FOURIER TRANSFORMS OF STATIONARY PROCESSES 1. k=1
|
|
- Deirdre Lewis
- 5 years ago
- Views:
Transcription
1 FOURIER TRANSFORMS OF STATIONARY PROCESSES WEI BIAO WU September 8, 003 Abstract. We consider the asymptotic behavior of Fourier transforms of stationary and ergodic sequences. Under sufficiently mild conditions, central limit theorems are established for almost all frequencies as well as for a given frequency. Applications to the widely used linear processes and iterated random functions are discussed. Our results shed new light on the foundation of spectral analysis in that the asymptotic distribution of periodogram, the fundamental quantity in the frequency-domain analysis, is obtained.. Introduction Let (X n ) n Z be a stationary and ergodic Markov chain on the state space X ; let g be a real-valued function on X such that E[g(X 0 )] = 0 and E[ g (X 0 ) ] <. Define the Fourier transform S n (θ) = S n (θ; g) = g(x k )e ikθ, θ R, () k= where i = is the imaginary unit. The setup () is general enough to allow one to consider S Y n (θ) = n k= Y ke ikθ for any stationary and ergodic process (Y n ) n Z by constructing a Markov chain X n = (..., Y n, Y n ) and g(x n ) = Y n. The quantity () is of fundamental importance in the spectral analysis of stationary processes. This paper considers asymptotic normality of (). history. This problem has a substantial Rosenblatt (Theorem 5.3, p 3, 985) considers mixing processes; Brockwell and Davis (Theorem 0.3.., p 347, 99), Walker (965) and Terrin and Hurvich (994) discuss linear processes. Other contributions can be found in Olshen (967), Rootzén (976), Yajima (989) and Walker (000) among others. Mathematical Subject Classification (99): Primary 60F05, 60F7; secondary 60G35 Key words and phrases. Spectral analysis, linear process, martingale Central Limit Theorem, periodogram, Fourier transformation, nonlinear time series
2 Under surprisingly weak conditions (cf ()), we show that the real and imaginary parts of S n (θ)/ n are asymptotically iid normal for almost all frequencies θ R. Thus the periodogram n S n (θ) is asymptotically distributed as a multiple of a χ () random variable. For a fixed θ, we present sufficient conditions for the asymptotic normality of S n (θ)/ n. Our general results go beyond earlier ones by providing mild and verifiable conditions. In our proof, we apply the celebrated Carleson s theorem in Fourier analysis to construct approximating martingales. Applications to special sequences such as linear processes and iterated random functions which are widely used in time series modelling are discussed.. Main Results Let (Ω, B(Ω), P) be the probability space on which the sequence X n is defined, where B(Ω) is a sigma algebra on Ω. For a random variable ξ on the space (Ω, B(Ω), P), we denote its L (P) norm by ξ = E( ξ ). Define the projection operator P k ξ = E(ξ..., X k, X k ) E(ξ..., X k ). Then by the Markovian property, P g(x n ) = E[g(X n ) X ] E[g(X n ) X 0 ] for n N. Let Rz and Iz be the real and the imaginary parts of the complex number z, namely z = Rz+iIz, and let z = (Rz) + (Iz). Denote by N(µ, Σ) the multivariate normal distribution with mean vector µ and covariance matrix Σ; denote by Id p the p p identity matrix. Theorems and concern the asymptotic normality for almost all θ and for a given θ respectively. Theorem. Assume that n= n E[g(X n) X 0 ] <. () Then (i) for almost all θ R (Lebesgue), there exists 0 σ(θ) < such that R S n(θ) N[0, σ (θ)id ]. (3) I n
3 (ii) Moreover, for almost all pairs (θ, ϕ) (Lebesgue), S n (θ)/ n and S n (ϕ)/ n are asymptotically independent. Theorem. For a given θ [0, π), suppose there exists ξ(θ, X 0, X ) L (P) such that as N, and N e inθ P g(x n ) ξ(θ, X 0, X ) (L (P)), (4) n= E[S n (θ) X 0 ] = o(n). (5) Then (i) if θ 0, π, R I S n(θ) n N[0, σ (θ)id ], (6) where σ (θ) = ξ(θ, X 0, X ) /, and (ii) S n (θ)/ n N(0, σ ) if θ = 0 or π, where σ (θ) = ξ(θ, X 0, X ). Proposition. Assume that P g(x n ) P g(x n+ ) <. (7) n= Then (4) and (5), and consequently (6), hold for all 0 < θ < π. Condition () is fairly mild and it imposes very weak decay rate of E[g(X n ) X 0 ]. Rootzén (976) obtained a central limit theorem under conditions which are not easily verifiable. In comparison, our Conditions (4), (5) and (7) are tractable in many cases. It is easily seen that () is equivalent to k= k k E[g(X i ) X 0 ] < (8) i= by exchanging the order of summation. Notice that E[g(X n ) X 0 ] is non-increasing in n in view of P n g(x 0 ) = E[g(X 0 ) X n ] E[g(X 0 ) X n ]. So another equivalent condition of () is k= E[g(X k) X 0] <. Inequality (8) is needed to establish a connection between σ(θ) in (3) and spectral densities (cf Proposition ). 3
