quadratic variation of martingales in riesz spaces national research foundation of south africa grant specific unique reference number uid 85672 and by gnampa of italy u 2012 000574 20 07 2012 and u 2012 000388 09 05 2012

unbounded order convergence and application to martingales without probability abstract a net x in a vector lattice x is unbounded order convergent uo convergent to x if x x y o 0 for each y x and is unbounded order cauchy uo cauchy if the net x x is uo convergent to 0 in the first part of this article we study uo convergent and uo cauchy nets in banach lattices and use them to characterize banach lattices with the positive schur property and kb spaces in the second part we use the concept of uo cauchy sequences to extend doob s submartingale convergence theorems to a measure free setting our results imply in particular that every norm bounded submartingale in l 1 f is almost surely uo cauchy in f where f is an order continuous banach lattice with a weak unit

averaging operators and positive contractive projections abstract the main topic of the present paper is a generalization of two theorems about positive contractive projections on c 0 x due to seever and kelley resp to the more general structure of algebras the theorems in question are the following if a is an archimedean semiprime algebra with the stone condition then every positive contractive projection t in a satisfies the identity t a tb t ta tb for all a b a seever furthermore t is averaging if and only if the range of t is a subalgebra kelley both theorems hold in particular for an archimedean unital algebra in contrast to the original proofs of seever and kelley in the c 0 x case our proofs are purely algebraic and order theoretical

completeness for vector lattices abstract a net x in a vector lattice x is unbounded order convergent uo convergent to x if x x y is order convergent to 0 for each y x and is unbounded order cauchy uo cauchy if the net x x is unbounded order convergent to 0 the vector space x is unbounded order complete uo complete if every uo cauchy net in x is uo convergent recently li and chen proved that a vector lattice having the countable sup property is universally complete if it is uo complete the main result of this paper states that this equivalence is still true without any further assumption on the vector lattice x which answers an open question asked by li and chen some applications to this are given too

inversion of an integral transform related to a general form of heat equation inversion algorithms have been derived by d v widder for the weierstrass transform and by d t haimo for the dual weierstrass laguerre transform for the genera1 form of heat equation l u u l a self adjoint operator an integral transform is introduced with kernel related to the fundamental solution of the equation an inversion formula for the transform is derived which includes the previous resuks as special cases cf 1988 academic press inc

ito s rule and levy s theorem in vector lattices abstract the change of variable formula or ito s rule is studied in a dedekind complete vector lattice e with weak order unit e using the functional calculus we prove that for a holder continuous semimartingale x t x a m t b t t j and a twice continuously differentiable function f the formula 0 1 f x t f x a 0 t f x s d m s 0 t f x s d b s 1 2 0 t f x s d m s 0 s t j holds the first integral in the formula is an ito integral with reference to the local martingale m and the second and third integrals are dobrakov type integrals of a vector valued function with reference to a vector valued measure using the formula we prove levy s characterization of brownian motion as being a continuous martingale with compensator te the proof of this result yields a concrete description of abstract brownian motion defined in vector lattices

the hajek renyi chow maximal inequality and a strong law of large numbers in riesz spaces abstract in this paper we generalize the hajek renyi chow maximal inequality for submartingales to l p type riesz spaces with conditional expectation operators as applications we obtain a submartingale convergence theorem and a strong law of large numbers in riesz spaces along the way we develop a riesz space variant of the clarkson s inequality for 1 p 2

conditional expectations on riesz spaces abstract conditional expectations operators acting on riesz spaces are shown to commute with a class of principal band projections using the above commutation property conditional expectation operators on riesz spaces are shown to be averaging operators here the theory of f algebras is used when defining multiplication on the riesz spaces this leads to the extension of these conditional expectation operators to their so called natural domains i e maximal domains for which the operators are both averaging operators and conditional expectations the natural domain is in many aspects analogous to l 1

near epoch dependence in riesz spaces abstract the abstraction of the study of stochastic processes to banach lattices and vector lattices has received much attention by grobler kuo labuschagne stoica troitsky and watson over the past fifteen years by contrast mixing processes have received very little attention in particular mixingales were generalized to the riesz space setting in kuo et al 2013 12 the concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in kuo et al 2017 11 in the present work we formulate the concept of near epoch dependence for riesz space processes and show that if a process is near epoch dependent and either strong or uniform mixing then the process is a mixingale giving access to a law of large numbers the above is applied to autoregressive processes of order 1 in riesz spaces

