CONVERGENCE OF RANDOM FOURIER SERIES

Transcription

1 CONVERGENCE OF RANDOM FOURIER SERIES MICH HILL Abstract. his paper will study Fourier Series with random coefficients. We begin with an introduction to Fourier series on the torus and give some of the most important results. We then give some important results from probability theory, and build on these to prove a variety of theorems that deal with the convergence or divergence of general random series. In the final section, the focus is placed on random Fourier series, and we combine results from the previous sections to prove our main theorem. he main result of this paper gives a simple condition for the almost-everywhere convergence or divergence of a random trigonometric series, and proves that divergence implies that the coefficients cannot be the Fourier series of any function. Contents. Introduction 2. Fourier Series 2 3. Summability Kernels and the Convergence of Fourier Series 5 4. Probability heory 9 5. Series of Random Vectors 6. he Paley-Zygmund heorem 6 Acknowledgments 20 References 20. Introduction he goal of this paper will be to study the behavior of the random trigonometric series. Y n e int n= where the Y n s are independent but not necessarily identically distributed complex-valued random variables for t [0,. he partial sums of this series represent random functions on the torus, and we are interested in exploring when these partial sums converge for some, almost all, or all t. In particular, if these partial sums converge almost-everywhere, then. represents a function on whose Fourier coefficients are given by the Y n. o simplify the problem, we will focus on the series.2 X n cosnt + Φ n n=0 where X n and Φ n are real-valued and X n e iφn are independent, and the results obtained for this case can be translated to the general case with a few minor modifications. Date: August 26, 202.

2 2 MICH HILL he main result of this paper comes in the form of the Paley-Zygmund heorem, which gives a simple condition for the almost-sure convergence or divergence of.2. It is stated below. heorem.3. Paley-Zygmund If n=0 EX2 n <,.2 converges almost-surely almost-everywhere to a function ft such that f L p for p <. If n=0 E [X2 n] =,.2 diverges almost-surely almost-everywhere, and the sequence {X n } almost-surely does not represent the Fourier coefficients of a function in L p for p <. his theorem divides random Fourier series into two different convergence classes. If the X n s satisfy the conditions of the Paley-Zygmund heorem,.2 converges to a function which is in L p for arbitrarily large p, but not necessarily to a bounded function. Once boundedness is satisfied, however, the continuity of the function immediately follows, although we will not prove that result in this paper. 2. Fourier Series We start with some fundamental results about Fourier series. hroughout this paper we will restrict ourselves to complex-valued functions on, the onedimensional torus. Recognizing that can be thought of as the group R/Z under addition, we define the Lebesgue measure on in the natural way. Definition 2.. he integral of a function f : C is defined as ft dt = fx dx so that the Lebesgue measure dt is the restriction of the Lebesgue measure dx on R to the interval [0,. he factor / is present so that dt = 0 he most important property of the measure dt is translation invariance, which means for any t 0, we have 2.2 ft t 0 dt = ft dt his can easily be seen by recognizing that a function on can be thought of as a -periodic function on R. We now define some important Banach spaces on. Definition 2.3. he following are Banach spaces: L p = {f : C ft p dt < } for p < with norm /p f L p = ft p dt

3 CONVERGENCE OF RANDOM FOURIER SERIES 3 C = {f : C f is continuous} with norm f = sup ft t Note that, for p < q, we have C L q L p. he above Banach spaces are important because they satisfy the following properties, which are not necessarily true in the case of a general Banach space. Lemma 2.4. Consider f B, where B is one of the Banach spaces above, and τ, τ 0. Let f τ t = ft τ. he following properties are satisfied: f τ B = f B lim f τ f τ0 B = 0 τ τ 0 ranslation Invariance in Norm Continuity of ranslation in Norm Proof. 2.5 is a consequence of the translation invariance of the measure dt. 2.6 is obvious if f is a continuous function. Now consider f L p for p <. Recalling that continuous functions are dense in L p, take some ɛ > 0 and continuous function g such that f g B < ɛ/2. hen f τ f τ0 B f τ g τ B + g τ g τ0 B + g τ0 f τ0 B = f g τ B + g τ g τ0 B + g f τ0 B ɛ + g τ g τ0 B Since g τ g τ0 B vanishes as τ τ 0, and ɛ is arbitrary, 2.6 is proven. In this paper we will also refer to B, the space of bounded functions with norm f = sup t ft. However, it is important to note that this space is not a Banach space because it is not complete. Next, we define an important family of functions on that are closely related to Fourier series. Definition 2.7. A trigonometric polynomial on is a function P of the form N 2.8 P t = a n e int n= N he degree of P is the largest integer m such that a m + a m 0. Definition 2.9. A trigonometric series on is an expression of the form 2.0 S a n e int n= Note that this definition makes no assumptions about the convergence of the series for any t. Remark 2.. Recall the random trigonometric series n= Y n e int introduced at the beginning of the paper. Using the well-known identity e it = cos t + i sin t and writing Y n = Z n e iθn with Z n and Θ n real, we have Y n e int = Z n e iθn+nt + Z n e iθ n nt n= = n=0 n=0 n= Z n cosnt + Θ n + i Z n sinnt + Θ n n=0 + Z n cos nt + Θ n + i Z n sin nt + Θ n n= n=