4 Let r k = E[g(X 0 )g(x k )] be the covariance function and introduce the spectral distribution function F (θ), 0 θ π via Herglotz s Theorem r k = π 0 exp(ikθ)df (θ). Assume that F is absolutely continuous with the spectral density function f, namely F (θ) = θ 0 f(u)du. Proposition. Assume (). Then for almost all θ [0, π] (Lebesgue), f(θ) = σ (θ)/π. Example. Iterated Random Functions. Let (X, ρ) be a complete and separable metric space and let X n = F εn (X n ), where F ε ( ) = F (, ε) is the ε-section of a jointly measurable function F : X Υ X and ε, ε n, n Z are iid random variables which take values in a second measurable space Υ. Define L ε = sup x x ρ[f ε (x), F ε (x )]/ρ(x, x ). Diaconis and Freedman (999) prove that X n admits a unique stationary distribution (say Π) if E(L α ε ) <, E(log L ε ) < 0, and E[ρ α (x 0, F ε (x 0 ))] < (9) for some α > 0 and x 0 X. Let g (δ) = sup{ [g(x) g(x )] [ρ(x,x ) δ] : X, X Π}. Corollary. Assume (9), E[g(X )] = 0 and E[ g(x ) l ] < for some l >. (a) If then () holds. (b) If / 0 / Then n= E[g(X n) X ] < and hence (7) hold. 0 g(t) dt <, (0) t log t g (t) dt <, () t Proof. Let X 0 Π and X 0 be independent of X 0 and (ε k ) k Z. For n let X n = F εn F εn... F ε (X 0). Then (9) implies that there exists β, C > 0 and 0 < r < such that E[ρ β (X n, X n)] Cr n hold for all n 0 (cf Lemma 3 in Wu and Woodroofe (000)). Let p = l/, q = p/(p ) and δ n = (Cr n ) /(q+β). Since E[g(X n) X 0 ] = 0, E[g(X n ) X 0 ] [g(x n ) g(x n)] ρ(xn,x n)<δ n + [g(x n ) g(x n)] ρ(xn,x n) δ n g (δ n ) + [g(x n ) g(x n)] / p [P(ρ(X n, X n) δ n )] q g (δ n ) + C δ / n 4
5 So () follows since (0) entails n= g(δ n )/n <. (b) similarly follows. Example. Linear processes. Let ε k, k Z be iid random variables with mean 0 and finite variance; let X n = (..., ε n, ε n ) and g(x n ) = i=0 a iε n i, where a i are real numbers such that i=0 a i <. Then (7) is reduced to i= a i a i < and the central limit theorem (6) holds for all 0 < θ < π. Notice that this can not be extended to θ = 0 if a n is not summable. For example, a n = n β, / < β < for n. 3. Proofs Let U be the uniform probability measure on Θ = [0, π) and B(Θ) be the class of the Borel sets of Θ. Denote by (Θ Ω, B(Θ) B(Ω), U P) the product probability space of (Θ, B(Θ), U) and (Ω, B(Ω), P). Lemma. Assume (). Then for almost all θ Θ (U), n E[S n (θ) X 0 ] 0 almost surely (P). Proof of Lemma. Let A B(Θ) B(Ω) be the set on which N n= n e inθ E[g(X n ) X 0 ] does not converge as N. Let A ω and A θ be the ω section and θ-section of A respectively. Define Ω 0 = { ω Ω : n= } n E[g(X n) X 0 ] <. () By (), P(Ω 0 ) =. For ω Ω 0, by Carleson s theorem U(A ω ) = 0. Thus, by Fubini s theorem, for almost all θ Θ (U), P(A θ ) = 0, i.e., n= einθ E[g(X n ) X 0 ]/ n converges almost surely (P). Hence by Kronecker s lemma (cf. Lemma 5.., Chow and Teicher 988), as N, N N n= e inθ E[g(X n ) X 0 ] 0 a.s. (P). 5
6 Lemma. For almost all θ Θ (U), there exists ξ(θ, X 0, X ) L such that P S n (θ) ξ(θ, X 0, X ) almost surely (P). (3) Proof of Lemma. Since the sequence E[g(X 0 ) X n ] E[g(X 0 ) X n ], n = 0,,... forms martingale differences, P g(x n ) = n= P n g(x 0 ) = g(x 0 ) <. n= Hence for almost all ω Ω (P), n= P g(x n ) <, which yields (3) by Carleson s theorem and the same argument as in Lemma. Next we show that ξ(θ, X 0, X ) L. To this end, since the trigonometric bases e in, n Z are orthogonal, P k g(x 0 ) = k= By Fatou s lemma, g(x 0 ) lim inf E P S n (θ) U(dθ) n Θ E lim inf P S n (θ) U(dθ) = n Θ Hence for almost all θ Θ (U), ξ(θ, X 0, X ) <. Lemma 3. Assume (). Then for almost all θ Θ (U), Θ E P S n (θ) U(dθ). Θ E[ ξ(θ, X 0, X ) ]U(dθ). E[ξ(θ, X 0, X ) X 0 ] = 0 a.s. (P) (4) Proof of Lemma 3. The almost sure convergence (3) in Lemma alone does not guarantee the L convergence. To prove (4), we shall show below that, under (), the Cesàro average N n= P S n (θ)/n converges to ξ(θ, X 0, X ) in L. The relation (4) reveals the martingale structure and we can apply the martingale central limit theorem. Let the tail R N (θ, X 0, X ) = n=n+ e inθ {E[g(X n ) X ] E[g(X n ) X 0 ]}. 6
7 Then R 0 (θ, X 0, X ) = ξ(θ, X 0, X ). Again by the orthogonality of the bases e in, R N (θ, X 0, X ) U(dθ) = P g(x n ) <. Θ n=+n Hence by () N= E R N (θ, X 0, X ) U(dθ) = N Θ = N= N= N n=+n E P g(x n ) N E[g(X 0) X N ] <. So for almost all θ Θ (U), N= N E R N(θ, X 0, X ) <. It follows from Kronecker s lemma that as N which yields N N E R j (θ, X 0, X ) 0, (5) j= E N N R j (θ, X 0, X ) j= by Cauchy s inequality. In other words, 0 s N (θ, X 0, X ) := N N P S n (θ) ξ(θ, X 0, X ) in L. n= Clearly, E[ s N (θ, X 0, X ) X 0 ] = 0 a.s. (P). Hence as N, E E[ξ(θ, X 0, X ) X 0 ] = E E[ξ(θ, X 0, X ) s N (θ, X 0, X ) X 0 ] E ξ(θ, X 0, X ) s N (θ, X 0, X ) ξ(θ, X 0, X ) s N (θ, X 0, X ) 0, which proves (4). 7
