Departure from normality of increasing-dimension martingales

In this paper, we consider sequences of vector martingale differences of increasing dimension. We show that the Kantorovich distance from the distribution of the k(n)-dimensional average of n martingale differences to the corresponding Gaussian distribution satisfies certain inequalities. As a consequence, if the growth of k(n) is not too fast, then the Kantorovich distance converges to zero. Two applications of this result are presented. The first is a precise proof of the asymptotic distribution of the multivariate portmanteau statistic applied to the residuals of an autoregressive model and the second is a proof of the asymptotic normality of the estimates of a finite autoregressive model when the process is an AR(∞) and the order of the model grows with the length of the series.     

A short proof of the Collapsing Walls Lemma

In this short note we give a simple geometric proof of the Collapsing Walls Lemma given by Pak and Pinchasi (2012) in [1]. Our arguments work in spaces of constant curvature.     

Every minor-closed property of sparse graphs is testable

Suppose G is a graph of bounded degree d, and one needs to remove ?n of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G^′. In fact, a similar result is proved for any minor-closed property of bounded degree graphs.The main motivation of the above result comes from theoretical computer-science. Using our main result we infer that for any minor-closed property P, there is a constant time algorithm for detecting if a graph is “far” from satisfying P. This, in particular, answers an open problem of Goldreich and Ron [STOC 1997] [20], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.     

Pathwise superhedging for time-dependent barrier options on càdlàg paths—Finite or infinite tradeable European,One-Touch,lookback or forward starting options

We establish pathwise duality using simple predictable trading strategies for the robust hedging problem associated with a barrier option whose payoff depends on the terminal level and the infimum of a càdlàg strictly positive stock price process, given tradeable European options at all strikes at a single maturity. The result allows for a significant dimension reduction in the computation of the superhedging cost, via an alternate lower-dimensional formulation of the primal problem as a convex optimization problem, which is qualitatively similar to the duality which was formally sketched using linear programming arguments in Duembgen and Rogers [10] for the case where we only consider continuous sample paths. The proof exploits a simplification of a classical result by Rogers (1993) which characterizes the attainable joint laws for the supremum and the drawdown of a uniformly integrable martingale (not necessarily continuous), combined with classical convex duality results from Rockefellar (1974) using paired spaces with compatible locally convex topologies and the Hahn–Banach theorem. We later adapt this result to include additional tradeable One-Touch options using the Kertz and Rösler (1990) condition. We also compute the superhedging cost when in the more realistic situation where there is only finite tradeable European options; for this case we obtain the full duality in the sense of quantile hedging as in Soner (2015), where the superhedge works with probability $1?\epsilon$ where $\epsilon$ can be arbitrarily small), and we obtain an upper bound for the true pathwise superhedging cost. In Section 5, we extend our analysis to include time-dependent barrier options using martingale coupling arguments, where we now have tradeable European options at both maturities at all strikes and tradeable forward starting options at all strikes. This set up is designed to approximate the more realistic situation where we have a finite number of tradeable Europeans at both maturities plus a finite number of tradeable forward starting options.¹     

An elementary proof of the <Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript> bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A _p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L ¹ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A ² bounds in that setting.     

On the chromatic number of random d-regular graphs

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k−1)log(k−1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k−1 or k. If moreover d>(2k−3)log(k−1), then the value k−1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of equitable k-colourings, where a colouring is equitable if the number of vertices of each colour is equal.     

A representation of the Kantorovich-Functional

This paper establishes the representation of the generalizedN-dimensional Wasserstein distance (Kantorovich-Functional) $$W_c (P_1 ,...,P_N ): = \inf \left\{ {\int_{S^N } {c(x_1 ,...,x_N )d\mu } (x_1 ,...,x_N ):\pi _i \mu = P_i ,i = 1,...,N} \right\}$$ in the form ofW _c(P ₁,...,P _N )=sup{∑ _i=1 ^N }∫sf _i dP _i . The conditions we impose onP _i ,c andf _i enable us to follow those classical lines of arguments which lead to the Kantorovich-Rubinstein Theorem: By elementary methods we show how the result for an arbitrary metric space (S, d) can be derived from the case of finiteS. We also apply this result and the techniques of its proof in order to obtain a fairly simple proof of Strassen's Theorem.     

Martingale Transforms and the Hardy-Littlewood-Sobolev Inequality for Semigroups

We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev (HLS) inequality based on this representation. The proof rests on a new inequality for the fractional Littlewood-Paley g–function.     

A stochastic calculus model of continuous trading: Complete markets

A paper by the same authors in the 1981 volume of Stochastic Processes and Their Applications presented a general model, based on martingales and stochastic integrals, for the economic problem of investing in a portfolio of securities. In particular, and using the terminology developed therein, that paper stated that every integrable contingent claim is attainable (i.e., the model is complete) if and only if every martingale can be represented as a stochastic integral with respect to the discounted price process. This paper provides a detailed proof of that result as well as the following: The model is complete if and only if there exists a unique martingale measure.     

A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.     

Asymptotics of orthogonal polynomials beyond the scope of Szegő’s theorem

First, we give a simple proof of a remarkable result due to Videnskii and Shirokov: let B be a Blaschke product with n zeros; then there exists an outer function φ, φ(0) = 1, such that ‖(Bφ)′‖ ? Cn, where C is an absolute constant. Then we apply this result to a certain problem of finding the asymptotics of orthogonal polynomials.     

