scholarly journals Limit Theorems for a Generalized ST Petersburg Game

2010 ◽  
Vol 47 (3) ◽  
pp. 752-760 ◽  
Author(s):  
Allan Gut
Keyword(s):  
Limit Theorems ◽  
Convergence In Distribution ◽  
Stable Law ◽  
Geometric Size

The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr(k-1)/α) = pqk-1, k = 1, 2,…, where p + q = 1, s = 1 / p, r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2n-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.

scholarly journals Limit Theorems for a Generalized ST Petersburg Game

2010 ◽  
Vol 47 (03) ◽  
pp. 752-760 ◽  
Author(s):  
Allan Gut
Keyword(s):  
Limit Theorems ◽  
Convergence In Distribution ◽  
Stable Law ◽  
Geometric Size

The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr (k-1)/α) = pq k-1, k = 1, 2,…, where p + q = 1, s = 1 / p, r = 1 / q, and 0 &lt; α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2 n -subsequence. The analog for 0 &lt; α &lt; 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.

scholarly journals Limit Theorems for a Generalized Feller Game

2013 ◽  
Vol 50 (1) ◽  
pp. 54-63 ◽  
Author(s):  
Keisuke Matsumoto ◽  
Toshio Nakata
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Convergence In Distribution ◽  
Stable Law ◽  
One Dimensional ◽  
Limit Distributions ◽  
Polynomial Size ◽  
Large Numbers ◽  
Geometric Size

In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game. Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0<α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.

Limit Theorems for a Generalized Feller Game

2013 ◽  
Vol 50 (01) ◽  
pp. 54-63 ◽  
Author(s):  
Keisuke Matsumoto ◽  
Toshio Nakata
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Convergence In Distribution ◽  
Stable Law ◽  
One Dimensional ◽  
Limit Distributions ◽  
Polynomial Size ◽  
Large Numbers ◽  
Geometric Size

In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game. Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0&lt;α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Stable Process ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Limit ◽  
Stable Law ◽  
Partial Sum Process ◽  
Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

scholarly journals Limit theorems for additive functionals of continuous time random walks

2020 ◽  
pp. 1-22
Author(s):  
Yuri Kondratiev ◽  
Yuliya Mishura ◽  
Georgiy Shevchenko
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Additive Functional ◽  
Stable Law ◽  
Additive Functionals ◽  
Lévy Motion ◽  
Continuous Time Random Walks ◽  
Poisson Shot Noise

Abstract For a continuous-time random walk X = {X t , t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

scholarly journals Critical markov branching process limit theorems allowing infinite variance

2010 ◽  
Vol 42 (02) ◽  
pp. 460-488 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Extreme Value Theory ◽  
Limit Theorems ◽  
Branching Process ◽  
Value Theory ◽  
Infinite Variance ◽  
Limit Laws ◽  
Stable Law ◽  
Markov Branching Process ◽  
Gumbel Law ◽  
Conditional Limit

This paper gives easy proofs of conditional limit laws for the population size Z t of a critical Markov branching process whose offspring law is attracted to a stable law with index 1 + α, where 0 ≤ α ≤ 1. Conditioning events subsume the usual ones, and more general initial laws are considered. The case α = 0 is related to extreme value theory for the Gumbel law.

LIMIT THEOREMS FOR SAMPLED DYNAMICAL SYSTEMS

2003 ◽  
Vol 03 (04) ◽  
pp. 477-497 ◽  
Author(s):  
NADINE GUILLOTIN-PLANTARD ◽  
DOMINIQUE SCHNEIDER
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Dynamical Systems ◽  
Limit Theorems ◽  
Probability Space ◽  
Convergence In Distribution

Let [Formula: see text] be a dynamical system where [Formula: see text] is a probability space and T an invertible transformation preserving the measure μ. Let (Sk)k≥0 be a transient ℤ-random walk. Let f ∈ L2(μ) and H ∈ ]0,1[, we study the convergence in distribution of the sequence [Formula: see text] We also study the case when the random walk (Sk)k≥0 is replaced by an increasing deterministic subsequence of integers.

A note on cumulative shock models

1988 ◽  
Vol 25 (01) ◽  
pp. 220-223 ◽  
Author(s):  
Kevin K. Anderson
Keyword(s):  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Shock Model ◽  
Time Intervals ◽  
Stable Law ◽  
Shock Models ◽  
Weak Limit Theorems ◽  
First Time ◽  
Intershock Interval

A shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered. Weak limit theorems are established for the cumulative magnitude of the shocks and the first time the cumulative magnitude exceeds z without any assumption on the dependence between the intershock interval and shock magnitude.

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