8 Lemma 4. Suppose (5) holds for θ for which ξ(θ, X 0, X ) L (P). Then R I S n(θ) E[S n (θ) X 0 ] n N[0, Σ(θ)], (6) where Σ(θ) = ξ(θ, X 0, X ) Id if θ 0, π, and Σ(θ) = ξ(θ, X 0, X ) 0 0 0, if θ = 0, π. Proof of Lemma 4. We generalize the arguments in Woodroofe (99) where the asymptotic normality for the case θ = 0 was obtained. Let ξ n (θ, x 0, x ) = E[S n (θ) X = x ] E[S n (θ) X 0 = x 0 ] and write S n (θ) E[S n (θ) X 0 ] = + e iθ(j ) ξ(θ, X j, X j ) j= e iθ(j ) [ξ n j+ (θ, X j, X j ) ξ(θ, X j, X j )]. (7) j= Now we claim that (5) implies that the second part in the proceeding display is o P ( n). Actually, by Lemma 3, E[ξ n j+ (θ, X j, X j ) ξ(θ, X j, X j ) X j ] = 0 almost surely P. Then it suffices to establish n ξ n j+ (θ, X j, X j ) ξ(θ, X j, X j ) 0, (8) j= which follows from (5) via the stationarity of X j. By Lemma 3, ξ(θ, X j, X j ), j Z is a martingale difference sequence, hence we can use the martingale central limit theorem to deduce (6) from (7). To this end, let ξ(θ, X j, X j ) = A j (θ) + ib j (θ), α j (θ) = A j (θ) cos[(j )θ] B j (θ) sin[(j )θ], β j (θ) = B j (θ) cos[(j )θ] + A j (θ) sin[(j )θ]. 8
9 Then A j (θ), B j (θ), α j (θ) and β j (θ) form martingale difference sequences, and e iθ(j ) ξ(θ, X j, X j ) = j= [α j (θ) + iβ j (θ)]. j= For a fixed 0 θ 0 < π let D j = α j (θ) cos θ 0 + β j (θ) sin θ 0. Then D j are again martingale differences. By the Cramer-Wold device, it suffices to show that D j N[0, σ (θ, θ 0 )], where σ (θ, θ 0 ) = (cos θ 0, sin θ 0 )Σ(θ) n j= cos θ 0 sin θ 0 We now apply the martingale central limit theorem to n j= D j/ n. The Lindeberg condition is automatically satisfied. Since E[ξ (θ, X j, X j ) F j ], j Z forms a stationary and ergodic sequence in L (P), by Lemma 5, the limit V (θ) = lim n n. E[ξ (θ, X j, X j ) F j ]e (j )iθ (9) j= exists and equals to 0 almost surely (P) if θ 0, π, and equals to ξ(θ, X 0, X ) if θ = 0 or π. After some algebra, Σ (θ) = lim n n Σ (θ) = lim n n Σ (θ) = lim n n E[αj(θ) F j ] = ξ(θ, X 0, X ) E[βj (θ) F j ] = ξ(θ, X 0, X ) IV (θ) E[α j (θ)β j (θ) F j ] = j= j= j= + RV (θ), RV (θ), almost surely (P), which completes the proof. Lemma 5. Let η n, n Z be a real-valued, stationary and ergodic sequence with E η <. Then for all 0 < α < π, almost surely and in L. n e itα η t 0 (0) t= 9
10 Proof of Lemma 5. This result seems to be implied but not explicitly stated in the literature (see for example Doob (953, pp ), Rozanov (967, pp 60 6), Wiener and Wintner (94)). Since n t= eitα E(η t )/n 0, we can assume without loss of generality that E(η t ) = 0. If α/(π) is a rational number, then (0) clearly follows from the classical ergodic theorem. In the case that α/(π) is not a rational number, define the rotation ρ α : Θ Θ by ρ α (θ) = θ + α mod π. Then ρ α is measure-preserving and ergodic. Let the random variable ϑ Uniform(Θ) be independent of η n, n Z. Since η t is stationary and ergodic, the sequence ϕ t = e itα+ϑ η t, t Z is then stationary (cf. Rozanov, p. 6) as well as ergodic on the product space (Θ Ω, B(Θ) B(Ω), U P). Therefore the classical ergodic theorem asserts that n n t= ϕ t 0 almost surely (U P) and in L, which clearly entails (0) since U is a uniform distribution. Proof of Theorem. (i) It follows from Lemmas and 4. (ii) By Lemmas, 3 and 4, for almost all θ, ϕ Θ (U), martingale differences ξ(τ, X t, X t ) = A t (τ) + ib t (τ), τ = θ, ϕ can be constructed and it suffices to show that n / n t= v t N[0, Σ(θ, ϕ)], where random vectors v t = (α t (θ), β t (θ), α t (ϕ), β t (ϕ)) and the covariance matrix Σ(θ, ϕ) = Diag(w(θ), w(θ), w(ϕ), w(ϕ)), w(τ) = ξ(τ, X 0, X ). To this end, by (i) and the Cramer-Wold device, it follows from α t(θ)α t (ϕ) n β t (θ)α t (ϕ) t= α t (θ)β t (ϕ) β t (θ)β t (ϕ) 0 a.s. (P), which in view of Lemma 5 is valid provided that ϕ θ and ϕ + θ π. Proof of Theorem. Observe that (6) in Lemma 4 holds for θ for which (4) is satisfied. Therefore (6) follows in view of (5). Proof of Proposition. Let h n = n t= eitθ. Since 0 < θ < π, h := sup n h n <. By (7), n= h n[p g(x n ) P g(x n+ )] converges in L (P). Thus (4) holds since P g(x n ) 0. To prove (5), let φ t = P g(x t ) P g(x t+ ) and Φ k = t=k φ t. Then 0
11 (7) entails j=0 [ n t= φ t+j] Φ j=0 n t= φ t+j = o(n). So (5) follows from P j S n (θ) h P j g(x t ) P j g(x t+ ) + h P j g(x n+ ), t= E[S n (θ) X 0 ] = j=0 P js n (θ) and j=0 P jg(x n+ ) g(x 0 ) by the orthogonality of P k. Proof of Proposition. Let K n (λ) = sin (nλ/)/[πn sin (λ/)] be the Fejér kernel. It is well known that S n (θ) /(πn) = π 0 K n (λ θ)f(λ)dλ (cf Theorem 8..7 in Anderson (97, p. 454)), which by the Fejér-Lebesgue Theorem (cf Bary (964, p. 39)) converges almost everywhere to f(θ). Therefore, by (7) and (8), for almost all θ (Lebesgue), lim n n E[S n(θ) X 0 ] = lim n n S n(θ) lim n n S n(θ) E[S n (θ) X 0 ] = πf(θ) ξ(θ, X 0, X ). () Notice that by (8), π k E[S k(θ) X 0 ] U(dθ) = k= 0 k= k k E[g(X i ) X 0 ] <. i= Hence by the Borel-Cantelli Lemma, for almost all θ (Lebesgue), k E[S k(θ) X 0 ] 0 as k, which by () entails πf(θ) = ξ(θ, X 0, X ) = σ (θ) almost surely. Acknowledgment The author is grateful to the referee for many helpful suggestions. REFERENCES Anderson, T. W. (97). The Statistical Analysis of Time Series. New York, Wiley. Bary, N. K. (964). A Treatise on Trigonometric Series. New York, Macmillan. Brockwell, P. J. and Davis, R. A. (99). Time Series: Theory and Methods. New York, Springer.
