Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (1) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

Conditional limit theorems for a left-continuous random walk

1973 ◽  
Vol 10 (01) ◽  
pp. 39-53 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Absorbing State ◽  
Conditional Limit Theorems ◽  
Recurrent Markov Chain ◽  
Continuous Random Walk ◽  
Condition C ◽  
Positive Recurrent ◽  
Positive Integers ◽  
Conditional Limit

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (2) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (02) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift. Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (01) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain,obtained by restarting the original chain at a fixed state after each absorption. The limiting age,A(j), is the weak limit of the timegivenXn=j(n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems forA(J) (J →∞) are given for these examples.

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.

Conditional Limit Theorems for the Terms of a Random Walk Revisited

2013 ◽  
Vol 50 (03) ◽  
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 of X 1 given S n =s n as n→ ∞, where the X i are independent and identically distributed (i.i.d.) random variables, S n =X 1+··· +X n , and s n /n converges or s n ≡ s is constant. We obtain convergence in total variation of P X 1∣ S n /n=s to a distribution associated to that of X 1 and of P nX 1∣ S n =s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

2000 ◽  
Vol 32 (01) ◽  
pp. 159-176 ◽  
Author(s):  
Markus Bachmann
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Number ◽  
Time Lag ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Branching Random Walk ◽  
Minimal Position ◽  
Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

Weakening the independence assumption on polar components: limit theorems for generalized elliptical distributions

2016 ◽  
Vol 53 (1) ◽  
pp. 130-145 ◽  
Author(s):  
Miriam Isabel Seifert
Keyword(s):  
Limit Theorems ◽  
Radial Component ◽  
Dependence Structure ◽  
Independence Assumption ◽  
Geometrical Meaning ◽  
Polar Components ◽  
Angular Component ◽  
Conditional Limit Theorems ◽  
Extreme Behavior ◽  
Conditional Limit

Abstract By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.

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