4 4 MICH HILL So we see that determining the convergence of. depends on the convergence of four different cases of.2, where Φ n = Θ n for the first term, Φ n = Θ n π 2 for the second, Φ n = Θ n for the third, and Φ n = Θ n + π 2 for the fourth. he Fourier series of a function is a trigonometric series with the a n chosen in such a way that they represent the function in a certain sense. o motivate the correct choice of these coefficients, observe that for a trigonometric polynomial P t, we have because for j Z, 2.2 a n = 0 0 e ijt dt = P te int dt { if j = 0 0 if j 0 herefore, we define the n th Fourier coefficient of a function f L by 2.3 ˆfn = fte int dt Definition 2.4. he Fourier series S[f] of a function f L is given by the trigonometric series 2.5 S[f] ˆfne int n= Again, no assumptions about convergence for any t are made. We write S N f, t for the partial sums of 2. up to degree N. Formally, n 2.6 S n f, t = ˆfke ikt his function is well defined for all t. k= n Earlier we claimed that the translation invariance of the measure dt was important, and to show why we define the convolution operation on. Definition 2.7. For two functions f and g in L, their convolution is given by 2.8 f gt = ft τgτ dτ It can be shown that f g L, and that the convolution operation is commutative, associative, and distributive with respect to addition of functions. he true importance of the convolution operation, however, comes from its relation to the Fourier coefficients of functions. heorem 2.9. Consider f, g L. hen for ht = f gt, we have 2.20 h L f L g L and 2.2 ĥt = ˆftĝt

5 CONVERGENCE OF RANDOM FOURIER SERIES 5 Proof. Let F t, τ = ft τgτ. hen we have F t, τ dt dτ = gτ f L dτ = f L g L By Fubini s heorem, 4π F t, τdtdτ = 2 4π F t, τdτdt, and using a well-known integral 2 inequality we get h L = 4π 2 F t, τdτ dt 4π 2 F t, τ dt dτ = f L g L which establishes 2.5. o prove 2.6, using Fubini s heorem once more, we get ĥn = hte int dt = 4π 2 ft τe int τ gτe inτ dt dτ = fte int dt gτe inτ dτ = ˆfnĝn 3. Summability Kernels and the Convergence of Fourier Series With the Fourier series of a function and the important convolution operation defined, the next topic to consider is the convergence of Fourier series. Although we might hope that the Fourier series of a function in L will converge to the function, either in the L norm or pointwise, this is not the case in general. Below we define two important kernels, the Fejer kernel and the Dirichlet kernel. hese kernels are functions that, after convolution with some function f, give some insight into the convergence of the Fourier series of f. Definition 3.. he n th Dirichlet kernel D n t is given by n 3.2 D n t = e ikt he n th Fejer kernel K n t is given by 3.3 K n t = n + Observe that, because of 2.2, 3.4 n D k t = k=0 k= n n k= n k e ikt n + D n t dt = K n t dt = Both of these kernels can be written in closed form, as the following lemma shows. Lemma D n t = sinn + 2 t sin 2 t 3.7 K n t = sin n+ 2 t n + sin 2 t 2

