Moment convergence in conditional limit theorems

2001 ◽  
Vol 38 (02) ◽  
pp. 421-437 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Normal Distribution ◽  
Random Forests ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Variables ◽  
Triangular Array ◽  
Moment Convergence ◽  
Conditional Limit Theorems ◽  
Occupancy Problems ◽  
Conditional Limit

Consider a sum ∑1 N Y i of random variables conditioned on a given value of the sum ∑1 N X i of some other variables, where X i and Y i are dependent but the pairs (X i ,Y i ) form an i.i.d. sequence. We consider here the case when each X i is discrete. We prove, for a triangular array ((X ni ,Y ni )) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

Moment convergence in conditional limit theorems

2001 ◽  
Vol 38 (2) ◽  
pp. 421-437 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Normal Distribution ◽  
Random Forests ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Variables ◽  
Triangular Array ◽  
Moment Convergence ◽  
Conditional Limit Theorems ◽  
Occupancy Problems ◽  
Conditional Limit

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

scholarly journals Conditional limit theorems for branching processes

1991 ◽  
Vol 4 (4) ◽  
pp. 263-292 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Conditional Limit Theorems ◽  
Total Progeny ◽  
Number Of Individuals ◽  
Conditional Limit

Let [ξ(m),m=0,1,2,…] be a branching process in which each individual reproduces independently of the others and has probability pj(j=0,1,2,…) of giving rise to j descendants in the following generation. The random variable ξ(m) is the number of individuals in the mth generation. It is assumed that P{ξ(0)=1}=1. Denote by ρ the total progeny, μ, the time of extinction, and τ, the total number of ancestors of all the individuals in the process. This paper deals with the distributions of the random variables ξ(m), μ and τ under the condition that ρ=n and determines the asymptotic behavior of these distributions in the case where n→∞ and m→∞ in such a way that m/n tends to a finite positive limit.

The stable pedigrees of critical branching populations

1984 ◽  
Vol 21 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Olle Nerman
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Branching Laws ◽  
Conditional Limit Theorems ◽  
First Results ◽  
Conditional Limit

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

The stable pedigrees of critical branching populations

1984 ◽  
Vol 21 (03) ◽  
pp. 447-463 ◽  
Author(s):  
Olle Nerman
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Branching Laws ◽  
Conditional Limit Theorems ◽  
First Results ◽  
Conditional Limit

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.

Conditional limit theorems for general branching processes

1977 ◽  
Vol 14 (3) ◽  
pp. 451-463 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Conditional Limit Theorems ◽  
Special Case ◽  
Conditional Limit

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

scholarly journals Conditional Limit Theorems for the Terms of a Random Walk Revisited

2013 ◽  
Vol 50 (3) ◽  
pp. 871-882
Author(s):  
Shaul K. Bar-Lev ◽  
Ernst Schulte-Geers ◽  
Wolfgang Stadje
Keyword(s):  
Random Walk ◽  
Total Variation ◽  
Gamma Distribution ◽  
Conditional Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Stable Distributions ◽  
Conditional Limit Theorems ◽  
Conditional Limit ◽  
Independent And Identically Distributed

In this paper we derive limit theorems for the conditional distribution ofX1givenSn=snasn→ ∞, where theXiare independent and identically distributed (i.i.d.) random variables,Sn=X1+··· +Xn, andsn/nconverges orsn≡sis constant. We obtain convergence in total variation of PX1∣Sn/n=sto a distribution associated to that ofX1and of PnX1∣Sn=sto a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

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