12 Carleson, L. (966). On convergence and growth of partial sums of Fourier series. Acta Math Chow, Y. S. and Teicher, H. (988). Probability Theory. nd ed. Springer, New York. Diaconis, P. and Freedman. D. (999). Iterated random functions. SIAM Rev Doob, J. (953). Stochastic Processes. Wiley. Gray, H. L., Zhang, N.-F. and Woodward, W.A. (989). On generalized fractional processes. J. Time Ser. Anal Ibragimov, I. A. and Yu. V. Linnik. (97). Independent and stationary sequences of random variables. Groningen, Wolters-Noordhoff. Meyn, S. P. and Tweedie, R. L. (993). Markov chains and stochastic stability. Springer, London; New York. Olshen, R. A. (967). Asymptotic properties of the periodogram of a discrete stationary process. J. Appl. Probab Rootzén, H. (976). Gordin s theorem and the periodogram. J. Appl. Probab Rosenblatt, M. (98). Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrsch. Verw. Gebiete Rosenblatt, M. (985). Stationary sequences and random fields. Birkhäuser, Boston. Rozanov, Yu. A. (967). Stationary random processes. San Francisco, Holden-Day. Terrin, N. and Hurvich, C. M. (994). An asymptotic Wiener-Ito representation for the low frequency ordinates of the periodogram of a long memory time series. Stochastic Process. Appl Tong, H. (990). Non-linear time series: a dynamical system approach. Oxford University Press. Walker, A. M. (965). Some asymptotic results for the periodogram of a stationary time series. J. Austral. Math. Soc Walker, A. M. (000). Some results concerning the asymptotic distribution of sample Fourier transforms and periodograms for a discrete-time stationary process with a continuous spectrum. J. Time Ser. Anal
13 Woodroofe, M. (99). A central limit theorem for functions of a Markov chain with applications to shifts. Stochastic Process. Appl Wu, W. B. and Woodroofe, M. (000). A central limit theorem for iterated random functions. J. Appl. Probab Yajima, Y. (989). A central limit theorem of Fourier transforms of strongly dependent stationary processes. J. Time Ser. Anal Department of Statistics The University of Chicago 5734 S. University Avenue Chicago, IL 60637, USA wbwu@galton.uchicago.edu 3
Additive functionals of infinite-variance moving averages. Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535
Additive functionals of infinite-variance moving averages Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535 Departments of Statistics The University of Chicago Chicago, Illinois 60637 June
More informationCentral Limit Theorem for Non-stationary Markov Chains
Central Limit Theorem for Non-stationary Markov Chains Magda Peligrad University of Cincinnati April 2011 (Institute) April 2011 1 / 30 Plan of talk Markov processes with Nonhomogeneous transition probabilities
More informationSTA205 Probability: Week 8 R. Wolpert
INFINITE COIN-TOSS AND THE LAWS OF LARGE NUMBERS The traditional interpretation of the probability of an event E is its asymptotic frequency: the limit as n of the fraction of n repeated, similar, and
More informationUNIT ROOT TESTING FOR FUNCTIONALS OF LINEAR PROCESSES
Econometric Theory, 22, 2006, 1 14+ Printed in the United States of America+ DOI: 10+10170S0266466606060014 UNIT ROOT TESTING FOR FUNCTIONALS OF LINEAR PROCESSES WEI BIAO WU University of Chicago We consider
More informationMi-Hwa Ko. t=1 Z t is true. j=0
Commun. Korean Math. Soc. 21 (2006), No. 4, pp. 779 786 FUNCTIONAL CENTRAL LIMIT THEOREMS FOR MULTIVARIATE LINEAR PROCESSES GENERATED BY DEPENDENT RANDOM VECTORS Mi-Hwa Ko Abstract. Let X t be an m-dimensional
More informationMOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES
J. Korean Math. Soc. 47 1, No., pp. 63 75 DOI 1.4134/JKMS.1.47..63 MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES Ke-Ang Fu Li-Hua Hu Abstract. Let X n ; n 1 be a strictly stationary
More informationA note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series
Publ. Math. Debrecen 73/1-2 2008), 1 10 A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series By SOO HAK SUNG Taejon), TIEN-CHUNG
More informationGaussian Processes. 1. Basic Notions
Gaussian Processes 1. Basic Notions Let T be a set, and X : {X } T a stochastic process, defined on a suitable probability space (Ω P), that is indexed by T. Definition 1.1. We say that X is a Gaussian
More informationGoodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach
Goodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach By Shiqing Ling Department of Mathematics Hong Kong University of Science and Technology Let {y t : t = 0, ±1, ±2,
More informationENEE 621 SPRING 2016 DETECTION AND ESTIMATION THEORY THE PARAMETER ESTIMATION PROBLEM
c 2007-2016 by Armand M. Makowski 1 ENEE 621 SPRING 2016 DETECTION AND ESTIMATION THEORY THE PARAMETER ESTIMATION PROBLEM 1 The basic setting Throughout, p, q and k are positive integers. The setup With
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationFaithful couplings of Markov chains: now equals forever
Faithful couplings of Markov chains: now equals forever by Jeffrey S. Rosenthal* Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 1A1 Phone: (416) 978-4594; Internet: jeff@utstat.toronto.edu
More informationRandom Bernstein-Markov factors
Random Bernstein-Markov factors Igor Pritsker and Koushik Ramachandran October 20, 208 Abstract For a polynomial P n of degree n, Bernstein s inequality states that P n n P n for all L p norms on the unit
More informationCitation Osaka Journal of Mathematics. 41(4)
TitleA non quasi-invariance of the Brown Authors Sadasue, Gaku Citation Osaka Journal of Mathematics. 414 Issue 4-1 Date Text Version publisher URL http://hdl.handle.net/1194/1174 DOI Rights Osaka University
More informationProbability and Measure
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability
More informationA CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN. Dedicated to the memory of Mikhail Gordin
A CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN Dedicated to the memory of Mikhail Gordin Abstract. We prove a central limit theorem for a square-integrable ergodic
More informationA NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES
Bull. Korean Math. Soc. 52 (205), No. 3, pp. 825 836 http://dx.doi.org/0.434/bkms.205.52.3.825 A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES Yongfeng Wu and Mingzhu
More informationThe Codimension of the Zeros of a Stable Process in Random Scenery
The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar
More informationA GENERAL THEOREM ON APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION. Miljenko Huzak University of Zagreb,Croatia
GLASNIK MATEMATIČKI Vol. 36(56)(2001), 139 153 A GENERAL THEOREM ON APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION Miljenko Huzak University of Zagreb,Croatia Abstract. In this paper a version of the general
More informationThe Hilbert Transform and Fine Continuity
Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationA regeneration proof of the central limit theorem for uniformly ergodic Markov chains
A regeneration proof of the central limit theorem for uniformly ergodic Markov chains By AJAY JASRA Department of Mathematics, Imperial College London, SW7 2AZ, London, UK and CHAO YANG Department of Mathematics,
More informationTheorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension. n=1
Chapter 2 Probability measures 1. Existence Theorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension to the generated σ-field Proof of Theorem 2.1. Let F 0 be
More informationSoo Hak Sung and Andrei I. Volodin
Bull Korean Math Soc 38 (200), No 4, pp 763 772 ON CONVERGENCE OF SERIES OF INDEENDENT RANDOM VARIABLES Soo Hak Sung and Andrei I Volodin Abstract The rate of convergence for an almost surely convergent
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More informationPractical conditions on Markov chains for weak convergence of tail empirical processes
Practical conditions on Markov chains for weak convergence of tail empirical processes Olivier Wintenberger University of Copenhagen and Paris VI Joint work with Rafa l Kulik and Philippe Soulier Toronto,
More informationConditional independence, conditional mixing and conditional association
Ann Inst Stat Math (2009) 61:441 460 DOI 10.1007/s10463-007-0152-2 Conditional independence, conditional mixing and conditional association B. L. S. Prakasa Rao Received: 25 July 2006 / Revised: 14 May
More informationComplete Moment Convergence for Sung s Type Weighted Sums of ρ -Mixing Random Variables
Filomat 32:4 (208), 447 453 https://doi.org/0.2298/fil804447l Published by Faculty of Sciences and Mathematics, Uversity of Niš, Serbia Available at: http://www.pmf..ac.rs/filomat Complete Moment Convergence
More informationStochastic integration. P.J.C. Spreij
Stochastic integration P.J.C. Spreij this version: April 22, 29 Contents 1 Stochastic processes 1 1.1 General theory............................... 1 1.2 Stopping times...............................