6 6 MICH HILL Proof. Recall the formula for finite sums of geometric sequences: n ar k = a rn+ r k= n k=0 k=0 herefore, we can write n 2n r k = r n r k n r2n+ = r = r n 2 r 2n+ = r n 2 r n+ 2 r r 2 r r 2 r 2 Here r = e it. Recalling that sin t = 2i eit e it, we get n e ikt = e in+ 2 t e in+ 2 t = sinn + 2 t e 2 it e 2 it sin 2 t k=0 k= n which proves 3.6. o prove 3.7, we write n sink + 2 t n sin 2 t = e ik+ 2i sin 2 t 2 t e ik+ t 2 = = = = 2i sin 2 t 2i sin 2 t 2i sin 2 t 2i sin 2 t k=0 e 2 it n k=0 e ikt e 2 it n k=0 e ikt ] [e 2 e it in+t e 2 e it in+t e it e ik+t e ik+t e 2 it e 2 it e 2 it e 2 it 2 2 cosn + t 2i sin 2 t Since cos 2t = 2 sin 2 t, this proves the lemma. = e it = 2i sin 2 t 2 cosn + t sin 2 t2 2 e ik+t + e ik+t e 2 it e 2 it Since both kernels are functions on in particular, trigonometric polynomials, we can find their Fourier coefficients, which have a simple form. hey are { { if m n m ˆD n m = ˆK 0 otherwise n m = n+ if m n 0 otherwise he importance of the Dirichlet kernel can be seen as follows. By heorem 2.5, Moreover, which means D n fm = ˆD n m ˆfm = { ˆfm if m n 0 otherwise n k= n e ink τ fτ dτ = n k= n 3.8 D n ft = S n f, t e ikt fτe inτ dτ herefore, the limit as n of the convolution of the Dirichlet kernel and a function f is the Fourier series of the function. Recalling that the convolution operation is distributive with respect to addition, and writing σ n f, t = K n ft, we also have 3.9 σ n f, t = n + n S n f, t k=0

7 CONVERGENCE OF RANDOM FOURIER SERIES 7 he convolution of a function and the Fejer kernel gives the mean of the partial sums of its Fourier series up to order n. Now we are in a position to prove some important results about the convergence of Fourier series. heorem 3.0. Let B be a Banach space from Definition 2.3, and consider some f B. hen 3. lim n K n f = f in the norm of B. Proof. Choose ɛ > 0. It is easy to show that K n f = K n τft τ dτ in the norm of B. Now, for 0 < δ < π, using 3.4, we have K n τft τ dτ ft = δ δ δ + δ K n τ[ft τ ft] dτ B δ δ δ δ K n τ[ft τ ft] dτ K n τ ft τ ft B dτ sup ft τ ft B K n L τ δ By Lemma 2.4, we have lim τ 0 ft τ ft B = 0, so we can find δ > 0 such that sup τ δ ft τ ft B K n L < ɛ. Using this same δ, we write δ δ K n τ[ft τ ft] dτ sup ft τ ft B K n τ dτ B δ τ sin n+ By 3.7, for t 0,, lim n K n t = lim n n+ lim n δ which means we can bound sup τ ft τ ft B proof. Corollary 3.2. Uniqueness of Fourier Series If ˆfn = ĝn for all n N, then f = g. δ 2 t sin 2 t K n τ dτ = 0 δ δ δ 2 = 0, so K n τ dτ by ɛ as well, finishing the Proof. Again writing K n ft = σ n f, t, we have n σ n f g, t = j n + ˆfk ĝke ikt = 0 k= n for all n. Since σ n f g f g, we have f g = 0, or f = g. he Fejer sums of a function in a Banach space with certain important properties converge to the function in the norm of the Banach space, and this is an important foundation for many of the results that follow. Convergence of Dirichlet sums, however, is more problematic. We start with a lemma that gives a condition for the convergence of Dirichlet sums.

8 8 MICH HILL Definition 3.3. A Banach space B in definition 2.3 admits convergence in norm if for all f B. lim S nf f B = 0 n Lemma 3.4. A Banach space B admits convergence in norm if there exists some K > 0 such that 3.5 S n f B K f B for all f B and n N. Proof. Choose ɛ > 0. heorem 3.0 implies that trigonometric polynomials are dense in B, since K n f is a trigonometric polynomial for all n. So choose a trigonometric polynomial P t such that f P B < ɛ/2k. For n greater than the degree of P, S n P = P. hen we have S n f f B = S n f S n P + P f B S n f P B + P f B K ɛ 2K + ɛ 2K Recalling that S n f = D n f, we obtain the inequality which means 3.6 D n f B D n L f B S n f B f B D n L he numbers D n L = L n are called the Lebesgue constants. If they were bounded, then any Banach space from Definition 2.3 would admit convergence in norm, but as the next lemma shows, this is not the case. Lemma 3.7. L n as n Proof. L n = sinn + 2 t 0 sin 2 t dt = π sinn + 2 t π 0 sin 2 t dt Let fx = x sin x. f0 = 0 and f x = cos x 0 for x [0, π/2] which means that sin t/2 t/2 on [0, π]. So we have π sinn + 2 t π sin 2 t dt 2 π sinn + 2 t π t dt 2 n k+π n+ sinn t dt π kπ t Now, herefore, 0 k+π n+ 2 kπ n+ 2 sinn + 2 t t dt = 0 j+π jπ sin u u which can be made arbitrarily large as n. du π L n 4 n π 2 j + k= k= j + n+ 2 j+π jπ sin u du = 2 π j + We now prove that in the case of L and C, 3.6 becomes an equality, which means that these spaces do not admit convergence in norm. heorem 3.8. L and C do not admit convergence in norm.

