is not complete, instead, can be seen through a counterexample. C Incompleteness of C2 Consider the sequence of functions (n = 1, 2, ...) −1 if − 1 ≤ t < −1/n φn(t) = nt if − 1/n ≤ t < 1/n 1 if 1/n ≤ t ≤ 1 and assume that φn converges to a continuous function φ in the metric of C2. Let −1 if − 1 ≤ t < 0 f (t) = 1 if 0 ≤ t ≤ 1 Incompleteness of C2 (cont.) Clearly, (f (t) − φ(t))2dt 1/2 ≤ (f (t) − φn(t))2dt 1/2 + (φn(t) − φ(t))2dt Now the l.h.s. term is strictly positive, because f (t) is not continuous, while for n → ∞ we have (f (t) − φn(t))2dt → 0. Therefore, contrary to what assumed, φn cannot converge to φ in the metric of C2. 1/2 . Completion of a metric space Given a metric space IR with closure IR, a complete metric space IR∗ is called a completion of IR if IR ⊂ IR∗ and IR = IR∗. Examples 1. The space of real numbers is the completion of the space of rational numbers. 2. Let K be a positive de nite Radial Basis Function then L2 is the completion the space of functions ( )= fx n =1 i ( ) ciK x − xi . Compact spaces A metric space is compact i it is totally bounded and complete. Let IR be a metric space and any positive number. Then a set A ⊂ IR is said to be an -net for a set M ⊂ IR if for every x ∈ M , there is at least one point a ∈ A such that ρ(x, a) < . Given a metric space IR and a subset M ⊂ IR suppose M has a nite -net for every > 0. Then M is said to be totally bounded. A compact space has a nite -net for all > 0. Examples 1. In Euclidean n-space, IRn, total boundedness is equivalent to boundedness. If M ⊂ IR is bounded then M is contained in some hypercube Q. We can partition this hypercube into smaller hypercubes with sides of length . The vertices of the little cubes from a nite √ n /2-net of Q. 2. This is not true for in nite-dimensional spaces. The unit sphere in l2 with constraint ∞ x2 n = 1, n=1 is bounded but not totally bounded. Consider the points e1 = (1, 0, 0, ...), e2 = (0, 1, 0, 0, ...), ..., where the n-th coordinate of en is one and all others are zero. These points lie on but the distance between √ any √ is 2. So cannot have a nite -net with two < 2/2. 3. In nite-dimensional spaces maybe totally bounded. Let be the set of points x = (x1, ..., xn, ..) in l2 satisfying the inequalities 1 1 |x1| < 1, |x2| < , ..., |xn| < n−1 , ... 2 2 The set called the Hilbert cube is an example of an in nite-dimensional totally bounded set. Given any > 0, choose n such that 1<, 2n+1 2 and with each point = (x1, ..., xn, ..) is associate the point x∗ = (x1, ..., xn, 0, 0, ...). (1) Then ∞ ∞ 1< 1 < . ∗) = 2< ρ(x, x xk 4k 2n−1 2 k=n+1 k=n The set ∗ of all points in that satisfy (1) is totally bounded since it is a bounded set in n-space. x 4. The RKHS induced by a kernel K with an in nite number of positive eigenvalues that decay exponentially is compact. In this case, our vector x = (x1, ..., xn, ..) can be written in terms of its basis functions, the eigenvectors of K . Now for the RKHS norm to be bounded |x1| < µ1, |x2| < µ2, ..., |xn| < µn, ... and we know that µn = O(n−α). So we have the case analogous to the Hilbert cube and we can introduce a point x∗ = (x1 , ..., xn, 0, 0, ...) (2) in a bounded n-space which can be made arbitrarily close to x. Compactness and continuity A family of functions φ de ned on a closed interval is said to be uniformly bounded if for K > 0 |φ(x)| < K for all x ∈ a, b] and all φ ∈ . a, b A family of functions φ is equicontinuous of for any given > 0 there exists δ > 0 such that |x − y | < δ implies |φ(x) − φ(y )| < for all x, y ∈ a, b] and all φ ∈ . Arzela's theorem: A necessary and su cient condition for a family of continuous functions de ned on a closed interval a, b] to be (relatively) compact in C a, b] is that is uniformly bounded and equicontinuous. Linear space A set L of elements x, y, z, ... is a linear space if the following three axioms are satis ed: 1. Any two elements x, y ∈ L uniquely determine a third element in x + y ∈ L called the sum of x and y such that (a) x + y = y + x (commutativity) (b) (x + y) + z = x + (y + z) (associativity) (c) An element 0 ∈ L exists for which x + 0 = x for all x∈L (d) For every x ∈ L there exists an element with the property x + (−x) = 0 −x ∈ L 2. Any number α and any element x ∈ L uniquely determine an element αx ∈ L called the product such that (a) α(βx) = β (αx) (b) 1x = x 3. Addition and multiplication follow two distributive laws (a)(α + β )x = αx + βx (b)α(x + y) = αx + αy Linear functional A functional, F , is a function that maps another function to a real-value F : f → IR. A linear functional de ned on a linear space L, satis es the following two properties 1. Additive: F (f + g) = F (f ) + F (g) for all f, g ∈ L 2. Homogeneous: F (αf ) = αF (f ) Examples 1. Let IRn be a real n-space with elements x = (x1, ..., xn), and a = (a1, ..., an) be a xed element in IRn. Then n ( )= Fx =1 aixi i is a linear functional 2. The integral ( )] = is a linear functional Ff x b a ()() f x p x dx 3. Evaluation functional: another linear functional is the Dirac delta function ( )] = f (t). δt f · Which can be written δt f (·)] = b a ()( ) f x δ x − t dx. 4. Evaluation functional: a positive de nite kernel in a RKHS Ft f (·)] = (Kt, f ) = f (t). This is simply the reproducing property of the RKHS. Normed space A normed space is a linear (vector) space N in which a norm is de ned. A nonnegative function · is a norm i ∀f, g ∈ N and α ∈ IR 1. f≥ 0 and 2. f +g ≤f 3. αf = |α| f + f =0 i f =0 g . Note, if all conditions are satis ed except f = 0 i then the space has a seminorm instead of a norm. f =0 Measuring distances in a normed space In a normed space N , the distance ρ between f and g, or a metric, can be de ned as ρ(f, g ) = g − f . Note that ∀f, g, h ∈ N 1. ρ(f, g) = 0 i f = g. 2. ρ(f, g) = ρ(g, f ). 3. ρ(f, h) ≤ ρ(f, g) + ρ(g, h). Example: continuous functions A norm in C a, b] can be established by de ning f = max |f (t)|. a≤tb The distance between two functions is then measured as ρ(f, g ) = max |g (t) − f (t)|. a≤t≤b With this metric, C a, b] is denoted as C . Examples (cont.) A norm in L1 a, b] can be established by de ning = a |f (t)|dt. The distance between two functions is then measured as b ρ(f, g ) = |g (t) − f (t)|dt. a With this metric, L1 a, b] is denoted as L1. b f Examples (cont.) A norm in C2 a, b] and L2 a, b] can be established by de ning 1/2 = a f 2(t)dt . The distance between two functions now becomes b f 1/2 ( ) = a (g(t) − f (t))2dt . With this metric, C2 a, b] and L2 a, b] are denoted as C2 and L2 respectively. b ρ f, g Convergence revisited A sequence of functions fn converge to a function f almost everywhere i lim fn(x) = f (x) n→+∞ A sequence of functions fn converge to a function measure i ∀ > 0 lim µ{x : |fn(x) − f (x)| ≥ } = 0. n→+∞ f in A sequence of functions fn converge to a function f uniformly i lim sup(fn(x) − f (x)) = 0 n→+∞ x Relationship between di erent types of convergence In the case of bounded intervals: uniform convergence (C ) implies • • convergence in the quadratic mean (L2) which implies convergence in the mean (L1) which implies convergence in measure almost everywhere convergence which implies convergence in measure. Relationship between di erent types of convergence That uniform convergence implies all other type of convergence is clear. Consider L2 over a bounded interval of width A. Keeping in mind that the function g = 1 belongs to L2 and that g L2 = A, convergence in the quadratic mean implies convergence in the mean because for every function f ∈ L2 we have f L1 = |f |dx = |f | · 1dx ≤ f L2 1 L2 = A f L2 A A and hence that f ∈ L1. Any convergence implies convergence in measure Convergence in measure is obtained by convergence in the mean through Chebyshev's inequality: For any real random variable X and t > 0, P (|X | ≥ t) ≤ E X 2 /t2]. The proof that almost everywhere convergence implies convergence in measure is somewhat more complicated. Almost everywhere convergence does not imply convergence in the (quadratic) mean Over the interval 0, 1] let fn be n x ∈ (0, 1/n] fn = 0 otherwise Clearly fn → 0 for all x ∈ 0, 1]. Note that each fn is not a continuous function and that the convergence is not uniform (the closer the x to 0, the larger n must be for fn(x) = 0). However, 1 |fn(x)|dx = 1 for all n, 0 in both the Riemann or the Lebesgue sense. Convergence in the quadratic mean does not imply convergence at all! Over