scholarly journals Stochastic comparison of point random fields

1997 ◽  
Vol 34 (4) ◽  
pp. 868-881 ◽  
Author(s):  
Hans-Otto Georgii ◽  
Torsten Küneth
Keyword(s):  
Point Process ◽  
Random Fields ◽  
Alternative Proof ◽  
Probability Measures ◽  
Stochastic Comparison ◽  
Sufficient Condition ◽  
Stochastic Domination ◽  
Positive Correlations ◽  
Increasing Functions

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.

scholarly journals Stochastic comparison of point random fields

1997 ◽  
Vol 34 (04) ◽  
pp. 868-881 ◽  
Author(s):  
Hans-Otto Georgii ◽  
Torsten Küneth
Keyword(s):  
Point Process ◽  
Random Fields ◽  
Alternative Proof ◽  
Probability Measures ◽  
Stochastic Comparison ◽  
Sufficient Condition ◽  
Stochastic Domination ◽  
Positive Correlations ◽  
Increasing Functions

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.

scholarly journals Three Essays on Stopping

Risks ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 105
Author(s):  
Eberhard Mayerhofer
Keyword(s):  
First Passage Time ◽  
Passage Time ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Diffusion Characteristic ◽  
Reflected Brownian Motion ◽  
First Passage ◽  
Brownian Motion With Drift ◽  
Spectrally Negative Lévy Process ◽  
Closed Form Formula

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This corrects a formula by Perry et al. (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown, if and only if the diffusion characteristic μ / σ 2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum before a fixed drawdown is exponentially distributed for any spectrally negative Lévy process, a result due to Mijatović and Pistorius (2012). Our proof is similar, but simpler than Lehoczky (1977) or Landriault et al. (2017).

Seneta constants for the supercritical Bellman–Harris process

1982 ◽  
Vol 14 (04) ◽  
pp. 732-751
Author(s):  
H.-J. Schuh
Keyword(s):  
Point Process ◽  
Law Of Large Numbers ◽  
Random Variable ◽  
Alternative Proof ◽  
Large Numbers ◽  
Watson Process

Let be a supercritical Bellman-Harris process with finite offspring mean. Cohn [17] has shown that there always exist constants Ct such that lim t→∞ Zt /Ct = W almost surely for some non-degenerate random variable W. In this paper we give an alternative proof, based on the study of (Zt ) as a point process. Our methods are to some extent analytical and parallel Seneta's [18] and Heyde's [11] approaches in the case of the Galton–Watson process. We further identify Ct as 1/(–log Ft (–1)(γ)), where Ft (γ) = E(γ z t), i.e. the norming constants found by Seneta [18] for the Galton–Watson process, apply also to the Bellman-Harris process. Finally we derive a weak law of large numbers for W, prove that W is continuous on (0,∞) and show that W has [0,∞) as its support.

scholarly journals Tail asymptotics of signal-to-interference ratio dis­tribution in spatial cellular network models

2018 ◽  
Vol 37 (2) ◽  
pp. 431-453 ◽  
Author(s):  
Naoto Miyoshi ◽  
Tomoyuki Shirai
Keyword(s):  
Point Process ◽  
Upper Bound ◽  
Cellular Network ◽  
Path Loss ◽  
Network Models ◽  
Base Stations ◽  
Sufficient Condition ◽  
Tail Asymptotics ◽  
Tail Distribution ◽  
Stationary Point Process

TAIL ASYMPTOTICS OF SIGNAL-TO-INTERFERENCE RATI ODISTRIBUTION IN SPATIAL CELLULAR NETWORK MODELSWe consider a spatial stochastic model of wireless cellular networks, where the base stations BSs are deployed according to a simple and stationary point process on Rd, d > 2. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio SIR, which is a key quantity in wireless communications. In the case where the pathloss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based on a Poisson point process and on a determinantal point process meet the sufficient condition, we also give a counterexample violating it. In the case of bounded path-loss functions, we derive a logarithmically asymptotic upper bound on the SIR tail distribution for the Poisson-based and -Ginibrebased models. A logarithmically asymptotic lower bound with the same order as the upper bound is also obtained for the Poisson-based model.

scholarly journals On a Class of Tensor Markov Fields

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 451
Author(s):  
Enrique Hernández-Lemus
Keyword(s):  
Graphical Models ◽  
Random Fields ◽  
Markov Random Fields ◽  
Random Variables ◽  
Probability Measures ◽  
Gibbs Fields ◽  
Markov Random ◽  
Statistical Dependency ◽  
Markov Fields ◽  
Dependency Structures

