The limit distribution of a random variable used in the construction of confidence bands

Biometrika ◽  
1973 ◽  
Vol 60 (2) ◽  
pp. 431-432 ◽  
Author(s):  
R. SRINIVASAN ◽  
R. M. WHARTON
Keyword(s):  
Limit Distribution ◽  
Random Variable ◽  
Confidence Bands

On the first zero of an empirical characteristic function

1991 ◽  
Vol 28 (3) ◽  
pp. 593-601 ◽  
Author(s):  
H. U. Bräker ◽  
J. Hüsler
Keyword(s):  
Characteristic Function ◽  
Limit Distribution ◽  
Random Variable ◽  
Empirical Characteristic Function ◽  
The Real ◽  
Exponential Decrease ◽  
Characteristic Process

We deal with the distribution of the first zero Rn of the real part of the empirical characteristic process related to a random variable X. Depending on the behaviour of the theoretical real part of the underlying characteristic function, cases with a slow exponential decrease to zero are considered. We derive the limit distribution of Rn in this case, which clarifies some recent results on Rn in relation to the behaviour of the characteristic function.

scholarly journals Limit distribution of distances in biased random tries

2006 ◽  
Vol 43 (02) ◽  
pp. 377-390 ◽  
Author(s):  
Rafik Aguech ◽  
Nabil Lasmar ◽  
Hosam Mahmoud
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Limit Distribution ◽  
Analytical Techniques ◽  
Random Variable ◽  
Wasserstein Metric ◽  
Contraction Method ◽  
Point Solution ◽  
Mean And Variance

Thetrieis a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (03) ◽  
pp. 640-654
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres K n of some volume V(K n ) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form L n (s) molecules with exactly s atoms K n. The random variable L n(s) has a limit distribution if V(K n ) tends to zero but nV(Kn ) tends to infinity at a certain rate: it is shown that for L n(s) is asymptotically Poisson.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (3) ◽  
pp. 640-654 ◽  
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for Ln(s) is asymptotically Poisson.

scholarly journals Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150324 ◽  
Author(s):  
Dmitry S. Shalymov ◽  
Alexander L. Fradkov
Keyword(s):  
Limit Distribution ◽  
Mass Conservation ◽  
Random Variable ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Stationary Processes ◽  
Entropy Principle ◽  
Gradient Principle ◽  
Probability Density Function Pdf ◽  
Speed Gradient

We propose dynamics equations which describe the behaviour of non-stationary processes that follow the maximum Rényi entropy principle. The equations are derived on the basis of the speed-gradient principle originated in the control theory. The maximum of the Rényi entropy principle is analysed for discrete and continuous cases, and both a discrete random variable and probability density function (PDF) are used. We consider mass conservation and energy conservation constraints and demonstrate the uniqueness of the limit distribution and asymptotic convergence of the PDF for both cases. The coincidence of the limit distribution of the proposed equations with the Rényi distribution is examined.

On the continuity and the positivity of the finite part of the limit distribution of an irregular branching process with infinite mean

1980 ◽  
Vol 17 (3) ◽  
pp. 696-703 ◽  
Author(s):  
H. Cohn ◽  
H.-J. Schuh
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Random Variable ◽  
Finite Part ◽  
Positive Distribution ◽  
Large Numbers

It is shown that the limiting random variable W(si) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si) < ∞}. This implies that for all branching processes (Zn) with infinite mean there exists a function U such that the distribution of V = limnU(Zn)e–n a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si) is derived.

scholarly journals Tools to Estimate the First Passage Time to a Convex Barrier

2005 ◽  
Vol 42 (1) ◽  
pp. 61-81
Author(s):  
Ola Hammarlid
Keyword(s):  
Limit Distribution ◽  
Sequential Analysis ◽  
Large Deviation ◽  
First Passage Time ◽  
Stopping Time ◽  
Random Variable ◽  
Passage Time ◽  
First Passage ◽  
Linear Barrier ◽  
Asymptotic Normal

The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear barrier is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

On the waiting times to repeated hits of cells by particles for the polynomial allocation scheme

2020 ◽  
Vol 30 (6) ◽  
pp. 409-415
Author(s):  
Boris I. Selivanov ◽  
Vladimir P. Chistyakov
Keyword(s):  
Limit Distribution ◽  
Waiting Times ◽  
Random Variable ◽  
Random Polynomial ◽  
Allocation Scheme

AbstractWe consider random polynomial allocations of particles over N cells. Let τk, k ≥ 1, be the minimal number of trials when k particles hit the occupied cells. For the case N → ∞ the limit distribution of the random variable $\tau_k/\sqrt{N}$is found. An example of application of τk is given.

scholarly journals Limit distribution of distances in biased random tries

2006 ◽  
Vol 43 (2) ◽  
pp. 377-390 ◽  
Author(s):  
Rafik Aguech ◽  
Nabil Lasmar ◽  
Hosam Mahmoud
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Limit Distribution ◽  
Analytical Techniques ◽  
Random Variable ◽  
Wasserstein Metric ◽  
Contraction Method ◽  
Point Solution ◽  
Mean And Variance

The trie is a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

scholarly journals Limit Distribution of a Generalized Ornstein -- Uhlenbeck Process

2016 ◽  
Vol 6 (1) ◽  
pp. 24
Author(s):  
Andriy Yurachkivsky
Keyword(s):  
Characteristic Function ◽  
Random Process ◽  
Bounded Variation ◽  
Limit Distribution ◽  
Random Variable ◽  
Uhlenbeck Process ◽  
Independent Increments ◽  
Ornstein Uhlenbeck Process

Let an $\bR^d$-valued random process $\xi$ be the solution of an equation of the kind $\xi(t)=\xi(0)+\int_0^tA(u)\xi(u)\rd\iota(u)+S(t),$ where $\xi(0)$ is a random variable measurable w.\,r.\,t. some $\sigma$-algebra $\cF(0)$,  $S$ is a random process with $\cF(0)$-conditionally independent increments, $\iota$ is a continuous numeral random process of locally bounded variation, and $A$ is a matrix-valued random process such that for any $t>0$ $\int_0^t\|A(s)\|\ |\rd\iota(s)|<\iy.$ Conditions guaranteing existence of the limiting, as $t\to\iy$, distribution of $\xi(t)$ are found. The characteristic function of this distribution is written explicitly.

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