9 CONVERGENCE OF RANDOM FOURIER SERIES 9 Proof. First consider L. Recalling that K N L =, Since lim N K N D n = D n, S n f sup L f L f L S nk N L K n L = K N D n L S n f sup L f L f L S n f D n L sup L f L f L = D n L By the previous lemma, D n L = L n can be made as large as we like, so by Lemma 3.4, L does not admit convergence in norm. Now consider C. Consider a set of functions ψ n C such that ψ n and ψ n t = sgnd n t except near points of discontinuity of sgnd n t. If the sum of the lengths of the intervals where ψ n t ± is less than ɛ/2n, then S n f sup S n ψ n, 0 = f C f D n tψ n t dt > L n ɛ While these two important spaces do not admit convergence in norm, it can be shown that L p does admit convergence in norm for < p <. We will not prove this result, but the proof has to do with the a properties of a function s conjugate Fourier series in C. 4. Probability heory After establishing some fundamental properties of Fourier series, we now change gears and give some basic results from probability theory that will be important later. We will use the usual definition of probability space, random variables mapping outcomes from the probability space to the real numbers, and the expected value of a random variable. We will use Ω to denote the set of outcomes and ω to denote an element in Ω. For an event A Ω, we write A for the random variable defined so that { if ω A A ω = 0 if ω / A Note that E A = PA. We say an event that occurs with probability occurs almost-surely. Definition 4.. A Rademacher sequence is a sequence of independent random variables {ɛ n } such that Pɛ n = = 2 Pɛ n = = 2 Sometimes we will also use {ɛ n } to refer to a sequence of constants with value or. Definition 4.2. A symmetric random vector X is a vector such that X and X have the same distribution. If {X n } is a sequence of independent symmetric random vectors, then it has the same distribution as {ɛ n X n }. he next three lemmas that we will prove deal with events that depend on an infinite number of outcomes. hese lemmas are very useful since we are trying to study the convergence or divergence of a random series where each term in the series is determined by a random outcome. In particular, these lemmas show that under certain conditions, an event that depends on an infinite number of outcomes occurs either with probability 0 or with probability. Lemma 4.3. Suppose {X n } is a sequence of random variables with X n 0 and n= EX n <. hen n= X n < almost-surely.

10 0 MICH HILL Proof. Since the random variables are positive, by Lebesgue s Monotone Convergence heorem the sum of expected values is the expected value of the sum, so we have E X n = EX n < M n= n= for some M > 0. Let Y = n= X n, so Y 0. Choose ɛ > 0 and take N > large enough so that M/N < ɛ. EY = hen 0 y µ Y dy < M N y µ Y dy < M PY N, = µ Y dy < y µ Y dy < M N N N N < ɛ herefore ɛ PY [0, N]. Since ɛ is arbitary, this proves the lemma. Definition 4.4. Consider an infinite sequence of events A, A 2,..., A n,... hen 4.5 lim A n = k= n k Informally, lim A n holds when infinitely many A n occur. A n Lemma 4.6. Borel-Cantelli Consider an infinite sequence A, A 2,..., A n,... of independent events. If PA n <, then Plim A n = 0 If PA n =, then Plim A n = Proof. Recall E An = PA n. If n= P A n = E n= A n <, then by Lemma 4.3, n= An < almost-surely. herefore, it is almost sure that only finitely many A n hold, so Plim A n = 0. Now suppose PA n =. Plim A n = P Since the A n are independent, we have which means Plim A n =. k= n k P n k A n = A c n = lim k P PA n n=k exp PA n n=k = exp n k A n PA n = 0 Lemma 4.7. Zero-One Law Let X = A be a random variable defined on Ω = n= Ω n such that Xω, ω 2,..., ω n,... = Xω, ω 2,..., ω n,... whatever the particular values of ω might be. In other words, X is a tail event that does not depend on any finite number of outcomes. hen either PA = or PA = 0. n=k