More informationOn the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables
On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables Deli Li 1, Yongcheng Qi, and Andrew Rosalsky 3 1 Department of Mathematical Sciences, Lakehead University,
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationMixing time for a random walk on a ring
Mixing time for a random walk on a ring Stephen Connor Joint work with Michael Bate Paris, September 2013 Introduction Let X be a discrete time Markov chain on a finite state space S, with transition matrix
More informationWeighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes
More informationBERNOULLI DYNAMICAL SYSTEMS AND LIMIT THEOREMS. Dalibor Volný Université de Rouen
BERNOULLI DYNAMICAL SYSTEMS AND LIMIT THEOREMS Dalibor Volný Université de Rouen Dynamical system (Ω,A,µ,T) : (Ω,A,µ) is a probablity space, T : Ω Ω a bijective, bimeasurable, and measure preserving mapping.
More informationAsymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals
Acta Applicandae Mathematicae 78: 145 154, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands. 145 Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals M.
More informationGAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n
GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional
More informationWittmann Type Strong Laws of Large Numbers for Blockwise m-negatively Associated Random Variables
Journal of Mathematical Research with Applications Mar., 206, Vol. 36, No. 2, pp. 239 246 DOI:0.3770/j.issn:2095-265.206.02.03 Http://jmre.dlut.edu.cn Wittmann Type Strong Laws of Large Numbers for Blockwise
More informationIndependence of some multiple Poisson stochastic integrals with variable-sign kernels
Independence of some multiple Poisson stochastic integrals with variable-sign kernels Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological
More informationLecture 12. F o s, (1.1) F t := s>t
Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationAn almost sure central limit theorem for the weight function sequences of NA random variables
Proc. ndian Acad. Sci. (Math. Sci.) Vol. 2, No. 3, August 20, pp. 369 377. c ndian Academy of Sciences An almost sure central it theorem for the weight function sequences of NA rom variables QUNYNG WU
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationWeak invariance principles for sums of dependent random functions
Available online at www.sciencedirect.com Stochastic Processes and their Applications 13 (013) 385 403 www.elsevier.com/locate/spa Weak invariance principles for sums of dependent random functions István
More informationConvergence Rates for Renewal Sequences
Convergence Rates for Renewal Sequences M. C. Spruill School of Mathematics Georgia Institute of Technology Atlanta, Ga. USA January 2002 ABSTRACT The precise rate of geometric convergence of nonhomogeneous
More informationMaximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments
Austrian Journal of Statistics April 27, Volume 46, 67 78. AJS http://www.ajs.or.at/ doi:.773/ajs.v46i3-4.672 Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments Yuliya
More informationON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY
J. Korean Math. Soc. 45 (2008), No. 4, pp. 1101 1111 ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY Jong-Il Baek, Mi-Hwa Ko, and Tae-Sung
More informationAn almost sure invariance principle for additive functionals of Markov chains
Statistics and Probability Letters 78 2008 854 860 www.elsevier.com/locate/stapro An almost sure invariance principle for additive functionals of Markov chains F. Rassoul-Agha a, T. Seppäläinen b, a Department
More informationPROBABILISTIC METHODS FOR A LINEAR REACTION-HYPERBOLIC SYSTEM WITH CONSTANT COEFFICIENTS
The Annals of Applied Probability 1999, Vol. 9, No. 3, 719 731 PROBABILISTIC METHODS FOR A LINEAR REACTION-HYPERBOLIC SYSTEM WITH CONSTANT COEFFICIENTS By Elizabeth A. Brooks National Institute of Environmental
More informationThe main results about probability measures are the following two facts:
Chapter 2 Probability measures The main results about probability measures are the following two facts: Theorem 2.1 (extension). If P is a (continuous) probability measure on a field F 0 then it has a
More informationLogarithmic scaling of planar random walk s local times
Logarithmic scaling of planar random walk s local times Péter Nándori * and Zeyu Shen ** * Department of Mathematics, University of Maryland ** Courant Institute, New York University October 9, 2015 Abstract
More information3. WIENER PROCESS. STOCHASTIC ITÔ INTEGRAL
3. WIENER PROCESS. STOCHASTIC ITÔ INTEGRAL 3.1. Wiener process. Main properties. A Wiener process notation W W t t is named in the honor of Prof. Norbert Wiener; other name is the Brownian motion notation
More informationChapter 4. The dominated convergence theorem and applications
