On the Dispersion of Time-Dependent Means of a Stationary Stochastic Process

1961 ◽  
Vol 6 (1) ◽  
pp. 87-93 ◽  
Author(s):  
V. P. Leonov
Keyword(s):  
Stochastic Process ◽  
Time Dependent ◽  
Stationary Stochastic Process

scholarly journals Coagulation Processes with Gibbsian Time Evolution

2012 ◽  
Vol 49 (03) ◽  
pp. 612-626
Author(s):  
Boris L. Granovsky ◽  
Alexander V. Kryvoshaev
Keyword(s):  
Stochastic Process ◽  
Recurrence Relation ◽  
Time Evolution ◽  
Nonnegative Function ◽  
Gibbs Distribution ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Gibbs Distributions ◽  
Giant Cluster ◽  
Positive Integers

We prove that a stochastic process of pure coagulation has at any timet≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i,j) of single coagulations are of the form ψ(i;j) =if(j) +jf(i), wherefis an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the functionf. For the three corresponding models, we study the probability of coagulation into one giant cluster by timet> 0.

Stochastic Kriging for Crashworthiness Optimization Accounting for Simulation Noise

2020 ◽  
Author(s):  
Seyed Saeed Ahmadisoleymani ◽  
Samy Missoum
Keyword(s):  
Stochastic Process ◽  
Optimization Algorithm ◽  
Stationary Stochastic Process ◽  
Expected Improvement ◽  
Restraint System ◽  
Stochastic Kriging ◽  
Occupant Restraint System ◽  
Crashworthiness Optimization ◽  
Surrogate Based Optimization ◽  
Available Information

Abstract Vehicle crash simulations are notoriously costly and noisy. When performing crashworthiness optimization, it is therefore important to include available information to quantify the noise in the optimization. For this purpose, a stochastic kriging can be used to account for the uncertainty due to the simulation noise. It is done through the addition of a non-stationary stochastic process to the deterministic kriging formulation. This stochastic kriging, which can also be used to include the effect of random non-controllable parameters, can then be used for surrogate-based optimization. In this work, a stochastic kriging-based optimization algorithm is proposed with an infill criterion referred to as the Augmented Expected Improvement, which, unlike its deterministic counterpart the Expect Improvement, accounts for the presence of irreducible aleatory variance due to noise. One of the key novelty of the proposed algorithm stems from the approximation of the aleatory variance and its update during the optimization. The proposed approach is applied to the optimization of two problems including an analytical function and a crashwor-thiness problem where the components of an occupant restraint system of a vehicle are optimized.

Some comments on spectral representations of non-stationary stochastic processes

1973 ◽  
Vol 10 (04) ◽  
pp. 881-885 ◽  
Author(s):  
H. Tong
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Stationary Stochastic Process ◽  
Sufficient Condition ◽  
Stationary Stochastic Processes ◽  
Spectral Representations ◽  
Oscillatory Processes

The first part of the paper gives a multitude of essentially different representations of a stationary stochastic process. The second part gives a sufficient condition for the sum of two oscillatory processes to be again oscillatory.

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