Fluctuations of the free energy in the high temperature Hopfield model

We consider the Hopfield model of size N and with p∼tN patterns, in the whole high temperature (paramagnetic) region. Our result is that the partition function has log-normal fluctuations. It is obtained by extending to the present model the method of the interpolating Brownian motions used by Comets (Comm. Math. Phys. 166 (1995) 549–564) for the Sherrington–Kirkpatrick model. We view the load t of the memory as a dynamical parameter, making the partition function a nice stochastic process. Then we write some semi-martingale decomposition for the logarithm of the partition function, and we prove that all the terms in this decomposition converge. In particular, the martingale term converges to a Gaussian martingale.     

Remarks on the stochastic integral

In Karandikar-Rao [11], the quadratic variation [M, M] of a (local) martingale was obtained directly using only Doob’s maximal inequality and it was remarked that the stochastic integral can be defined using [M, M], avoiding using the predictable quadratic variation 〈M, M〉 (of a locally square integrable martingale) as is usually done. This is accomplished here- starting with the result proved in [11], we construct ∫ f dX where X is a semimartingale and f is predictable and prove dominated convergence theorem (DCT) for the stochastic integral. Indeed, we characterize the class of integrands f for this integral as the class L(X) of predictable processes f such that |f| serves as the dominating function in the DCT for the stochastic integral. This observation seems to be new.We then discuss the vector stochastic integral ∫ 〈f, dY〉 where f is ? ^d valued predictable process, Y is ? ^d valued semimartingale. This was defined by Jacod [6] starting from vector valued simple functions. Memin [13] proved that for (local) martingales M¹, … M ^d : If N ⁿ are martingales such that N _t ⁿ → N _t for every t and if ?f ⁿ such that N _t ⁿ = ∫ 〈f ⁿ , dM〉, then ?f such that N = ∫ 〈f, dM〉.Taking a cue from our characterization of L(X), we define the vector integral in terms of the scalar integral and then give a direct proof of the result due to Memin stated above.This completeness result is an important step in the proof of the Jacod-Yor [4] result on martingale representation property and uniqueness of equivalent martingale measure. This result is also known as the second fundamental theorem of asset pricing.     

ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS

By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r(A) such that its spectrum is the closed disc {z:|z| ≤ r(A)} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.     

Random regular graphs of non-constant degree: Concentration of the chromatic number

In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model G_n,d for d=o(n^1/5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G(n,p) with . Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for G(n,p) for p=n^−δ where δ>1/2. The main tool used to derive such a result is a careful analysis of the distribution of edges in G_n,d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model.     

An alternative approach to the Faber–Krahn inequality for Robin problems

We give a simple proof of the Faber–Krahn inequality for the first eigenvalue of the p-Laplace operator with Robin boundary conditions. The techniques introduced allow to work with much less regular domains by using test function arguments. We substantially simplify earlier proofs, and establish the sharpness of the inequality for a larger class of domains at the same time.     

The core of a simple game with ordinal preferences

Michael Dummett andRobin Farquharson [1961] provided a sufficient condition for ann-person simple majority game with ordinal preferences to have a nonempty core. In the present paper we generalize this result to an arbitrary proper simple game. It is proved that their condition is also sufficient for this game to have a nonempty core. Our proof of this theorem is much simpler than the proof given byDummett andFarquharson. Finally some applications of the theorem are presented.     

A unifying Radon-Nikodym theorem for vector measures

It is a known fact that certain derivation bases from martingales with a directed index set. On the other hand it is also true that the strong convergence of certain abstract martingales is a consequence of the Radon-Nikodym theory for vector measures (cf. Uhl, J. J., Jr., Trans. Amer. Math. Soc.145 1969, 271–285). Many other connections and applications of the latter theory with multidimensional problems in stochastic processes and representation theory are known (cf. Dinculeanu, N., Studia Math.25 1965, 181–205; Dinculeanu, N., and Foias, C., Canad. J. Math.13 1961, 529–556; Rao, M. M., Ann. Mat. pura et applicata76 1967, 107–132; Rybakov, V. I., Izv. Vys?. U?ebn. Zaved. Matematika19 1968, 92–101; Rybakov, V. I., Dokl. Akad. Nauk SSSR180 1968, 620–623). Starting from various vantage points, many authors have proposed several hypotheses for establishing abstract Radon-Nikodym theorems. In view of the great interest and importance of this problem in the areas mentioned above, it is natural to obtain a unifying result with a general enough hypothesis to deduce the various forms of the Radon-Nikodym theorem for vector measures. This should illuminate the Radon-Nikodym theory for vector measures and stimulate further work in abstract martingale problems. In this paper the first problem is attacked, leaving the martingale part and other applications for another study.The main result (Theorem 7 of Section 2) provides the desired unification and from if the Dunford-Pettis theorem, the Phillips theorem and several others are obtained. As martingale-type arguments are constantly present, a careful reader may note the easy translation of the hypothesis to the martingale convergence problem but we treat only the Radon-Nikodym problem using the language of measure theory and linear analysis.     

A simple proof e 2 is irrational

It is well known that e ² is irrational: this note presents a simple proof of it. The arguments stay within the realms of a first proof course in mathematical analysis offered for undergraduates.     

Stochastic Lotka-Volterra system with infinite delay

This paper investigates a stochastic Lotka-Volterra system with infinite delay, whose initial data comes from an admissible Banach space C_r. We show that, under a simple hypothesis on the environmental noise, the stochastic Lotka-Volterra system with infinite delay has a unique global positive solution, and this positive solution will be asymptotic bounded. The asymptotic pathwise of the solution is also estimated by the exponential martingale inequality. Finally, two examples with their numerical simulations are provided to illustrate our result.