the interval (0, 1], for every n = 1, 2, ..., and i = 1, ..., n let i 1 i−1 < x ≤ n n n fi = 0 otherwise Clearly the sequence n 122 n n n n f1 , f1 , f2 , ..., f1 , f2 , ...fn−1, fn , f1 +1, ..., converges to 0 both in measure and in the quadratic mean. However, the same sequence does not converge for any x! Convergence in probability and almost surely Any event with probability 1 is said to happen almost surely. A sequence of real random variables Yn converges almost surely to a random variable Y i P (Yn → Y ) = 1. A sequence Yn converges in probability to Y i for every > 0, limn→∞ P (|Yn − Y | > ) = 0. Convergence almost surely implies convergence in probability. A sequence X1, ...Xn satis es the strong law of large num1 bers if for some constant c, n n=1 Xi converges to c almost i surely. The sequence satis es the weak law of large num1 bers i for some constant c, n n=1 Xi converges to c in i probability. Euclidean space A Euclidean space is a linear (vector) space E in which a dot product is de ned. A real valued function (·, ·) is a dot product i ∀f, g, h ∈ E and α ∈ IR 1. (f, g ) = (g, f ) 2. (f + g, h) = (f, h∗) + (g, h) and (αf, g ) = α(f, g ) 3. (f, f ) ≥ 0 and (f, f ) = 0 i f = 0. A Euclidean space becomes a normed linear space when equipped with the norm f = (f, f ). Orthogonal systems and bases A set of nonzero vectors {xα} in a Euclidean space said to be an orthogonal system if (xα, xβ ) = 0 for α = β and an orthonormal system if (xα, xβ ) = 0 for α = β (xα, xβ ) = 1 for α = β. E is An orthogonal system {xα} is called an orthogonal basis if it is complete (the smallest closed subspace containing {xα} is the whole space E ). A complete orthonormal system is called an orthonormal basis. Examples 1. IRn is a real n-space, the set of n-tuples x = (x1, ..., xn), y = (y1 , ..., yn). If we de ne the dot product as (x, y) = n =1 xiyi i we get Euclidean n-space. The corresponding norms and distances in IRn are x ( ρ x, y )= x−y = = n =1 x2 i i n =1 i (xi − yi)2. The vectors e1 e2 = = ······························· = form an orthonormal basis in IRn. en (1, 0, 0, ...., 0) (0, 1, 0, ...., 0) (0, 0, 0, ...., 1) 2. The space l2 with elements x = (x1, x2, ..., xn, ....), y = (y1, y2, ..., yn, ....), ..., where ∞ =1 i x2 < ∞, i ∞ 2 yi < ∞, ..., ..., =1 i becomes an in nite-dimensional Euclidean space when equipped with the dot product ∞ (x, y) = xiyi. =1 i The simplest orthonormal basis in l2 consists of vectors e1 = (1, 0, 0, 0, ...) = (0, 1, 0, 0, ...) e2 e3 = (0, 0, 1, 0, ...) e4 = (0, 0, 0, 1, ...) ······························· there are an in nite number of these bases. 3. The space C2 a, b] consisting of all continuous functions on a, b] equipped with the dot product (f, g) = a f (t)g(t)dt is another example of Euclidean space. b An important example of orthogonal bases in this space is the following set of functions 2 2 1, cos b πnt , sin b πnt (n = 1, 2, ...). −a −a Hilbert space A Hilbert space is a Euclidean space that is complete, separable, and generally in nite-dimensional. A Hilbert space is a set H of elements f, g, ... for which 1. H is a Euclidean space equipped with a scalar product 2. H is complete with respect to metric ρ(f, g) = 3. is separable (contains a countable everywhere dense subset) H 4. (generally) H is in nite-dimensional. l2 f −g and L2 are examples of Hilbert spaces. The δ function We now consider the functional which returns the value of f ∈ C at the location t (an evaluation functional), f ] = f (t). Note that this functional is degenerate because it does not depend on the entire function f , but only on the value of f at the speci c location t. The δ(t) is not a functional but a distribution. The δ function (cont.) The same functional can be written as ∞ f ] = f (t) = f (s)δ (s − t)ds. −∞ No ordinary function exists (in L2) that behaves like δ(t), one can think of δ(t) as a function that vanishes for t = 0 and takes in nite value at t = 0 in such a way that ∞ δ (t)dt = 1. −∞ The δ function (cont.) The δ function can be seen as the limit of a sequence of ordinary functions. For example, if 1 r (t) = (U (t) − U (t − )) is a rectangular pulse of unit area, consider the limit ∞ lim −∞ f (s)r (s − t)ds. →0 By de nition of r this gives 1 t+ f (s)ds = f (t) lim t →0 because f is continuous. Fourier Transform The Fourier Transform of a real valued function f ∈ L1 is ~ the complex valued function f (ω) de ned as +∞ ~ F f (x)] = f (ω ) = f (x) e−jωxdx. −∞ ~ The FT f can be thought of as a representation of the information content of f (x). The original function f can be obtained through the inverse Fourier Transform as 1 +∞ f (ω) ejωxdω. ~ f (x) = 2π −∞ Properties () f ∗(t) F (t) f (t − t0) f (t)ejω0 t dnf (t) f at dtn (−jt)nf (t) ∞ f1(τ )f2 (t − τ )dτ −∞ ∞ f ∗(τ )f (t + τ )dτ −∞ ⇔ 1F |a| ⇔ F∗ ω ω a ⇔ () 2πf (−ω) F (ω )e−jt0 ω F (ω − ω0) (jω)nF (ω) dnF (ω ) ⇔ () () |F (ω )|2 ⇔ ⇔ ⇔ ⇔ dω n ⇔ F1 ω F2 ω Properties The box and the sinc f (t) = 1 if − a ≤ t ≤ a and 0 otherwise 2 sin(aω) . F (ω ) = ω 4 1.2 3.5 1 3 2.5 0.8 2 1.5 0.6 1 0.4 0.5 0 0.2 −0.5 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 −1 −10 −8 −6 −4 −2 0 2 4 6 8 10 Properties The Gaussian () = F (ω ) = −at2 e ft π −ω2/4a e . a 1 1.4 0.9 1.2 0.8 1 0.7 0.6 0.8 0.5 0.6 0.4 0.3 0.4 0.2 0.2 0.1 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 Properties The Laplacian and Cauchy distributions f (t) = e−a|t| 2a . F (ω ) = 2 a + ω2 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 Fourier Transform in the distribution sense With due care, the Fourier Transform can be de ned in the distribution sense. For example, we have () •δx ⇐⇒ 1 • cos(ω0x) ⇐⇒ π δ ω − ω0 • sin(ω0x) ⇐⇒ jπ δ ω () •Ux (( ) + δ(ω + ω0)) ( ( + ω0) − δ(ω − ω0)) () ⇐⇒ πδ ω − j/ω 2 • |x| ⇐⇒ − /ω 2 Parseval's formula If f is also square integrable, the Fourier Transform leaves the norm of f unchanged. Parseval's formula states that +∞ +∞ 2dx = 1 ~ |f (x)| |f (ω )|2dω. 2π −∞ −∞ Fourier Transforms of functions and distributions The following are Fourier transforms of some functions and distributions ( ) = δ(x) •fx ~( ) = 1 ⇐⇒ f ω ~( ) = π(δ(ω − ω0) + δ(ω + ω0)) ( ) = cos(ω0x) ⇐⇒ f ω ( ) = sin(ω0x) ⇐⇒ f ω •fx •fx ( ) = U (x) •fx ( ) = |x| •fx ~( ) = iπ(δ(ω + ω0) − δ(ω − ω0)) ~( ) = πδ(ω) − i/ω ⇐⇒ f ω ~( ) = −2/ω2. ⇐⇒ f ω Functional di erentiation In analogy with standard calculus, the minimum of a functional can be obtained by setting equal to zero the derivative of the functional. If the functional depends on the derivatives of the unknown function, a further step is required (as the unknown function has to be found as the solution of a di erential equation). Functional di erentiation The derivative of a functional f ] is de ned D f] f (t) + hδ (t − s)] − f (t)] . = h→0 lim Df (s) h Note that the derivative depends on the location s. For +∞ example, if f ] = −∞ f (t)g(t)dt +∞ D f] = −∞ g(t)δ(t − s)dt = g(s). Df (s) Intuition Let f : a, b] → IR, a = x1 and b = xN . The intuition behind this de nition is that the functional f ] can be thought of as the limit for N → ∞ of the function of N variables N= N (f1 , f2 , ..., fN ) with f1 = f (x1), f2 = f (x2), ... fN = f (xN ). For N → ∞, depends on the entire function f . The dependence on the location brought in by the δ function corresponds to the partial derivative with respect to the variable fk. Functional di erentiation (cont.) = f (t), the derivative is simply D f] Df (t = Df (s) = δ(t − s). Df (s) ) Similarly to ordinary calculus, the minimum of a functional f ] is obtained as the function solution to the equation D f] = 0. Df (s) If f Random variables We are given a random variable ξ ∼ F . To de ne a random variable you need three things: 1) a set to draw the values from, we'll call this 2) a σ-algebra of subsets of , we'll call this B 3) a probability measure F on B with F ( ) = 1 So ( , B, F ) is a probability space and a random variable is a masurable function X : → IR. Expectations Given a random variable ξ ∼ F the expectation is IEξ ≡ ξdF. Similarly the variance of the random variable σ2(ξ) is var(ξ) ≡ IE(ξ − IEξ)2. Law of large numbers The law of large numbers tells us: → IE lim 1 I →∞ If σ→c ( )=y]. x,y I f x σ→∞ If =1 i ( )=yi] f xi the Central Limit Theorem states: √1 ( √ I − IEI ) → N (0, 1), varI which implies 1 I − IEI ∼ √ . k the Central Limit Theorem implies 1 I − IEI ∼ k . ...
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