Covariance operator estimation of a functional autoregressive process with random coefficients

We present recursions for the total number,Sn, of mutations in a sample ofnindividuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rater>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/xis finite, we prove thatSn/(nr) converges in distribution to a limiting variable,S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2is finite, the distribution ofScoincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the formwith specific independent random coefficientsAandB. Examples are presented in which explicit representations for (the density of)Sare available. We conjecture thatSn/E(Sn)→1 in probability if the measure Λ(dx)/xis infinite.

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Testing the stability of the functional autoregressive process

Journal of Multivariate Analysis ◽

10.1016/j.jmva.2008.12.008 ◽

2010 ◽

Vol 101 (2) ◽

pp. 352-367 ◽

Cited By ~ 32

Author(s):

Lajos Horváth ◽

Marie Hušková ◽

Piotr Kokoszka

Keyword(s):

Autoregressive Process ◽

The Stability ◽

Functional Autoregressive Process

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On the number of segregating sites for populations with large family sizes

Advances in Applied Probability ◽

10.1239/aap/1158685000 ◽

2006 ◽

Vol 38 (3) ◽

pp. 750-767 ◽

Cited By ~ 14

Author(s):

M Möhle

Keyword(s):

Large Family ◽

Autoregressive Process ◽

Random Coefficients ◽

Exponential Integral ◽

Genealogical Tree ◽

Fixed Point Equation ◽

Point Equation ◽

Multiple Collisions ◽

Explicit Representations ◽

Large Families

We present recursions for the total number, Sn, of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate r>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/x is finite, we prove that Sn/(nr) converges in distribution to a limiting variable, S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2 is finite, the distribution of S coincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the form with specific independent random coefficients A and B. Examples are presented in which explicit representations for (the density of) S are available. We conjecture that Sn/E(Sn)→1 in probability if the measure Λ(dx)/x is infinite.

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