numerically efficient fourier based technique for calculating error probabilities with intersymbol interference we propose a numerically efficient technique for calculating the probability of symbol error for arbitrary coherent modulation schemes in the presence of intersymbol interference isi and additive noise the probability of error is formulated in terms of an inverse fourier transform of the windowed characteristic function of the random variable representing the interfering symbols and the noise process the integral is evaluated numerically using the sampling theorem

lp spaces with respect to conditional expectation on riesz spaces recently labuschagne and watson introduced the space l2 t for a conditional expectation t with natural domain a riesz space l1 t the main purpose of this paper is to carry on with this process by introducing and investigate the spaces lp t with p 1

unbounded order convergence in dual spaces abstract a net x in a vector lattice x is said to be unbounded order convergent or uo convergent for short to x x if the net x x y converges to 0 in order for all y x in this paper we study unbounded order convergence in dual spaces of banach lattices let x be a banach lattice we prove that every norm bounded uo convergent net in x is w convergent iff x has order continuous norm and that every w convergent net in x is uo convergent iff x is atomic with order continuous norm we also characterize among order complete banach lattices the spaces in whose dual space every simultaneously uo and w convergent sequence converges weakly in norm

on lp spectrum of elliptic operators abstract for elliptic operators a m a x d on rn and certain of their singular perturbations b m b x d relative compactness of b with respect to a is established this result applies to the study of lp spectra of elliptic operators for different p

the ito integral for martingales in vector lattices abstract in the setting of a dedekind complete riesz space on which a conditional expectation is defined we study the ito integral this integral obtains the integration of a stochastic process of a continuous parameter relative to a martingale a considerable part of the paper is devoted to aspects of integration of a vector valued function with respect to a vector measure here we use the dobrakov integral that is in our case a generalization of the well known bartle integral we discuss natural and predictable processes to prove a general version of the doob meyer decomposition of a submartingale this decomposition provides an essential tool used in the definition of the stochastic integral we derive the properties of the stochastic integral that are useful in developing the theory of stochastic processes in riesz spaces

on in dependence measures in riesz spaces abstract in this paper we introduce an independence measure between a vector f in a riesz space e and a conditional expectation operator t on e we show that this measure satisfies some postulates corresponding to these formulated by renyi 1959 for dependence measure between random variables the resulting dependence measure in e is also investigated some examples of application are given

a banach lattice approach to convergent integrably bounded set valued martingales and their positive parts denote by cf x the set of all nonempty convex closed subsets of a separable banach space x let be a complete probability space and denote by l1 cf x the complete metric space of equivalence classes of a e equal integrably bounded cf x valued functions for any preassigned filtration i we describe the space of convergent integrably bounded cf x valued martingales in terms of the closure of i 1 l1 i cf x in l1 cf x in particular we provide a formula to calculate the join of two such martingales and the positive part of such a martingale our object is achieved by considering the more general setting of a near vector lattice s d endowed with a riesz metric d by means of radstrom s embedding theorem for such spaces a link is established between the space of convergent martingales in s and the space of convergent martingales in the radstrom completion r s of s this link provides information about the former space of martingales via known properties of measure free martingales in riesz normed vector lattices applicable to r s we also apply our general results to the spaces of convergent ck x valued martingales where ck x denotes the set of all nonempty convex compact subsets of x

equisummability of eigenfunction expansions under analytic multipliers in this paper we present some abstract criteria for equisummability of expansions in eienfunctions of certain pairs of elliptic operators on general domains of w the criteria to be developed originate in some cases from general banach space arguments and in others from the more specific spatial nature of the differential operators involved the importance of our analysis of l equisummability of two operators is that the question of convergence of the two summability means is reduced essentially to showing that the difference of the modified resolvent operators is uniformly bounded we apply these criteria to give a simple proof of an equisummability result of gurarie and kon 4 for a certain class of elliptic operators whose leading terms are positive and lower order terms have coefficients which are singular on a nowhere dense set the key technique is to analyze kernels of the resolvents and use radial bounds of the resolvents developed in 3 the prototypical case for equisummability is found in the study of equiconvergence for differential operators classically haar 51 walsh lo birkhoff 2 and tamarkin lo have shown that the difference between expansions with respect to eigenfunctions of a sturm liouville operator or birkhoff series and the ordinary fourier series tends to zero uniformly in every finite interval in this case the multiplier or summator function is the characteristic function in some one dimensional cases where such equiconvergence fails stone 9 levitan and sargsjan 6 and benzinger 1 have showed equisummability for riesz typical means of eigenfunction expansions of differential operators where the summator function is