11 CONVERGENCE OF RANDOM FOURIER SERIES Proof. Write E n X = XωP dω P n dω Ω n Ω 2 Ω hen for each n, E n XE n X = 0 for each n, so EXE X = PA PA = 0. We now give two inequalities which will be important later. Lemma 4.8. For a > and X 0, PX aex a Proof. EX = EX X<aEX + X X aex 0 + EaEX X aex = aexpx aex Lemma 4.9. For 0 < λ <, Proof. First we define PX λex λ 2 EX2 EX 2 X ω = he Cauchy-Schwartz inequality gives We also have EX EX + λex, so { Xω if Xω λex 0 if Xω < λex EX 2 EX 2 PX 0 EX 2 PX λex λ 2 EX 2 EX 2 PX λex 5. Series of Random Vectors We now prove some results about general series of random vectors that will be needed to prove results about random trigonometric series. Definition 5.. A summation matrix is an infinite scalar matrix S = a nm with n, m N such that lim a nm = for all m N n An important summation matrix is a nm = sup0, m n. Note that this matrix is closely related to the Fejer kernel defined earlier. A series v n is S-summable if the sequence w, w 2,..., w n,... converges, where w n = a nm v m m= A series is S-bounded if w n converges for all n and the sequence w, w 2,..., w n,... is bounded. We define sup n w n as its S-bound.

12 2 MICH HILL Just as the Fejer kernel allowed us to study the Fourier series of a function even if the Fourier series itself did not converge, summation matrices allow us to prove results about properties of random series even when the series themselves are difficult to study. We will soon prove that for a random sequence, S-summability implies almost-sure convergence and S-boundedness implies almost-sure boundedness, but first we will prove some important lemmas. Lemma 5.2. Let X,..., X 2,..., X n,... be a sequence of symmetric random vectors. Let m Y m ω = X n ω and Mω = sup Y m ω m and let Y ω = X nω if the series is convergent. Suppose X n converges almost surely. hen for each r > 0, we have 5.3 PMω > r 2P Y ω > r Proof. Let Ω 0 be the set of ω such that Y ω is defined, and denote by A and B the subsets of Ω 0 where Mω > r and Y ω > r respectively. A can be divided as follows 5.4 A : Y > r A 2 : Y r, Y 2 > r A 3 : Y r, Y 2 r, Y 3 > r. so that m= A m = A and A m are disjoint. If ω A m, then at least one of the vectors 5.5 Z = Y m ω + n=m+ X nω Z = Y m ω n=m+ X nω lies outside the ball x r. Since the X n are symmetric, Y and Y have the same probability of being outside the ball. he union of these two events is A m, which means that each has a probability of at least 2 PA m of occurring. herefore, PB A m 2 PA m. Adding the probabilities over all disjoint A m proves the lemma. For the next lemma, we write Λ for an infinite set of integers and define M Λ ω = sup Y m ω m Λ Lemma 5.6. For each r > 0 and each set Λ, we have PM > r 2PM Λ > r Proof. Very similar to the proof above. Let A and B be the events M > r and M Λ > r. Let the events A m be defined as in 5.4. For ω A m, consider M Λ, m = sup Z v v Λ, v m M Λ, m = sup Z v v Λ, v m where Z v = Y m + v Λ, v m X v and Z v = Y m v Λ, v m X v. Again, the subevents of A m defined by M Λ, m and M Λ, m have the same probability, and their union is A m. Again, ω B A m M Λ, m > r, so PB A m 2 PA m, and addition of probability of disjoint events proves the lemma.

13 CONVERGENCE OF RANDOM FOURIER SERIES 3 heorem 5.7. Let X, X 2,..., X n,... be random vectors in a Banach space B, and S be a summation matrix. If the series X n is almost-surely S-summable, it converges almost-surely. If it is almost-surely S-bounded, then it is almost-surely bounded. Proof. Let S = a nm. By assumption, we have m= a nmx m = Z n almost surely for some Z n B, and lim n Z n = Z. Since lim n a nm =, there exists N p > 0 such that P a nm X m > 2 p < 2 p m p for n > N p. We may also suppose N < N 2 <.... By assumption, m= a nmx m converges almost surely, so there exists Q p such that P a NpmX m > 2 p < 2 p m>q for q Q p. Now write if m p b pm = a Npm if p < m Q p 0 if m > Q p his defines a new summation matrix, which we will call. he finite sums 5.8 Z p = b pm X m m= satisfy Z p Z Np < 22 p for p = v, v +,.... herefore, X n is almost-surely -summable. Now write p =, p j+ = Q pj for j N. Assuming X m = 0 when p j < m p j+ gives 5.9 Z p j = p j m= X m = herefore, if X m = 0 for when p j < m p j+ for an infinite set J of values j, 5.9 tends almost surely to a limit when j in J. p j+ m= X m Now we split X n into two parts, X n and X n, where X n = X n and X n = 0 if p 2j n < p 2j X n = 0 and X n = X n if p 2j n < p 2j+ Note that 2X n X n = ±X n and 2X n X n = ±X n, both have the same distribution as X n since the X n are symmetric, so 2X n X n and 2X n X n have the same almost-sure properties as X n. herefore X n and X n are almost-surely -summable. For these series, 5.9 implies the p j partial sums are convergent, so for X n, the p j partial sums are also convergent almost surely. Now we use Lemma 5.2. Since the p j partials sums are convergent in probability, for each η > 0 there exists a j such that P > η < η p j<m p k X m for k > j. According to lemma, for each j and k, 5.0 P sup p j<l p k p j<m l X m > η < 2η