Chapter 4. The dominated convergence theorem and applications The Monotone Covergence theorem is one of a number of key theorems alllowing one to exchange limits and [Lebesgue] integrals (or derivatives
More informationA Single-Pass Algorithm for Spectrum Estimation With Fast Convergence Han Xiao and Wei Biao Wu
4720 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 57, NO 7, JULY 2011 A Single-Pass Algorithm for Spectrum Estimation With Fast Convergence Han Xiao Wei Biao Wu Abstract We propose a single-pass algorithm
More informationSTAT 7032 Probability Spring Wlodek Bryc
STAT 7032 Probability Spring 2018 Wlodek Bryc Created: Friday, Jan 2, 2014 Revised for Spring 2018 Printed: January 9, 2018 File: Grad-Prob-2018.TEX Department of Mathematical Sciences, University of Cincinnati,
More informationExercises Measure Theoretic Probability
Exercises Measure Theoretic Probability 2002-2003 Week 1 1. Prove the folloing statements. (a) The intersection of an arbitrary family of d-systems is again a d- system. (b) The intersection of an arbitrary
More informationFunctional central limit theorem for super α-stable processes
874 Science in China Ser. A Mathematics 24 Vol. 47 No. 6 874 881 Functional central it theorem for super α-stable processes HONG Wenming Department of Mathematics, Beijing Normal University, Beijing 1875,
More informationPart II Probability and Measure
Part II Probability and Measure Theorems Based on lectures by J. Miller Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationSPACE AVERAGES AND HOMOGENEOUS FLUID FLOWS GEORGE ANDROULAKIS AND STAMATIS DOSTOGLOU
M P E J Mathematical Physics Electronic Journal ISSN 086-6655 Volume 0, 2004 Paper 4 Received: Nov 4, 2003, Revised: Mar 3, 2004, Accepted: Mar 8, 2004 Editor: R. de la Llave SPACE AVERAGES AND HOMOGENEOUS
More informationarxiv: v1 [math.pr] 1 May 2014
Submitted to the Brazilian Journal of Probability and Statistics A note on space-time Hölder regularity of mild solutions to stochastic Cauchy problems in L p -spaces arxiv:145.75v1 [math.pr] 1 May 214
More informationSUPPLEMENT TO PAPER CONVERGENCE OF ADAPTIVE AND INTERACTING MARKOV CHAIN MONTE CARLO ALGORITHMS
Submitted to the Annals of Statistics SUPPLEMENT TO PAPER CONERGENCE OF ADAPTIE AND INTERACTING MARKO CHAIN MONTE CARLO ALGORITHMS By G Fort,, E Moulines and P Priouret LTCI, CNRS - TELECOM ParisTech,
More informationMartingale approximations for random fields
Electron. Commun. Probab. 23 (208), no. 28, 9. https://doi.org/0.24/8-ecp28 ISSN: 083-589X ELECTRONIC COMMUNICATIONS in PROBABILITY Martingale approximations for random fields Peligrad Magda * Na Zhang
More informationOn the Central Limit Theorem for an ergodic Markov chain
Stochastic Processes and their Applications 47 ( 1993) 113-117 North-Holland 113 On the Central Limit Theorem for an ergodic Markov chain K.S. Chan Department of Statistics and Actuarial Science, The University
More informationApplied Mathematics Letters
Applied Mathematics Letters 25 (2012) 545 549 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On the equivalence of four chaotic
More informationAlmost sure limit theorems for U-statistics
Almost sure limit theorems for U-statistics Hajo Holzmann, Susanne Koch and Alesey Min 3 Institut für Mathematische Stochasti Georg-August-Universität Göttingen Maschmühlenweg 8 0 37073 Göttingen Germany
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying
More informationZeros of lacunary random polynomials
Zeros of lacunary random polynomials Igor E. Pritsker Dedicated to Norm Levenberg on his 60th birthday Abstract We study the asymptotic distribution of zeros for the lacunary random polynomials. It is
More informationConvergence rates in weighted L 1 spaces of kernel density estimators for linear processes
Alea 4, 117 129 (2008) Convergence rates in weighted L 1 spaces of kernel density estimators for linear processes Anton Schick and Wolfgang Wefelmeyer Anton Schick, Department of Mathematical Sciences,
More informationA central limit theorem for randomly indexed m-dependent random variables
Filomat 26:4 (2012), 71 717 DOI 10.2298/FIL120471S ublished by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A central limit theorem for randomly
More informationSome Results on the Ergodicity of Adaptive MCMC Algorithms
Some Results on the Ergodicity of Adaptive MCMC Algorithms Omar Khalil Supervisor: Jeffrey Rosenthal September 2, 2011 1 Contents 1 Andrieu-Moulines 4 2 Roberts-Rosenthal 7 3 Atchadé and Fort 8 4 Relationship
More informationStochastic Optimization with Inequality Constraints Using Simultaneous Perturbations and Penalty Functions
International Journal of Control Vol. 00, No. 00, January 2007, 1 10 Stochastic Optimization with Inequality Constraints Using Simultaneous Perturbations and Penalty Functions I-JENG WANG and JAMES C.
