Renewal-type behavior of absorption times in Markov chains

Cutsem, Bernard Van; Ycart, Bernard

doi:10.2307/1427901

Cited by 11 publications

(16 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We are aware of only two papers, [36] and [39], which address the asymptotic behavior of M n as n tends to ∞ in the general setting when it is not assumed that π ij takes some particular form. The problem is simpler if either the probabilities π ij are given explicitly, or if they have some particular functional form.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the number of jumps of random walks with a barrier

Iksanov

Möhle

2008

Adv. Appl. Probab.

View full text Add to dashboard Cite

Let S0 := 0 and Sk := ξ1 + ··· + ξk for k ∈ ℕ := {1, 2, …}, where {ξk : k ∈ ℕ} are independent copies of a random variable ξ with values in ℕ and distribution pk := P{ξ = k}, k ∈ ℕ. We interpret the random walk {Sk : k = 0, 1, 2, …} as a particle jumping to the right through integer positions. Fix n ∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {Rk(n) : k = 0, 1, 2, …} which never reaches the state n. We call this process a random walk with barrier n. Let Mn denote the number of jumps of the random walk with barrier n. This paper focuses on the asymptotics of Mn as n tends to ∞. A key observation is that, under p1 > 0, {Mn : n ∈ ℕ} satisfies the distributional recursion M1 = 0 and for n = 2, 3, …, where In is independent of M2, …, Mn−1 with distribution P{In = k} = pk / (p1 + ··· + pn−1), k ∈ {1, …, n − 1}. Depending on the tail behavior of the distribution of ξ, several scalings for Mn and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps, Mn, with the first time, Nn, when the unrestricted random walk {Sk : k = 0, 1, …} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the number of jumps of random walks with a barrier

Iksanov

Möhle

2008

Adv. Appl. Probab.

View full text Add to dashboard Cite

show abstract

“…, {X (K) n , n ∈ N} are independent. Random recurrences (1), often in a simplified form with K = 1, arise in diverse areas of applied probability such as random regenerative structures [9,11], random trees [5,7,25,26], coalescent theory [6,10,12,16], absorption times in non-increasing Markov chains [13,2], recursive algorithms [15,24,27,28], random walks with barrier [17,18], to name but a few.…”

Section: Introduction and Main Resultmentioning

confidence: 99%

“…where {b n , n ∈ N} and {c nk , n ∈ N, k < n} are given numeric sequences. The purpose of the present paper is to propose a new method of obtaining the first-order asymptotics of solutions to (2), as n → ∞.…”

Section: Introduction and Main Resultmentioning

confidence: 99%

On the asymptotics of moments of linear random recurrences

Marynych¹

2009

Preprint

View full text Add to dashboard Cite

We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the Poisson-Dirichlet coalescent behave like the powers of the "log star" function which grows slower than any iteration of the logarithm, and thereby prove a weak law of large numbers. Finally, we discuss merits and limitations of the method and give several examples related to beta coalescents, recursive algorithms and random trees.

show abstract

“…In such cases it is natural to expect that, given M 0 = n for large n, the trajectory of (M k ) k≥0 stays close to the trajectory of (n − S k ) k≥0 for a suitable zero-delayed random walk (S k ) k≥0 with positive increments, implying that Q n is close to the law of the first passage time inf{k ≥ 0 : n − S k < a} and therefore to some stable law after normalization. Using such a "renewal approximation", the simplest case of a random walk with increments having finite variance, was treated in [27], where some sufficient conditions were derived for the convergence of T n , properly centered and normalized, to the standard normal law. Finally, we mention a short note [25], where a representation of T n as a sum of independent indicators was provided under the assumption that the increment distribution can be decomposed as the product of a function of the current state and a function of the jump size.…”

Section: Bibliographic Notesmentioning

confidence: 99%

“…where Λ * t ≤ Λ t for all t ∈ R by (32) has been utilized for the penultimate equality, and ( 26), (27) plus subsequent remark for the final estimate.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Renewal approximation for the absorption time of a decreasing Markov chain

Alsmeyer

Marynych

2016

J. Appl. Probab.

View full text Add to dashboard Cite

We consider a Markov chain (M n ) n≥0 on the set N 0 of nonnegative integers which is eventually decreasing, i.e. P{M n+1 < M n |M n ≥ a} = 1 for some a ∈ N and all n ≥ 0. We are interested in the asymptotic behaviour of the law of the stopping time T = T (a) := inf{k ∈ N 0 : M k < a} under P n := P(·|M 0 = n) as n → ∞. Assuming that the decrements of (M n ) n≥0 given M 0 = n possess a kind of stationarity for large n, we derive sufficient conditions for the convergence in minimal L p -distance of P n ((T − a n )/b n ∈ ·) to some non-degenerate, proper law and give an explicit form of the constants a n and b n .

show abstract

Renewal-type behavior of absorption times in Markov chains

Cited by 11 publications

References 25 publications

On the number of jumps of random walks with a barrier

On the number of jumps of random walks with a barrier

On the asymptotics of moments of linear random recurrences

Renewal approximation for the absorption time of a decreasing Markov chain

Contact Info

Product

Resources

About