14 4 MICH HILL Writing 5.0 for η = η k = 2 k, j = jη k = j k, and k = jη κ+ = j κ+, κ = µ, µ +,..., and adding, we obtain P X m η k > 4η k p j<m l when p jκ < l p jκ+. Since the p jκ partial sums converge almost surely, the series X m converges with probabiliy as near to as we please. he case for boundedness is similar, and uses lemma 5.6 instead of lemma 5.2. proof. We omit the As a consequence of this proof, if X m is not almost surely convergent, then there exists an η > 0 and sequences of integers m, m 2,... and m, m 2,... with m < m < m 2 < m 2 <... such that 5. P X n > η > η for k N m k <n<m k We now go on to prove a variety of theorems that deal with the conditions under which the random series converge or diverge almost surely. hese theorems are the backbone upon which our study of random Fourier series is built, and many of the results given for random Fourier series are just special cases of these more general theorems. heorem 5.2. If X n L 2 Ω and EX n = 0 for each n, then P sup X + X X n > r < n=, 2,, N r 2 VX + VX VX N Proof. Let Y n = n m= X m, and again consider the disjoint events 5.3 A : Y > r A 2 : Y r, Y 2 > r A 3 : Y r, Y 2 r, Y 3 > r. with A = N A n. Our goal is to estimate PA = N PA n. By the definition of A n, we have r 2 PA n E An X + + X n 2 Now, we write X = An X + + X n and Y = X n+ + + X N. he events X and Y are independent, since they depend on the sum of sets of events independent from each other. Recall that CovX, Y = EX Y EXEY. herefore, CovX, Y = 0. Since X = 0 lies outside of A n, we also have CovX, Y = CovX, An Y = 0. herefore, so that E X + An Y 2 = E X 2 + E An Y 2 E X 2 r 2 PA n E X + An Y 2 = E An X + + X N 2 Adding over n =,..., N gives the desired inequality, since VX n = E X n 2 by the assumption EX n = 0. heorem 5.4. Suppose X n L 2 Ω and EX n = 0 for all n and moreover VX n <. hen the series X n converges almost-surely. In particular, if u n 2 <, then ɛ nu n converges almost surely.

15 CONVERGENCE OF RANDOM FOURIER SERIES 5 Proof. By the previous theorem, for each r > 0, P sup X m + + X m+j > r < j r 2 VX n Since VX n <, we have P lim sup m j n=m X m + + X m+j > r = 0 Since r > 0 can be made arbitrarily small, this proves the theorem. heorem 5.5. Paley-Zygmund Inequality Suppose X n L 4 Ω, EX n = 0, and E X n 4 CVX n 2 for each n. Let 0 < λ <. hen 5.6 P X + + X ν > λvx + + VX ν /2 > η where η = min 3, /C λ2 2 and ν is an arbitrary integer. In particular, if {u n } is a sequence of vectors and {ɛ n } is a Rademacher sequence, 5.7 P ɛ u + + ɛ ν u ν > λ u u ν 2 /2 > 3 λ2 2 Proof. Consider the random variable X = X + + X ν 2. Using lemma 4.9, we have PX λ 2 EX λ 2 2 EX2 EX 2 Since the X n are independent, we have EX = VX + + VX ν. Manipulating the properties of the inner products gives EX 2 = E[ X n, X n2 X n3, X n4 ] n, n 2, n 3, n 4 If n n 2 or n 3 n 4 then E[ X n, X n2 X n3, X n4 ] = 0 by independence. So we can write ν EX 2 = E X 4 [ + 2 Re E Xn, X m 2 which proves the result. n= n<m v +E X n, X m 2 + E X n 2 E X m 2 ] ν C VX n VX n VX m n= ν sup3, c VX n n= n<m v 2 heorem 5.8. Suppose X n L 4 Ω, EX n = 0, and E X n 4 CVX n 2 for each n. Given a summation matrix S, suppose that X n is S-bounded. hen VX n <. In particular, if ɛ nu n is almost-surely S-bounded, then u n 2 <. Proof. Write S = a nm. Also write ν ν /2 E nν = a nm X m > λ a 2 nmvx m m= E n = lim ν E nν = E = lim n E n p= ν p E nν m=

