Independent Increment Processes Research Articles

Quantum independent increment processes on superalgebras

On introduit la notion de processus quantique independant a increment stationnaire sur des superalgebres et on demontre un theoreme de reconstruction qui etablit une correspondance un-a-un entre ces processus et leurs generateurs infinitesimaux

Statistical analysis of the virkler data on fatigue crack growth

The extensive data set obtained by Virkler et al. for fatigue crack growth under homogeneous cyclic stressing is the object of statistical analysis. It is based on a previously published stochastic model of the crack growth. The statistical scatter of the experimental data is made up of a random “between” specimen variation and a random “within” specimen variation, the former being of the finite dimensional random vector type and the latter of random independent increment process type. The main results of the statistical analysis are 1. (1) that a random equation of the ParisErdogan type that allows for random material inhomogeneities fits very well to the data and 2. (2) that the distribution type for the number of stress cycles needed to grow a crack by a given length is convincingly described as being inverse Gaussian. Within the basic stochastic model this distribution type is asymptotically correct for large cycle number increments. Furthermore 3. (3) the random vector variation between specimens is reasonably well described by a joint normallog normal distribution. Numerical differentiation of the (basically non-difierentiable) experimental crack growth curves to obtain crack growth rates is avoided in the present model. In place the model points at direct maximum likelihood estimation of the parameters in the Paris-Erdogan equation.

Performance of an Adaptive Array Processor Subjected to Time-Varying Interference

Array processors for passive detection of directional wide-band Signals are commonly used in sonar, seismic work, and radio astronomy. The performance of such processors is degraded by directional noise sources referred to as interferences and adaptive processors to eliminate the effect of interferences have been used for many years. The issue addressed in this paper is movement of the interference and the effect that this has on adaptive loop design. The processor considered in the paper is an array of adjustable FIR filters using the Widrow LMS algorithm to adjust the filter weights. Changes in interference patterns affect both the covariance matrix of the observed signal and the optimum weight vector. For the purposes of this paper the optimum weight vector is modelled as an independent-increment process. This model permits the rigorous derivation of the optimum adaptive-loop gain parameter and of worst-case performance. In general it is found. that interference movement results in two error components: a gradient-noise component and a tracking component. Increases in adaptive loop gain decrease the tracking component but increase the gradient-noise component. Thus for a given set of interference-motion statistics an optimum gain can be found.

Gaps in the range of nearly increasing processes with stationary independent increments

For a class of stationary independent increments processes we find a necessary and sufficient integral test on a function 0 0, where R= range of X, Rt = range of X up to epoch t.

Harmonizable stable processes

This paper examines properties of a class of complex-valued stable processes which have spectral representation by means of independent-increments processes. A representation is derived by an application of Schilder's stochastic integral. Also, another construction of harmonizable stable processes by means of generalized stochastic processes is given, and its relation to the stochastic integral is shown. Some limit theorems of the Fourier transform of a sample from harmonizable stable processes are provided. Moreover, a linear prediction theory which pertains to those processes is suggested as an extension of that of second-order stationary processes.

The martingales of an independent increment process

A simple proof is given of the representation of martingales adapted to the sigma fields of a process with stationary, independent increments.

Optimal sampling of independent increment processes

A sample function drawn from a large class of independent increment processes with one of two sets of parameters is observed at evenly spaced time instants. The error probabilities associated with a standard hypothesis testing problem are studied as a function of the time interval between observations, holding the total number of observations fixed. A partial solution is obtained to the problem of choosing the optimal sampling rate.

Small-time limit theorems for stationary independent increment processes

Let X(t), t≧0, be a real-valued stochastic process with stationary independent increments, and X(0) = 0. Theorems analogous to the classical theorems on domains of attraction (see Feller [1]) are proved, and used to give necessary and sufficient conditions for the existence of weak limits of the form These are applied to extend the results of [2].

Second Order Moments and Prediction for Doubly-Reflected Symmetric Independent Increment Processes

Let $U(t)$ be a real-valued continuous alphabet random process with stationary independent increments and symmetric properties, i.e., \[ \Pr [U(t) - U(s) \in (a,b)] = \Pr [(U(t) - U(s)) \in ( - b, - a)],\quad t > s,\quad a < b. \] Let $x(t)$ be the process formed by reflection of $U(t)$ between barriers at 0 and $b > 0$. We use the Poisson summation formula to obtain series expansions for the conditional expectation, covariance, and power spectral density of $x(t)$. For the special case of the reflected Wiener process, a closed form expression is obtained for the power spectral density. Finally, the expected squared error for the nonlinear minimum mean-squared-error predictor is shown to depend only on the covariance and to strongly resemble the expected error for the optimal linear predictor.

