A note on the subcritical generalized age-dependent branching process

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

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A note on the age-dependent branching process with immigration

Journal of Applied Probability ◽

10.1017/s0021900200033167 ◽

1975 ◽

Vol 12 (01) ◽

pp. 130-134 ◽

Cited By ~ 1

Author(s):

Norman Kaplan

Keyword(s):

General Class ◽

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Direct Generalization ◽

Age Dependent ◽

General Class Of Functions ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Class Of Functions

Let {Z(t)}t0be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for whenE{Φ (Z(t))} <∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

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A note on the age-dependent branching process with immigration

Journal of Applied Probability ◽

10.2307/3212414 ◽

1975 ◽

Vol 12 (1) ◽

pp. 130-134 ◽

Cited By ~ 2

Author(s):

Norman Kaplan

Keyword(s):

General Class ◽

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Direct Generalization ◽

Age Dependent ◽

General Class Of Functions ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Class Of Functions

Let {Z(t)}t0 be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for when E{Φ (Z(t))} <∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

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On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.2307/3213010 ◽

1977 ◽

Vol 14 (2) ◽

pp. 387-390 ◽

Cited By ~ 3

Author(s):

Harry Cohn

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Limit Distribution Function ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

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THE COMPATIBLE BOND-STOCK MARKET WITH JUMPS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024911006449 ◽

2011 ◽

Vol 14 (05) ◽

pp. 723-755 ◽

Cited By ~ 1

Author(s):

DEWEN XIONG ◽

MICHAEL KOHLMANN

Keyword(s):

Stock Market ◽

Marked Point ◽

Quadratic Functions ◽

Marked Point Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

No Arbitrage ◽

The Mean ◽

Necessary And Sufficient ◽

Mean Variance

We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market. We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks [Formula: see text], we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market [Formula: see text]. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim [Formula: see text] is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks [Formula: see text] is computed.

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A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means

Stochastic Processes and their Applications ◽

10.1016/0304-4149(88)90070-1 ◽

1988 ◽

Vol 28 (1) ◽

pp. 123-139 ◽

Cited By ~ 19

Author(s):

David Tanny

Keyword(s):

Random Environment ◽

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient

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The weak law of large numbers for nonnegative summands

Advances in Applied Probability ◽

10.1017/apr.2018.83 ◽

2018 ◽

Vol 50 (A) ◽

pp. 241-252

Author(s):

Eugene Seneta

Keyword(s):

Branching Process ◽

Law Of Large Numbers ◽

Random Variables ◽

Sufficient Condition ◽

Specific Function ◽

Slowly Varying Function ◽

Necessary And Sufficient Condition ◽

Large Numbers ◽

Norming Sequence ◽

Necessary And Sufficient

Abstract Khintchine's (necessary and sufficient) slowly varying function condition for the weak law of large numbers (WLLN) for the sum of n nonnegative, independent and identically distributed random variables is used as an overarching (sufficient) condition for the case that the number of summands is more generally [cn],cn→∞. Either the norming sequence {an},an→∞, or the number of summands sequence {cn}, can be chosen arbitrarily. This theorem generalizes results from a motivating branching process setting in which Khintchine's sufficient condition is automatically satisfied. A second theorem shows that Khintchine's condition is necessary for the generalized WLLN when it holds with cn→∞ and an→∞. Theorem 3, which is known, gives a necessary and sufficient condition for Khintchine's WLLN to hold with cn=n and an a specific function of n; it is extended to general cn subject to a growth restriction in Theorem 4. Section 6 returns to the branching process setting.

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On the identification of continuous-time Markov chains with a given invariant measure

Journal of Applied Probability ◽

10.1017/s0021900200099435 ◽

1994 ◽

Vol 31 (04) ◽

pp. 897-910

Author(s):

P. K. Pollett

Keyword(s):

Invariant Measure ◽

Continuous Time ◽

Branching Process ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Birth Process ◽

Continuous Time Markov Chains ◽

Branching Process With Immigration ◽

Birth Processes ◽

Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

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Fixation in Conditional Branching Process Models in Population Genetics

Journal of Applied Probability ◽

10.1239/jap/1197908828 ◽

2007 ◽

Vol 44 (4) ◽

pp. 1103-1110 ◽

Cited By ~ 1

Author(s):

Thomas Prince ◽

Neville Weber

Keyword(s):

Population Genetics ◽

Process Model ◽

Branching Process ◽

Process Models ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Alternative Version ◽

Necessary And Sufficient ◽

Insight Into

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.

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Some generalized variability orderings among life distributions with reliability applications

Journal of Applied Probability ◽

10.1017/s0021900200039759 ◽

1991 ◽

Vol 28 (02) ◽

pp. 374-383 ◽

Cited By ~ 4

Author(s):

M. C. Bhattacharjee

Keyword(s):

Random Variables ◽

Parallel Systems ◽

Reliability Theory ◽

Dominance Relation ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Closure Properties ◽

Life Distributions ◽

The Mean ◽

Necessary And Sufficient

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

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