16 6 MICH HILL By the previous theorem, PE nν > η, so PE n > η and PE > η. Since by assumption X n is S-bounded, there is some ω E and b > 0 such that a nm X m ω converges and a nm X m ω < b m= for n N. For each n with ω E n, we also have ω E nν for infinitely many ν, so λ 2 m= m= a 2 nmvx m < b 2 which holds for infinitely many n. By the definition of a summation matrix, lim n a nm =, which completes the proof. heorem 5.9. Suppose that U n L 2 Ω and sup n EUn/EU 2 n 2 <. hen U n converges or diverges almost-surely according to whether EU n converges or diverges. Proof. When EU n < we already know the result by lemma 4.3. If U n =, we use lemma 4.9 and write PU + + U ν > λeu + + U ν > λ 2 EU + + U ν 2 EU + + U ν 2 EU + +U ν 2 EU + +U ν 2 Since EU n U m = EU n EU m and by assumption, EUn 2 CEU n 2 for some C, is bounded below as ν. herefore P U n = > 0, so it is true almost surely by the Zero-One law. 6. he Paley-Zygmund heorem We are finally in a position to prove the first of the two main results of this paper, the Paley- Zygmund heorem. Suppose you have a random sequence of independent random vectors {X n e iφn }, with X n 0 and Φ n [0,. Consider the random trigonometric series 6. X n cosnt + Φ n n=0 and the Fejer sums of this series, given by 6.2 σ N t = N n=0 n X n cosnt + Φ n N By heorem 3.0, if 6. represents a function in L p then 6.2 converges to 6. in L p. Since the sums σ N represent a summation matrix for each N, by heorem 5.7 the convergence or boundedness of 6.2 implies the convergence or boundedness of 6. almost surely. We state this in the following proposition. Proposition L p almost-surely lim N σ N converges in L p for p < almost-surely 6. L p almost-surely sup N σ N L p < for p < almost-surely Lemma 6.4. If {x n } and {φ n } are two real sequences and x2 n =, then x2 n cos 2 nt+φ n = for almost every t.

17 CONVERGENCE OF RANDOM FOURIER SERIES 7 Proof. If the lemma is false, then there exists a set E of positive measure E where the series is bounded, so we can can write x 2 n cos 2 nt + φ n < b for t E Integrating gives x 2 n E cos 2 nt + φ n dt < b E Now, cos 2 nt + φ n = e2int+φn + e 2int+φn, so cos 2 nt + φ n dt = E E 2 dt + e 2int+φn dt + e 2int+φn dt 4 E E For any set E, lim n E eint dt = 0, so lim n E cos2 nt + φ n dt = 2 E. herefore, there exists an N such that, for n N, cos 2 nt + φ n dt > 3 E E which means n=n x n < 3b, contradicting our assumption and proving the lemma. he results of the following three lemmas will be based on the assumption that either 0 X2 n = almost surely or 0 X2 n < almost surely. We already know conditions under which the Xn 2 converge or diverge by heorem 5.9 of the previous section, and we will use these conditions in the final proof of this section. Lemma 6.5. If 0 X2 n = almost surely, 6. / L p almost surely for p <. Moreover, the sequence {X n } almost-surely does not represent a Fourier series of a function in L p for p <. Proof. Consider the Fejer sums Writing σ N t = p N t = N n=0 N n=0 n ɛ n X n cosnt + φ n N n /2 2 X 2 N n cos 2 nt + φ n and choosing 0 < λ <, η = 3 λ2 2, the Paley-Zygmund inequality gives Pσ N t λp N t η By the previous lemma, there exists a sequence ρ N and a subset of the circle such that dt = π and p N t ρ N for each t. Consider some probability space Ω. Let E = E N be the set of ω, t Ω such that σ N t λρ N. For fixed ω, let E ω by the set of t such that ω, t E, and for fixed t let E t be the set of ω such that ω, t E. Let F = F N be the event E ω η. hen we have E = PE t dt πη E = F + Ω\F E ω Pdω