the kolmogorov centsov theorem and brownian motion in vector lattices abstract the well known kolmogorov centsov theorem is proved in a dedekind complete vector lattice riesz space with weak order unit on which a strictly positive conditional expectation is defined it gives conditions that guarantee the holder continuity of a stochastic process in the space we discuss the notion of independence of projections and elements in the vector lattice and use this together with the kolmogorov centsov theorem to give an abstract definition of brownian motion in a vector lattice this definition captures the fact that the increments in a brownian motion are normally distributed and that the paths are continuous

mixing inequalities in riesz spaces abstract various topics in stochastic processes have been considered in the abstract setting of riesz spaces for example martingales martingale convergence ergodic theory amarts markov processes and mixingales here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes the two mixing coefficients which will be considered are the strong and uniform mixing coefficients we conclude with mixing inequalities for these types of processes in order to facilitate this development the study of generalized l 1 and l spaces begun by kuo labuschagne and watson will be extended

spaces of regular abstract martingales abstract in 15 10 the authors introduced and studied the space m r of regular martingales on a vector lattice and the space m r of bounded regular martingales on a banach lattice in this note we study these two spaces from the vector lattice point of view we show in particular that these spaces need not be vector lattices however if the underlying space is order complete then m r is a vector lattice and m r is a banach lattice under the regular norm

simple and accurate methods for outage analysis in cellular mobile radio systems a unified approach two unified expressions for computing the refined outage criterion which considers the receiver noise in cellular mobile radio systems are derived using the laplace and fourier inversion formulas since these expressions do not impose any restrictions on the signal statistics while being easy to program they provide a powerful tool for outage analysis over generalized fading channels we also assess compatibility and applicability of previously published approaches that treat noise as cochannel interference noise limited model or consider a minimum detectable receiver signal threshold and receiver noise the outage probability in an interference limited case can be evaluated directly by setting the minimum power threshold to zero the analysis of correlated interferers is presented results are also developed for a random number of interferers several new closed form expressions for the outage probability are also derived some previous studies have suggested approximating rician desired signal statistics by a nakagami m 1960 model with positive integer fading severity index to circumvent the difficulty in evaluating the outage in rician fading the suitability of this approximation is examined by comparing the outage performance under these two fading conditions surprisingly some basic results for nakagami m channel have been overlooked which has led to misleadingly optimistic results with the nakagami m approximation model however similar approximation for the interferer signals is valid

ergodic theory and the strong law of large numbers on riesz spaces abstract in w c kuo c c a labuschagne b a watson discrete time stochastic processes on riesz spaces indag math n s 15 3 2004 435 451 we introduced the concepts of conditional expectations martingales and stopping times on riesz spaces here we formulate and prove order theoretic analogues of the birkhoff hopf and wiener ergodic theorems and the strong law of large numbers on riesz spaces vector lattices

a useful technique for interference analysis in nakagami fading this paper presents a unified analytical method for computing the average e g x sigmak 1 k yk b of some arbitrary function g when x is a gamma random variable and independent of the arbitrary random variables y1 y2 yk this leads to new explicit expressions for the averages of some specific functions including erfc radicz exp z zn and in 1 z in terms of the moment generating functions of y1 y2 yk

kernels of elliptic operators bounds and summability in this paper we shall study the kernel of the resolvent r c a and of some other related functions of a e g e for elliptic operators a on f or more generally for perturbations of elliptic operators it turns out that the resolvent consequently all other related functions are given by an integral kernel which is bounded by a convolution with a radial decreasing l function this result has numerous applications bounds for l spectrum of a closedness semigroup generation essential selfadjointness summability etc in gkl we studied this problem for perturbation of constant coefficient elliptic operators and established the following bound on the kernel r x y of a ir w i c w w ix vi p ici7 1 where h is the radial l function h z i izl s

