Stability of perpetuities

We consider multivariate stationary processes (X t ) satisfying a stochastic recurrence equation of the formwhere (Q t ) are iid random vectors andare iid diagonal matrices and (M t ) are iid random variables. We obtain a full characterization of the Vector Scaling Regular Variation properties of (X t ), proving that some coordinates X t,i and X t,j are asymptotically independent even though all coordinates rely on the same random input (M t ). We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK-ARCH and CCC-GARCH) processes and to log-returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.

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“…However, divergence of S n implies divergence of X 1,n , see e.g. [16,Theorem 2.1]. Hence we have the identity…”

Section: Proof Of the Main Resultsmentioning

confidence: 92%

Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model

Mentemeier

Wintenberger

2022

Journal Time Series Analysis

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“…A sufficient condition for the almost sure (a.s.) absolute convergence of the random series k≥0 b S k η k+1 with fixed b ∈ (0, 1) is Eξ ∈ (0, ∞) and E log + |η| < ∞, see, for instance, Theorem 2.1 in [13]. This sufficient condition holds, that is, the discounted perpetuity is well-defined for all b ∈ (0, 1), under the assumptions of all our results to be formulated soon.…”

Section: Introductionmentioning

confidence: 75%

“…Although a random power series is a toy example of perpetuities, transferring results from the former to the latter may be a challenge. To justify this claim, we only mention that while necessary and sufficient conditions for the a.s. convergence of random power series can be easily obtained (just use the Cauchy root test in combination with the Borel-Cantelli lemma), the corresponding result for perpetuities is highly non-trivial, EJP 26 (2021), paper 131. see Theorem 2.1 in [13] and its proof. The reason is clear: the random power series is a weighted sum of independent random variables, whereas it is not the case for perpetuities.…”

Section: Related Literaturementioning

confidence: 99%

“…Assuming, for the time being, that ξ and η are a.s. positive, we can interpret η k and e −ξ k as the planned payment and the discount factor (risk) for year k, respectively. Then k≥0 e −S k η k+1 can be thought of as 'the present value of a permanent commitment to make a payment ... annually into the future forever' (the phrase borrowed from p. 1196 in [13]). When studying the aforementioned random series from a purely mathematical viewpoint, the one-sided assumptions are Discounted convergent perpetuities normally omitted whereas the term 'perpetuity' is still used.…”

Section: Introductionmentioning

confidence: 99%

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Limit theorems for discounted convergent perpetuities

Iksanov

Nikitin

Самойленко

2021

Electron. J. Probab.

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Let (ξ1, η1), (ξ2, η2), . . . be independent identically distributed R 2 -valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for the convergent perpetuities k≥0 b ξ 1 +...+ξ k η k+1 as b → 1−. Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.

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“…The moment assumptions in Theorem 2.1 cannot be essentially weakened; see Goldie and Maller (1997). Of course, the Markov chain (2.4) can be defined when A n is expanding rather than contracting, but different normings are required for convergence.…”

Section: Motivating Examplementioning

confidence: 99%

Iterated random functions

Diaconis¹,

Freedman²

2017

Stochastic Processes

104

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A bstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. Introduction.The applied probability literature is nowadays quite daunting. Even relatively simple topics, like Markov chains, have generated enormous complexity. This paper describes a simple idea that helps to unify many arguments in Markov chains, simulation algorithms, control theory, queuing, and other branches of applied probability. The idea is that Markov chains can be constructed by iterating random functions on the state space S. More specifically, there is a family {f θ : θ ∈ Θ} of functions that map S into itself, and a probability distribution µ on Θ. If the chain is at x ∈ S, it moves by choosing θ at random from µ, and going to f θ (x). For now, µ does not depend on x.The process can be written aswhere θ 1 , θ 2 , . . . are independent draws from µ. The Markov property is clear: given the present position of the chain, the conditional distribution of the future does not depend on the past.

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Stability of perpetuities

Cited by 132 publications

References 19 publications

Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model

Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model

Limit theorems for discounted convergent perpetuities

Iterated random functions

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