More informationTwo remarks on normality preserving Borel automorphisms of R n
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 3, No., February 3, pp. 75 84. c Indian Academy of Sciences Two remarks on normality preserving Borel automorphisms of R n K R PARTHASARATHY Theoretical Statistics
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationSome basic elements of Probability Theory
Chapter I Some basic elements of Probability Theory 1 Terminology (and elementary observations Probability theory and the material covered in a basic Real Variables course have much in common. However
More informationWeak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij
Weak convergence and Brownian Motion (telegram style notes) P.J.C. Spreij this version: December 8, 2006 1 The space C[0, ) In this section we summarize some facts concerning the space C[0, ) of real
More informationON CLASSES OF EVERYWHERE DIVERGENT POWER SERIES
ON CLASSES OF EVERYWHERE DIVERGENT POWER SERIES G. A. KARAGULYAN Abstract. We prove the everywhere divergence of series a n e iρn e inx, and 1 [ρn] a n cos nx, for sequences a n and ρ n satisfying some
More informationThe Kadec-Pe lczynski theorem in L p, 1 p < 2
The Kadec-Pe lczynski theorem in L p, 1 p < 2 I. Berkes and R. Tichy Abstract By a classical result of Kadec and Pe lczynski (1962), every normalized weakly null sequence in L p, p > 2 contains a subsequence
More informationPointwise convergence rates and central limit theorems for kernel density estimators in linear processes
Pointwise convergence rates and central limit theorems for kernel density estimators in linear processes Anton Schick Binghamton University Wolfgang Wefelmeyer Universität zu Köln Abstract Convergence
More informationEmpirical Processes: General Weak Convergence Theory
Empirical Processes: General Weak Convergence Theory Moulinath Banerjee May 18, 2010 1 Extended Weak Convergence The lack of measurability of the empirical process with respect to the sigma-field generated
More informationON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary
More informationHan-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek
J. Korean Math. Soc. 41 (2004), No. 5, pp. 883 894 CONVERGENCE OF WEIGHTED SUMS FOR DEPENDENT RANDOM VARIABLES Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek Abstract. We discuss in this paper the strong
More informationOPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS
APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze
More informationSome functional (Hölderian) limit theorems and their applications (II)
Some functional (Hölderian) limit theorems and their applications (II) Alfredas Račkauskas Vilnius University Outils Statistiques et Probabilistes pour la Finance Université de Rouen June 1 5, Rouen (Rouen
More informationJOINT LIMIT THEOREMS FOR PERIODIC HURWITZ ZETA-FUNCTION. II
Annales Univ. Sci. Budapest., Sect. Comp. 4 (23) 73 85 JOIN LIMI HEOREMS FOR PERIODIC HURWIZ ZEA-FUNCION. II G. Misevičius (Vilnius Gediminas echnical University, Lithuania) A. Rimkevičienė (Šiauliai State
More informationThe Essential Equivalence of Pairwise and Mutual Conditional Independence
The Essential Equivalence of Pairwise and Mutual Conditional Independence Peter J. Hammond and Yeneng Sun Probability Theory and Related Fields, forthcoming Abstract For a large collection of random variables,
More informationCHAPTER 3: LARGE SAMPLE THEORY
CHAPTER 3 LARGE SAMPLE THEORY 1 CHAPTER 3: LARGE SAMPLE THEORY CHAPTER 3 LARGE SAMPLE THEORY 2 Introduction CHAPTER 3 LARGE SAMPLE THEORY 3 Why large sample theory studying small sample property is usually
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationOn Linear Processes with Dependent Innovations
The University of Chicago Department of Statistics TECHNICAL REPORT SERIES On Linear Processes with Dependent Innovations Wei Biao Wu, Wanli Min TECHNICAL REPORT NO. 549 May 24, 2004 5734 S. University
More informationOn Kesten s counterexample to the Cramér-Wold device for regular variation
On Kesten s counterexample to the Cramér-Wold device for regular variation Henrik Hult School of ORIE Cornell University Ithaca NY 4853 USA hult@orie.cornell.edu Filip Lindskog Department of Mathematics
More informationCentral-limit approach to risk-aware Markov decision processes
Central-limit approach to risk-aware Markov decision processes Jia Yuan Yu Concordia University November 27, 2015 Joint work with Pengqian Yu and Huan Xu. Inventory Management 1 1 Look at current inventory
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationAN EXTENSION OF THE HONG-PARK VERSION OF THE CHOW-ROBBINS THEOREM ON SUMS OF NONINTEGRABLE RANDOM VARIABLES
J. Korean Math. Soc. 32 (1995), No. 2, pp. 363 370 AN EXTENSION OF THE HONG-PARK VERSION OF THE CHOW-ROBBINS THEOREM ON SUMS OF NONINTEGRABLE RANDOM VARIABLES ANDRÉ ADLER AND ANDREW ROSALSKY A famous result
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationERRATA: Probabilistic Techniques in Analysis
ERRATA: Probabilistic Techniques in Analysis ERRATA 1 Updated April 25, 26 Page 3, line 13. A 1,..., A n are independent if P(A i1 A ij ) = P(A 1 ) P(A ij ) for every subset {i 1,..., i j } of {1,...,
More informationApplied Mathematics Letters. Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system
Applied Mathematics Letters 5 (1) 198 1985 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Stationary distribution, ergodicity
More informationInvariance principles for fractionally integrated nonlinear processes
IMS Lecture Notes Monograph Series Invariance principles for fractionally integrated nonlinear processes Xiaofeng Shao, Wei Biao Wu University of Chicago Abstract: We obtain invariance principles for a
More informationLONG TIME BEHAVIOUR OF PERIODIC STOCHASTIC FLOWS.
LONG TIME BEHAVIOUR OF PERIODIC STOCHASTIC FLOWS. D. DOLGOPYAT, V. KALOSHIN AND L. KORALOV Abstract. We consider the evolution of a set carried by a space periodic incompressible stochastic flow in a Euclidean
More informationOn Reparametrization and the Gibbs Sampler
On Reparametrization and the Gibbs Sampler Jorge Carlos Román Department of Mathematics Vanderbilt University James P. Hobert Department of Statistics University of Florida March 2014 Brett Presnell Department
More informationOptimal Stopping under Adverse Nonlinear Expectation and Related Games
Optimal Stopping under Adverse Nonlinear Expectation and Related Games Marcel Nutz Jianfeng Zhang First version: December 7, 2012. This version: July 21, 2014 Abstract We study the existence of optimal
More information