mixingales on riesz spaces abstract a mixingale is a stochastic process which combines properties of martingales and mixing sequences mcleish introduced the term mixingale at the 4th conference on stochastic processes and application at york university toronto 1974 in the context of l 2 in this paper we generalize the concept of a mixingale to the measure free riesz space setting this generalizes all of the l p 1 p variants and prove that a weak law of large numbers holds for riesz space mixingales in the process we also generalize the concept of uniform integrability to the riesz space setting

the evaluation of error probabilities for intersymbol and cochannel interference an efficient series that can be used to calculate the probability of error in a binary symmetric channel with intersymbol interference and additive noise is derived the series is derived by representing the noise complementary probability distribution function by an exact or approximate fourier series bounds on the accuracy of the estimate are derived for gaussian noise examples show that only a small numerical effort may be required to compute error probabilities of interest using the series applications to both finite and infinite intersymbol interference systems are discussed a similar technique is used to derive series representations for the probability of error of additive noise channels with cochannel interference or with combined intersymbol and cochannel interferences the accuracy of the results is bounded for gaussian noise

the ito integral for brownian motion in vector lattices part 2 abstract the ito integral for brownian motion in a vector lattice as constructed in part 1 of this paper is extended to accommodate a larger class of integrands this extension provides an analogue of the indefinite ito integral in the classical setting which yields a local martingale the assumption is that there exists a conditional expectation operator on the vector lattice and the construction does not depend on a probability measure space the classical case of the extended ito integral is a special case of the constructed integral in the vector lattice

an infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of rayleigh random variables the properties of the series are studied for both bounded and unbounded random variables the technique is used to find efficient series for computation of the distributions of sums of uniform random variables and sums of rayleigh random variables a useful closed form expression for the characteristic function of a rayleigh random variable is presented and an efficient method for computing a confluent hypergeometric function is given an infinite series for the probability density function of a sum of independent random variables is also derived the inversion of characteristic functions a trapezoidal rule for numerical integration and the sampling theorem in the frequency domain are related to and interpreted in terms of the results

a description of norm convergent martingales on vector valued lp spaces abstract norm convergent martingales on tensor products of banach spaces are considered in a measure free setting as a consequence we obtain the following characterization for convergent martingales on vector valued l p spaces let be a probability space x a banach space and n an increasing sequence of sub algebras of in order for f n n n 1 to be a convergent martingale in l p x 1 p it is necessary and sufficient that for each i n there exists a convergent martingale x i n n n 1 in l p and y i x such that for each n n we have f n s i 1 x i n s y i for all s where i 1 lim n x i n l p and lim i y i 0

radial bounds for perturbations of elliptic operators abstract elliptic operators a m b x d a multi index with leading term positive and constant coefficient and with lower order coefficients b x l r l with n r defined on r n or a quotient space r n r n u u r n are considered it is shown that the lp spectrum of a is contained in a parabolic region of the complex plane enclosing the positive real axis uniformly in p outside the kernel of the resolvent of a is shown to be uniformly bounded by an l1 radial convolution kernel some consequences are a can be closed in all lp 1 p and is essentially self adjoint in l2 if it is symmetric a generates an analytic semigroup e ta in the right half plane strongly lp and pointwise continuous at t 0 a priori estimates relating the leading term and remainder are obtained and summability ea e 0 0 with analytic is proved for l p with convergence in lp and on the lebesgue set of more comprehensive summability results are obtained when a has constant coefficients

new multiplier methods for summing classical eigenfunction expansions on unifie et on generalise un certain nombre de resultats en theorie de sommabilite classique pour des developpements de sturm liouville regulier et singulier

operator martingale decompositions and the radon nikodým property in banach spaces abstract we consider submartingales and uniform amarts of maps acting between a banach lattice and a banach lattice or a banach space in this measure free setting of martingale theory it is known that a banach space y has the radon nikod m property if and only if every uniformly norm bounded martingale defined on the chaney schaefer l tensor product e l y where e is a suitable banach lattice is norm convergent we present applications of this result firstly an analogues characterization for banach lattices y with the radon nikod m property is given in terms of a suitable set of submartingales supermartingales on e l y secondly we derive a riesz decomposition for uniform amarts of maps acting between a banach lattice and a banach space this result is used to characterize banach spaces with the radon nikod m property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent in the case 1 p our results yield l p y space analogues of some of the well known results on uniform amarts in l 1 y spaces