Necessary and sufficient conditions of weak convergence of stochastic processes

1976 ◽  
Vol 17 (2) ◽  
pp. 364-366 ◽  
Author(s):  
V. M. Borodikhin
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Asymptotic behavior of the records of multivariate random sequences in a norm sense

2020 ◽  
Vol 70 (6) ◽  
pp. 1457-1468
Author(s):  
Haroon M. Barakat ◽  
M. H. Harpy
Keyword(s):  
Asymptotic Behavior ◽  
Weak Convergence ◽  
Sufficient Conditions ◽  
Record Values ◽  
Necessary And Sufficient Conditions ◽  
Random Sequences ◽  
Necessary And Sufficient

AbstractIn this paper, we investigate the asymptotic behavior of the multivariate record values by using the Reduced Ordering Principle (R-ordering). Necessary and sufficient conditions for weak convergence of the multivariate record values based on sup-norm are determined. Some illustrative examples are given.

Weak limits of sample range

1974 ◽  
Vol 11 (04) ◽  
pp. 836-841 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Weak Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weak Limits ◽  
Sample Range ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

scholarly journals On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

2009 ◽  
Vol 46 (03) ◽  
pp. 732-755 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Weak Convergence ◽  
Detailed Analysis ◽  
Sufficient Conditions ◽  
Interesting Case ◽  
Necessary And Sufficient Conditions ◽  
Limit Behavior ◽  
Gamma Process ◽  
Lévy Measure ◽  
Necessary And Sufficient ◽  
Significant Class

Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let X ε be the sum of jumps not exceeding ε, and let µ(ε)=E[X ε(1)]. We study the question of weak convergence of X ε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of X ε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

scholarly journals Weak convergence of first-rare-event times for semi-Markov processes

2020 ◽  
Author(s):  
AISDL
Keyword(s):  
Weak Convergence ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Rare Event ◽  
Necessary And Sufficient Conditions ◽  
Finite Set ◽  
In Series ◽  
Event Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions for weak convergence of first-rareevent times for semi-Markov processes with finite set of states in series of schemes are obtained.

Weak limits of sample range

1974 ◽  
Vol 11 (4) ◽  
pp. 836-841 ◽  
Author(s):  
Laurens De Haan
Keyword(s):  
Weak Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weak Limits ◽  
Sample Range ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

scholarly journals The factorial moments of additive functions with rational argument

2006 ◽  
Vol 81 (3) ◽  
pp. 425-440
Author(s):  
J. Šiaulys ◽  
G. Stepanauskas
Keyword(s):  
Weak Convergence ◽  
Limit Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Additive Functions ◽  
Factorial Moments ◽  
Rational Argument ◽  
Necessary And Sufficient ◽  
The Given

AbstractWe consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

WEAK CONVERGENCE TO LÉVY STABLE PROCESSES IN DYNAMICAL SYSTEMS

2010 ◽  
Vol 10 (02) ◽  
pp. 263-289 ◽  
Author(s):  
MARTA TYRAN-KAMIŃSKA
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Stable Process ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Processes ◽  
Skorohod Space ◽  
Necessary And Sufficient

We study convergence of normalized ergodic sum processes to Lévy stable process in the Skorohod space with J1-topology. Our necessary and sufficient conditions allow us to prove or disprove such convergence for specific examples.

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