Here, we introduce a class of Tensor Markov Fields intended as probabilistic graphical models from random variables spanned over multiplexed contexts. These fields are an extension of Markov Random Fields for tensor-valued random variables. By extending the results of Dobruschin, Hammersley and Clifford to such tensor valued fields, we proved that tensor Markov fields are indeed Gibbs fields, whenever strictly positive probability measures are considered. Hence, there is a direct relationship with many results from theoretical statistical mechanics. We showed how this class of Markov fields it can be built based on a statistical dependency structures inferred on information theoretical grounds over empirical data. Thus, aside from purely theoretical interest, the Tensor Markov Fields described here may be useful for mathematical modeling and data analysis due to their intrinsic simplicity and generality.

scholarly journals On Some Criteria for a Set to be of Class N

1961 ◽  
Vol 19 ◽  
pp. 189-194
Author(s):  
Nobuyuki Suita
Keyword(s):  
Hausdorff Measure ◽  
Measure Zero ◽  
Plane Domain ◽  
Alternative Proof ◽  
Compact Set ◽  
Sufficient Condition ◽  
Closed Set ◽  
One Dimensional ◽  
Dimensional Measure

Let D be a plane domain containing the point at infinity and E its complementary closed set. As to a sufficient condition for a compact set E to be of class N, Pfluger-Mori’s criterion is well-known (Pfluger [10], Mori [6]). Various relations between the conditions of this type and the Hausdorff measure of the set E have been investigated recently by Kuroda and Ozawa (Kuroda [5], Ozawa and Kuroda [8], Ozawa [7]). For example they showed that Pfluger-Mori’s condition implies that the set E is of one dimensional measure zero under some additional conditions (cf. [7], [8]). In the present paper we shall give an alternative proof of Pfluger-Mori’s criterion and another criterion using analytic module and, further, prove some criteria for the set E to be of one dimensional measure zero.

scholarly journals Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample

10.37236/6773 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Karl Heuer
Keyword(s):  
Finite Graph ◽  
Infinite Graphs ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Locally Finite ◽  
Finite Graphs ◽  
The Third ◽  
Locally Finite Graph ◽  
Locally Finite Graphs ◽  
A Minor

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.

A sample-path approach to Palm probabilities

1994 ◽  
Vol 31 (2) ◽  
pp. 430-437
Author(s):  
Shaler Stidham
Keyword(s):  
Path Analysis ◽  
Function Space ◽  
Point Process ◽  
Sample Path ◽  
Probability Measures ◽  
Sample Path Analysis ◽  
Inversion Formulas ◽  
One Dimensional ◽  
And Inversion ◽  
Palm Probabilities

Previous papers have established sample-path versions of relations between marginal time-stationary and event-stationary (Palm) state probabilities for a process with an imbedded point process. This paper extends the use of sample-path analysis to provide relations between frequencies for arbitrary (measurable) sets in function space, rather than just marginal (one-dimensional) frequencies. We define sample-path analogues of the time-stationary and event-stationary (Palm) probability measures for a process with an imbedded point process, and then derive sample-path versions of the Palm transformation and inversion formulas.

scholarly journals Invariant coupling of determinantal measures on sofic groups

2014 ◽  
Vol 36 (2) ◽  
pp. 574-607 ◽  
Author(s):  
RUSSELL LYONS ◽  
ANDREAS THOM
Keyword(s):  
Bernoulli Shift ◽  
Probability Measures ◽  
Stochastic Domination ◽  
Positive Contraction ◽  
Spanning Forests ◽  
Invariant Coupling ◽  
Sofic Groups ◽  
Dependent Processes ◽  
Monotone Coupling ◽  
Explicit Proof

To any positive contraction $Q$ on $\ell ^{2}(W)$, there is associated a determinantal probability measure $\mathbf{P}^{Q}$ on $2^{W}$, where $W$ is a denumerable set. Let ${\rm\Gamma}$ be a countable sofic finitely generated group and $G=({\rm\Gamma},\mathsf{E})$ be a Cayley graph of ${\rm\Gamma}$. We show that if $Q_{1}$ and $Q_{2}$ are two ${\rm\Gamma}$-equivariant positive contractions on $\ell ^{2}({\rm\Gamma})$ or on $\ell ^{2}(\mathsf{E})$ with $Q_{1}\leq Q_{2}$, then there exists a ${\rm\Gamma}$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination $\mathbf{P}^{Q_{1}}\preccurlyeq \mathbf{P}^{Q_{2}}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when ${\rm\Gamma}$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures $\mathbf{P}^{Q}$ as above are $\bar{d}$-limits of finitely dependent processes. Thus, when ${\rm\Gamma}$ is amenable, $\mathbf{P}^{Q}$ is isomorphic to a Bernoulli shift, which was known before only when ${\rm\Gamma}$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.

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