Some limit theorems for a subcritical branching process by immigration

1984 ◽  
Vol 21 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Y. S. Chow ◽  
K. F. Yu
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Total Population ◽  
Strong Law ◽  
Moment Conditions ◽  
Moment Convergence ◽  
Large Numbers ◽  
The Moment ◽  
Branching Process With Immigration

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

Some limit theorems for a subcritical branching process by immigration

1984 ◽  
Vol 21 (01) ◽  
pp. 50-57
Author(s):  
Y. S. Chow ◽  
K. F. Yu
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Total Population ◽  
Strong Law ◽  
Moment Conditions ◽  
Moment Convergence ◽  
Large Numbers ◽  
The Moment ◽  
Branching Process With Immigration

The strong law of large numbers of the Marcinkiewicz–Zygmund type is established for the total population in a subcritical branching process with immigration. The moment convergence of the total population is obtained under appropriate moment conditions on the offspring distribution and the immigration distribution.

COMPLETE CONVERGENCE FOR ARRAYS AND THE LAW OF THE SINGLE LOGARITHM

2017 ◽  
Vol 96 (2) ◽  
pp. 333-344
Author(s):  
ALLAN GUT ◽  
ULRICH STADTMÜLLER
Keyword(s):  
Limit Theorems ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Almost Sure Convergence ◽  
Strong Law ◽  
Moment Conditions ◽  
Large Numbers ◽  
Distributed Arrays ◽  
The Law ◽  
Independent And Identically Distributed

The present paper is devoted to complete convergence and the strong law of large numbers under moment conditions near those of the law of the single logarithm (LSL) for independent and identically distributed arrays. More precisely, we investigate limit theorems under moment conditions which are stronger than $2p$ for any $p<2$, in which case we know that there is almost sure convergence to 0, and weaker than $E\,X^{4}/(\log ^{+}|X|)^{2}<\infty$, in which case the LSL holds.

One-sided strong laws forincrements of sumsof i.i.d. random variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 333-359 ◽  
Author(s):  
A. N. Frolov
Keyword(s):  
Limit Theorems ◽  
Strong Approximation ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Moment Conditions ◽  
Second Moments ◽  
Large Numbers ◽  
Norming Sequence ◽  
Normal Law

We find a universal norming sequence in strong limit theorems for increments of sums of i.i.d. random variables with finite first moments and finite second moments of positive parts. Under various one-sided moment conditions our universal theorems imply the following results for sums and their increments: the strong law of large numbers, the law of the iterated logarithm, the Erdős-Rényi law of large numbers, the Shepp law, one-sided versions of the Csörgő-Révész strong approximation laws. We derive new results for random variables from domains of attraction of a normal law and asymmetric stable laws with index αЄ(1,2).

Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanqing Wang ◽  
Quansheng Liu
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Moderate Deviation ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Branching Process With Immigration

Abstract This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 p\geq 1 , and the boundedness of the harmonic moments E ⁢ W n - a \mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log ⁡ Z n \log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.

scholarly journals A NONCONVENTIONAL STRONG LAW OF LARGE NUMBERS AND FRACTAL DIMENSIONS OF SOME MULTIPLE RECURRENCE SETS

2012 ◽  
Vol 12 (03) ◽  
pp. 1150023 ◽  
Author(s):  
YURI KIFER
Keyword(s):  
Law Of Large Numbers ◽  
Fractal Dimensions ◽  
Growth Conditions ◽  
Multiple Recurrence ◽  
Strong Law ◽  
Moment Conditions ◽  
Vector Process ◽  
Regularity Properties ◽  
Large Numbers ◽  
Positive Functions

We provide conditions which yield a strong law of large numbers for expressions of the form [Formula: see text] where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and qi, i > m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q1(n), …, qℓ(n).

SOME LIMIT THEOREMS OF DELAYED AVERAGES FOR COUNTABLE NONHOMOGENEOUS MARKOV CHAINS

2019 ◽  
pp. 1-19
Author(s):  
Pingping Zhong ◽  
Weiguo Yang ◽  
Zhiyan Shi ◽  
Yan Zhang
Keyword(s):  
Information Theory ◽  
Markov Chains ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Strong Ergodicity ◽  
Large Numbers ◽  
Nonhomogeneous Markov Chains ◽  
Definition Of ◽  
Bivariate Functions

AbstractThe purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (3) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (03) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

scholarly journals Complete Moment Convergence for Sung’s Type Weighted Sums ofB-Valued Random Elements

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Wei Li ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Real Numbers ◽  
Strong Law ◽  
Moment Convergence ◽  
Large Numbers ◽  
Random Elements ◽  
Necessary And Sufficient

Letp≥1/αand1/2<α≤1.Let{X,Xn,  n≥1}be a sequence of independent and identically distributedB-valued random elements and let{ani,  1≤i≤n,  n≥1}be an array of real numbers satisfying∑i=1naniq=O(n)for someq>p.We give necessary and sufficient conditions for complete moment convergence of the form∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞,  ∀ε>0, where0<v<p.A strong law of large numbers for weighted sums of independentB-valued random elements is also obtained.

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