On a representation of local martingale additive functionals of symmetric diffusions

1981 ◽  
pp. 110-118 ◽  
Author(s):  
M. Fukushima
Keyword(s):  
Local Martingale ◽  
Additive Functionals

scholarly journals Local martingale and pathwise solutions for an abstract fluids model

2011 ◽  
Vol 240 (14-15) ◽  
pp. 1123-1144 ◽  
Author(s):  
Arnaud Debussche ◽  
Nathan Glatt-Holtz ◽  
Roger Temam
Keyword(s):  
Local Martingale ◽  
Pathwise Solutions

Large deviations of generalized renewal process

2020 ◽  
Vol 30 (4) ◽  
pp. 215-241
Author(s):  
Gavriil A. Bakay ◽  
Aleksandr V. Shklyaev
Keyword(s):  
Large Deviations ◽  
Renewal Process ◽  
Limit Theorems ◽  
Local Limit ◽  
Asymptotic Formulas ◽  
Additive Functionals ◽  
Wide Range ◽  
Finite Markov Chains ◽  
Independent Identically Distributed ◽  
Local Limit Theorems

AbstractLet (ξ(i), η(i)) ∈ ℝd+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, NT = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process ZT = $\begin{array}{} \sum_{i=1}^{N_T} \end{array}$ξ(i). Put IΔT(x) = {y ∈ ℝd : xj ≤ yj < xj + ΔT, j = 1, …, d}. We find asymptotic formulas for the probabilities P(ZT ∈ IΔT(x)) as ΔT → 0 and P(ZT = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

scholarly journals Second-order limit laws for occupation times of fractional Brownian motion

2017 ◽  
Vol 54 (2) ◽  
pp. 444-461 ◽  
Author(s):  
Fangjun Xu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Method Of Moments ◽  
Second Order ◽  
Hurst Index ◽  
Occupation Times ◽  
Limit Laws ◽  
Additive Functionals ◽  
Finite Dimensional ◽  
Functional Version

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (02) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

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