Limit distribution function of inhomogeneities in regions with random boundary. I

1992 ◽  
Vol 92 (1) ◽  
pp. 763-772
Author(s):  
M. Yu. Rasulova
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Random Boundary ◽  
Limit Distribution Function

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (2) ◽  
pp. 387-390 ◽  
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.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (02) ◽  
pp. 387-390 ◽  
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 &lt; m &lt; ∞ 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 /c n} converges in law to a proper limit distribution function F, with F(0 +) &lt; 1.

scholarly journals CONTINUED FRACTIONS WITH ODD PARTIAL QUOTIENTS

2012 ◽  
Vol 08 (06) ◽  
pp. 1541-1556 ◽  
Author(s):  
E. N. ZHABITSKAYA
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Continued Fractions ◽  
Functional Equations ◽  
Continued Fraction ◽  
Euclidean Algorithm ◽  
Partial Quotients ◽  
Limit Distribution Function

Every Euclidean algorithm is associated with a kind of continued fraction representation of a number. The representation associated with "odd" Euclidean algorithm we will call "odd" continued fraction. We consider the limit distribution function F(x) for sequences of rationals with bounded sum of partial quotients for "odd" continued fractions. In this paper we prove certain properties of the function F(x). Particularly this function is singular and satisfies a number of functional equations. We also show that the value F(x) can be expressed in terms of partial quotients of the "odd" continued fraction representation of a number x.

scholarly journals Longest Increasing Subsequences of Random Colored Permutations

10.37236/1445 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Alexei Borodin
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Main Idea ◽  
Random Permutations ◽  
Signed Permutations ◽  
Limit Distribution Function ◽  
Longest Increasing Subsequence ◽  
Colored Permutations

We compute the limit distribution for the (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In the two–colored case our method provides a different proof of a similar result by Tracy and Widom about the longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the 'colored' problem to the case of usual random permutations using certain combinatorial results and elementary probabilistic arguments.

A functional equation technique for obtaining wiener process probabilities associated with theorems of Kolmogorov-Smirnov type

1959 ◽  
Vol 55 (4) ◽  
pp. 328-332 ◽  
Author(s):  
J. Kiefer ◽  
D. V. Lindley
Keyword(s):  
Functional Equation ◽  
Distribution Function ◽  
Wiener Process ◽  
Limit Distribution ◽  
Sample Distribution ◽  
Probabilistic Computation ◽  
Limit Distributions ◽  
Kolmogorov Smirnov

1. Introduction. Since the first proofs by Kolmogorov (13) and Smirnov ((14), (15)) of their well-known results on the limit distribution of the deviations of the sample distribution function, many alternative proofs of these results have been given. For example, we may cite the various approaches of Feller (4), Doob (3), Kac (8), Gnedenko and Korolyuk(7), and Anderson and Darling (1). The approaches of (3), (8) and (1) rest on a probabilistic computation regarding the Wiener process, and are justified by the paper of Donsker (2) (see also (11)). Of all these approaches, only those of (8) and (1) can be extended to obtain the limit distributions of the ‘k–sample’ generalizations of the Kolmogorov-Smirnov statistics suggested in (9), and the author ((9), (10)) and Gihman(6) carried out such proofs.

scholarly journals Modeling the Dirichlet distribution using multiplicative functions

2020 ◽  
Vol 25 (2) ◽  
Author(s):  
Gintautas Bareikis ◽  
Algirdas Mačiulis
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Multiplicative Function ◽  
Remainder Term ◽  
Dirichlet Distribution ◽  
Distribution Functions ◽  
Multiplicative Functions ◽  
Cesàro Mean ◽  
Cesaro Mean ◽  
The One

For q,m,n,d ∈ N and some multiplicative function f > 0, we denote by T3(n) the sum of f(d) over the ordered triples (q,m,d) with qmd = n. We prove that Cesaro mean of distribution functions defined by means of T3 uniformly converges to the one-parameter Dirichlet distribution function. The parameter of the limit distribution depends on the values of f on primes. The remainder term is estimated as well. 

Limit distributions of the number of renewals and waiting times

1976 ◽  
Vol 13 (2) ◽  
pp. 301-312
Author(s):  
N. R. Mohan
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Random Variables ◽  
Stable Laws ◽  
Domains Of Attraction ◽  
Limit Distributions

Let {Xn} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi, i = 1, 2, …, be either F1 or F2 where F1 and F2 are distinct. Set Sn = X1 + X2 + … + Xn and for t > 0 define and Zt = SN(t)+1 – t. The limit distributions of N(t), Yt and Zt as t → ∞ are obtained when F1 and F2 are in the domains of attraction of stable laws with exponents α1 and α2, respectively and Sn properly normalised has the composition of these two stable laws as its limit distribution.

Limit distributions of the number of renewals and waiting times

1976 ◽  
Vol 13 (02) ◽  
pp. 301-312
Author(s):  
N. R. Mohan
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Random Variables ◽  
Stable Laws ◽  
Domains Of Attraction ◽  
Limit Distributions

Let {X n} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi , i = 1, 2, …, be either F 1 or F 2 where F 1 and F 2 are distinct. Set Sn = X 1 + X 2 + … + Xn and for t &gt; 0 define and Zt = SN (t)+1 – t. The limit distributions of N(t), Yt and Zt as t → ∞ are obtained when F 1 and F 2 are in the domains of attraction of stable laws with exponents α 1 and α 2 , respectively and Sn properly normalised has the composition of these two stable laws as its limit distribution.

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