The asymmetric Cauchy process

Among the independent increment processes in R, the strictly stable processes of index a have been extensively studied. Information about stable processes which are not strictly stable is much harder to establish because they do not satisfy the scaling property-that r-'l/X(rt) is a version of X(t) for every r > 0. This lecture describes some results, recently obtained in collaboration with W. E. Pruitt, about the general Cauchy process, which is the most interesting of the stable processes which are not strictly stable. In R1, the process is determined by

On the innovation theorem

Let z ( t ) , 0 ⩽ t ⩽ T z(t),0 \leqslant t \leqslant T , be the signal process and y ( t ) = ∫ 0 t z ( r ) d r + w ( t ) y(t) = \smallint _0^tz(r)\;dr + w(t) be the observation process where w ( t ) w(t) is a process of independent increments. It is shown that, under certain conditions, the innovation process v ( t ) = y ( t ) − ∫ 0 t E ( z ( r ) | y ( u ) , 0 ⩽ u ⩽ r ) d r v(t) = y(t) - \smallint _0^tE(z(r)|y(u), 0 \leqslant u \leqslant r)\;dr , has the same probability law as w ( t ) w(t) .

Distinguishing Stable Probability Measures Part II: Continuous Time

A sample function from one of two stable, stationary, independent-increment processes is observed for a finite time interval. For differing location, characteristic index, skewness, or scale, the probabilities measures induced by the process under either hypothesis are found to be mutually orthogonal. By suitably modifying the Levy measure associated with each probability measure, continuous-time tests for differing characteristic indices, skewness, or scale parameters can be posed as nonsingular detection problems; distinguishing location remains a singular detection problem. For the nonsingular problems, the likelihood functional is found explicitly, and performance limitations are determined. As an alternative approach, the observed sample function is sampled at discrete time instants over a finite time interval, and the performance of log likelihood test is studied as a function of sample spacing with a fixed, total number of observations.

Orthogonal functionals of independent-increment processes

In analogy with the Wiener-Itô theory of multiple integrals and orthogonal polynominals, a set of functionals of general square-integrable martingales is presented which, in the case of independent-increments processes, is orthogonal and complete in the sense that every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L^{2}</tex> -functional of the independent-increment process can be represented as an infinite sum of these elementary functionals. The functionals are iterated integrals of the basic martingales, similar to the multiple iterated integrals of Itô and can be also thought of as being the analogs of the powers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,x,x^{2}, \cdots</tex> of the usual calculus. The analogy is made even clearer by observing that expanding the Doleans-Dade formula for the exponential of the process in a Taylor-like series leads again to the above elementary functionals. A recursive formula for these functionals in terms of the basic martingale and of lower order functionals is given, and several connections with the theory of reproducing kernel Hilbert spaces associated with independent-increment processes are obtained.

Radon-Nikodym Derivatives with Respect to Measures Induced by Discontinuous Independent-Increment Processes

We obtain representation formulas for the Radon-Nikodym derivatives of measures absolutely continuous with respect to measures induced by processes with stationary independent increments. The proofs of these formulas, which have applications in signal detection and estimation problems, call heavily upon recent results in martingale theory, especially a general formula of Doleans-Dade for the logarithm of a strictly positive martingale in terms of a function measuring its jumps.

A Representation of Independent Increment Processes without Gaussian Components

A Representation of Independent Increment Processes without Gaussian Components

On the maximum of a stationary independent increment process

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.

On the maximum of a stationary independent increment process

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.

A note on optimal nonlinear prediction for independent increment processes

In this note, some earlier results of Fisher and Stear on optimal nonlinear filtering for independent increment processes are extended to include the problem of optimal non-linear prediction. Specifically, it is shown that the prediction problem requires solution of the filtering problem plus extrapolation of the last filter output through the nonlinear system dynamics, and it is shown that the conditional density function of the system state during the extrapolation operation satisfies the Fokker-Planck-Kolmogorov-Feller equation. Finally, some earlier results obtained by Fisher and Stear on the approximate solution of the optimal non-linear filtering problem for independent increment processes are extended to the case of prediction and, as a by-product, an earlier result of Kalman is extended to the case of independent increment processes.

A note on filtering for independent increment processes (Corresp.)

Using the equations of Fisher and Stear, a suboptimal filter for estimating the state of a linear system subject to non-Gaussian noise is developed. Some qualitative properties of this filter are derived.

The response of a particular nonlinear system with feedback to each of two random processes

A discrete time, nonlinear system composed of an integrator preceded by a binary quantizer with integrated negative feedback, which can model a tracking loop or a single integrating delta modulation communication system, is discussed with regard to the input-output statistics for two types of input processes: independent inputs and independent increments inputs. A recursion on time for the joint distribution of input and output is obtained for the independent inputs process and explicitly solved for the time asymptotic distribution, when it exists. The solution is examined in greater detail for the special case of IID normal inputs. When the system is excited by a process of independent increments, the asymptotic behavior of the input and output (they diverge) is of less interest than that of the difference between input and output, the tracking error. A recursion in time for the characteristic function of the error is developed and the time asymptotic solution found, The tracking error is interpreted by decomposition into static and dynamic parts, and an exponential bound to its distribution is provided. The particular case of normal increments input is discussed in additional detail.