On the tradeoff between almost sure error tolerance and its mean deviation frequency in martingale convergence

Convergence in L2 of Stochastic Ising Models: Jump Processes and Diffusions

We consider a simple estimator of the density mode using the frequency polygon estimate. We investigate strong consistency of the estimator for strong mixing sequence of real variables under mild assumptions. We study the almost sure rate of convergence and we show that the estimator can achieve optimal almost sure rates of convergence for appropriate choices of the bin widths. The asymptotic normality of the simple estimator is given and a simulation study is performed. Our asymptotic results are obtained without any differentiability condition assumed on the density around the mode.

Rate of convergence in the strong law of large numbers for a class of U-statistics and von Mises statistics

Almost sure convergence in noncommutative L2-spaces.- Individual ergodic theorems in L2 over a von neumann algebra.- Asymptotic formulae.- Convergence of iterates of contractions.- Convergence of orthogonal series and strong laws of large numbers.- Convergence of conditional expectations and martingales.- Miscellaneous results.

Under certain mixing conditions expressed in terms of absolutely joint cumulant summability, we establish two kinds of strong laws of large numbers for general two-dimensional (2-D) processes and give their convergence rates. Using these basic results, we derive the convergence rate of higher-order moment and cumulant estimators for 2-D stationary processes, and discuss the parameter estimation for 2-D harmonics in multiplicative and additive noise.

In this paper we provide the asymptotic theory of the general of φ-divergences measures, which include the most common divergence measures : R´enyi and Tsallis families and the Kullback-Leibler measure. We are interested in divergence measures in the discrete case. One sided and two-sided statistical tests are derived as well as symmetrized estimators. Almost sure rates of convergence and asymptotic normality theorem are obtained in the general case, and next particularized for the R´enyi and Tsallis families and for the Kullback-Leibler measure as well. Our theoretical results are validated by simulations.

Eweda and Macchi [IEEE Trans. Automat. Control, 29 (1984), pp. 119–127] and Watanabe [IEEE Trans. Inform. Theory, 30 (1984), pp. 134–140] show that thee sequence of random vectors generated by a stochastic gradient adaptive filtering algorithm converges almost surely and in $L_p $ (for p an even integer) to the solution of the associated Wiener–Hopf equation when the driving data process is stationary and weakly dependent. Under strong (i.e., Rosenblatt or $\alpha $) and $\psi $ -mixing conditions, together with various moment bounds on the driving data process, an almost sure functional invariance principle is obtained that approximates the sample paths of the random process generated by the stochastic gradient algorithm with the sample paths of a particular Gauss–Markov process. Almost sure rates of convergence in the form of laws of the iterated logarithm follow from the functional invariance principle. As a byproduct a functional central limit theorem is also obtained for a sequence of processes derived by suitably scaling the sequence of iterations generated by the algorithm.

ON THE ALMOST SURE RATE OF CONVERGENCE OF TEMPORAL-DIFFERENCE LEARNING ALGORITHMS

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. https://doi.org/10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is Hin (0,1) and the drift is mathcal {C}^alpha , alpha in [0,1] and alpha >1-1/(2H), we show the strong L_p and almost sure rates of convergence to be ((1/2+alpha H)wedge 1) -varepsilon , for any varepsilon >0. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. https://doi.org/10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence 1/2-varepsilon of the Euler–Maruyama scheme for mathcal {C}^alpha drift, for any varepsilon ,alpha >0.

In this paper, a notion of negative side ϱ-mixing (ϱ −-mixing) which can be regarded as asymptotic negative association is defined, and some Rosenthal type inequalities for ϱ −-mixing random fields are established. The complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are also discussed for ϱ −-mixing random fields. The results obtained extend those for negatively associated sequences and ϱ * random fields.

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n−1/2βnlog 3/2n in which βn can be particularly taken as (log log n)1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n−1/2(log log n)1/2log 2n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi:10.1155/2008/385362 (2008) for the case mentioned above, and derive the convergence rate n−1/2βnlog 1/2n for the above βn under the given covariance function, which improves the relevant one n−1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.

Let $S_n$ denote the partial sums of i.i.d. random variables with mean zero and moment generating function existing in some neighborhood of the origin. We give explicit upper bounds for $P_m^+ = P(S_n \geqq a + bn$ for some $n \geqq m)$ and $P_m = P(|S_n| \geqq a + bn$ for some $n \geqq m), a \geqq 0, b > 0$. These bounds immediately give the rate of convergence for the strong law of large numbers. An application is also made to a sequential selection procedure.

This chapter is devoted to the Monte Carlo method for the computation of expectation of simulable random variables. After a presentation of the Strong Law of Large Numbers (without proof) and the different ways to measure its rate of convergence (quadratic mean, Central Limit theorem, Law of the Iterated Logarithm), we introduce the notion of confidence interval at a given confidence level, illustrated by a simple application to the pricing of a multi-asset European option. Then, we propose a first approach to sensitivities computation - known as the Greeks when dealing with derivative financial products - by Monte Carlo simulation

We consider the random continued fraction where s n are independent random variables with the same gamma distribution. Every realization of the sequence defines a Stieltjes function that can be expressed as for some measure σ on the positive half-line. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Padé approximants of the function S . Using the Dyson–Schmidt method for an equivalent one-dimensional disordered system and the results of Marklof et al. , we obtain explicit formulae (in terms of modified Bessel functions) for the almost sure rate of convergence of these approximants, and for the almost sure distribution of their poles.

Asymptotic robustness of tests of overidentification and predeterminedness