Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (4) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (04) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

scholarly journals Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials

2016 ◽  
Vol 53 (3) ◽  
pp. 747-764 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Zhenxia Liu ◽  
Xiangfeng Yang
Keyword(s):  
Large Deviations ◽  
Laplace Transform ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Precise Asymptotics ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Large Deviation Principles ◽  
Logarithmic Asymptotics ◽  
The Moment

AbstractThe longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log1∕pn and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

On the connection between heat waves and large deviations of temperature

2020 ◽  
Author(s):  
Jeroen Wouters ◽  
Vera Melinda Galfi ◽  
Valerio Lucarini
Keyword(s):  
Large Deviations ◽  
Extreme Events ◽  
General Circulation ◽  
Large Deviation ◽  
Heat Waves ◽  
Spatial Scales ◽  
Near Surface ◽  
Large Deviation Theory ◽  
Rate Functions ◽  
Deviation Theory

<p>We use large deviation theory to study persistent extreme events of temperature, like heat waves or cold spells. We consider the mid-latitudes of a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature averages over different spatial scales. We find that, in order to represent persistent extreme events based on large deviation theory, one has to look at temporal averages of spatially averaged observables. The spatial averaging scale is crucial, and has to correspond with the scale of the event of interest. Accordingly, the computed rate functions indicate substantially different statistical properties of temperature averages over intermediate spatial scales (larger, but still of the order of the typical scale), as compared to the ones related to any other scale. Thus, heat waves (or cold spells) can be interpreted as large deviations of temperature averaged over intermediate spatial scales. Furthermore, we find universal characteristics of rate functions, based on the equivalence of temporal, spatial, and spatio-temporal rate functions if we perform a re-normalisation by the integrated auto-correlation.</p>

scholarly journals On the longest runs in Markov chains

2018 ◽  
Vol 38 (2) ◽  
pp. 407-428 ◽  
Author(s):  
Zhenxia Liu ◽  
Xiangfeng Yang
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Generating Function ◽  
Cumulative Distribution Function ◽  
Moment Generating Function ◽  
Cumulative Distribution ◽  
Global Estimate ◽  
Success Run ◽  
Longest Success Run ◽  
The Moment

In the first n steps of a two-state success and failure Markov chain, the longest success run Ln has been attracting considerable attention due to its various applications. In this paper, we study Ln in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of Ln proposed in this paper, and the proofs of the large deviations include the Gärtner–Ellis theorem and the moment generating function.

LARGE DEVIATIONS OF MULTIFRACTAL MEASURES

Fractals ◽  
2002 ◽  
Vol 10 (01) ◽  
pp. 117-129 ◽  
Author(s):  
DANIELE VENEZIANO
Keyword(s):  
Probability Distribution ◽  
Large Deviations ◽  
Large Deviation ◽  
Single Point ◽  
Dimension Function ◽  
Large Deviation Theory ◽  
Exponential Term ◽  
Scaling Relationships ◽  
Deviation Theory ◽  
Multifractal Measures

We analyze the extremes of stationary multifractal measures using large deviation theory. We consider various cases involving discrete multiplicative cascades: scalar or vector cascades with dependent or independent generators, bare or dressed measures, and marginal (single-point) or joint (multi-point) extremes. In each case, we obtain the scaling behavior of the probability of large deviations as the resolution of the cascade diverges. Existing rough exponential limits for scalar cascades are confirmed, whereas for other cases our scaling relationships differ from previously published results. For scalar cascades, we refine the rough limits by obtaining the asymptotic pre-factor to the exponential term. Based on these refined asymptotics, we propose a variant to the Probability Distribution/Multiple Scaling (PDMS) technique to estimate the co-dimension function c(γ).

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