18 8 MICH HILL hen we can write Also, πη πpf + η PF P F N σ N t dt σ N t dt λρ N E ω E ω π η π η Since PF N > c for some c > 0 for all N, we have P F N =, so by Borel-Cantelli, lim F N happens almost surely. herefore, we can write lim N σ N t dt = herefore by Proposition 6.3, we have 6. / L almost surely, so 6. / L p almost surely for p <. o prove the second part of the lemma, assume that {X n } represents the Fourier series of a function f L p. hen by heorem 3.0, lim N σ N = f in the L p norm, but we have already seen that lim N σ N L =, which means that it cannot converge to f in L p, since by definition f L p is finite. Next we will show that 0 X2 n = almost surely implies divergence for almost every t. Lemma 6.6. If 0 X2 n = almost surely, 6. diverges almost surely almost everywhere. Proof. By lemma 6.4, we have almost surely that Xn 2 cos 2 nt + φ n = Let a nm = rn m for 0 < r n <, and lim n r n =. hen S = a nm is a summation matrix. Since x2 n cos 2 nt + φ n =, by heorem 5.8 we have m= ±rm N X m cosmt + φ m = almost-surely for each r N. aking the limit as N proves the result. Lemma 6.7. If 0 X2 n < almost surely, then 6. converges almost surely almost everywhere to a function in L p for p <. Proof. By heorem 5.4, 6. converges almost surely for any given t, and therefore converges almost-surely almost-everywhere. Let F t be the function that 6. converges to. hen for λ > 0, we have Ee λf t = E exp λx n ɛ n cosnt + φ n Since cosh u e u2 /2, we have = = EexpλX n ɛ n cosnt + φ n coshλx n cosnt + φ n Ee λf t e λ2 r/2

19 CONVERGENCE OF RANDOM FOURIER SERIES 9 where r = X2 n. By symmetry, we have EF 2n+ t = 0 for n = 0,, 2,..., so λ 2n 2n! EF 2n t e λ 2 r/2 Choosing λ 2 = 2n/r gives for some constant C. herefore n=0 EF 2n t 2n! Ee λf 2 t C n 2n e n Cn!2r n r 2λr n < whenever λ < /2r. Since this holds for almost every t, we have E e λf 2 t dt = Ee λf 2 t dt < 0 which means eλf 2 t dt < almost surely whenever λ < /2r. his implies that F L p almost surely for p <, since the exponential function grows faster than any power of F t. It is not possible to go further and claim that F t is bounded, which is shown as follows. 0 0 Construction 6.8. here exists a trigonometric series gt = 0 p ncosnt + φ n such that g L p for p <, but g is unbounded. Let n, n 2, n 3,... be an increasing sequence of integers such that, n j+ > 6n j for some ɛ > 0, and let φ, φ 2,... be a sequence of real numbers. Claim: For k N, there exists a connected interval I k such that meas I k = 3π 4n k cosn k t + φ k > α for t I k for some small α > 0. and Proof. We will use a proof by induction. Clearly this is true for k =. Assume the claim is true for some j N. Consider g j+ t = cosn j+ t + φ k+. Periodg j+ = < π < n j+ 3n j 2 I j herefore, g j+ completes two entire periods on I k, and the conclusion follows from the properties of cosine. aking the limit as k shows that there is some t 0 such that cosn k t 0 + φ k > α for all k N. Letting p k = /k, it follows that p k cosn k t 0 + φ k = even though p2 k <, which shows that p k cosn k t + φ k is not bounded even though it is almost surely in L p for p < by Lemma 6.7. Now we combine all the results from this section into our main result, the Paley-Zygmund heorem. heorem 6.9. Suppose that sup n EX 4 n/ex 2 n <. hen, If 0 EX2 n <, 6. converges almost surely almost everywhere to a function F t such that F t L p for p <.

20 20 MICH HILL If 0 EX2 n =, 6. diverges almost surely almost everywhere and {X n } almost surely does not represent the Fourier series of a function in L p for p <. Proof. If 0 EX2 n <, then by heorem X2 n < almost surely, and the conclusion follows from Lemma 6.7. If 0 EX2 n =, then by heorem X2 n = almost surely, and the conclusion follows from Lemma 6.5 and Lemma 6.6. Acknowledgments. I would like to thank my mentor Bobby Wilson for his time, knowledge and guidance, and Peter May for organizing the REU program that gave me the opportunity to write this paper. References [] Kahane, Jean-Pierre. Some Random Series of Functions, Second Edition. Cambridge University Press, 985. [2] Katznelson, Yitzhak. An Introduction to Harmonic Analysis, hird Edition. Cambridge University Press, [3] Wade, William R. An Introduction to Analysis, Fourth Edition. Prentice Hall, 2009