Maxima of partial sums of random variables and suprema of stable processes

1973 ◽  
Vol 5 (1) ◽  
pp. 5-6 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Stable Processes ◽  
Sums Of Random Variables

On the maximum of sums of random variables and the supremum functional for stable processes

1969 ◽  
Vol 6 (2) ◽  
pp. 419-429 ◽  
Author(s):  
C.C. Heyde
Keyword(s):  
Limit Theorem ◽  
Stable Distribution ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Law ◽  
Sums Of Random Variables ◽  
Symmetric Stable Distribution ◽  
General Limit ◽  
General Limit Theorem

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.

Absorption Probabilities for Sums of Random Variables Defined on a Finite Markov Chain

1962 ◽  
Vol 58 (2) ◽  
pp. 286-298 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Markov Chain ◽  
Asymptotic Expression ◽  
Random Variables ◽  
Partial Sums ◽  
Finite Markov Chain ◽  
Initial State ◽  
Main Result Theorem ◽  
Sums Of Random Variables ◽  
Result Theorem ◽  
Image Position

SummaryThis paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.

Precise large deviations for sums of random variables with consistently varying tails

2004 ◽  
Vol 41 (01) ◽  
pp. 93-107 ◽  
Author(s):  
Kai W. Ng ◽  
Qihe Tang ◽  
Jia-An Yan ◽  
Hailiang Yang
Keyword(s):  
Large Deviations ◽  
Risk Model ◽  
Random Variables ◽  
Tail Probability ◽  
Arrival Process ◽  
Partial Sums ◽  
Precise Large Deviations ◽  
Sums Of Random Variables ◽  
Doubly Stochastic ◽  
Common Distribution Function

Let {X k , k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ &gt; 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S n and the random sums S N(t), where N(·) is a counting process independent of the sequence {X k , k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.

Limit theorems in fluctuation theory

1973 ◽  
Vol 5 (03) ◽  
pp. 554-569 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Limit Theorems ◽  
Recurrent Events ◽  
Random Variables ◽  
Fluctuation Theory ◽  
Partial Sums ◽  
Sums Of Random Variables ◽  
Independent Increments

We shall be concerned here with limit theorems arising in the fluctuation theory of random walks, processes with stationary independent increments, recurrent events and regenerative phenomena. In Section 1 on discrete time, we consider limit theorems for ladder-points (Theorem 1) and for maxima of partial sums of random variables (Theorem 2), and discuss some related questions. In Section 2 (Theorems 3 to 6) we consider the analogues of these results in continuous time.

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