General limit theorem for n phase barker sequences

1996 ◽  
Vol 32 (15) ◽  
pp. 1364 ◽  
Author(s):  
W.H. Mow
Keyword(s):  
Limit Theorem ◽  
General Limit ◽  
General Limit Theorem

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.

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

1969 ◽  
Vol 6 (02) ◽  
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 S 0= 0, S n = Σ i=1 n Xi , n ≧ 1, and Mn = max0 ≦ k ≦ n Sk . In the case where the Xi are such that Σ1 ∞ n −1Pr(Sn &gt; 0) &lt; ∞, we have lim n→∞M n = M which is finite with probability one, while in the case where Σ1 ∞ n −1Pr(Sn &lt; 0) &lt; ∞, 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 &lt; 0) &lt; ∞, Σ1 ∞ n −1Pr(Sn &gt; 0) &lt; ∞ (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.

scholarly journals NON-RELATIVISTIC LIMIT OF A DIRAC–MAXWELL OPERATOR IN RELATIVISTIC QUANTUM ELECTRODYNAMICS

2003 ◽  
Vol 15 (03) ◽  
pp. 245-270 ◽  
Author(s):  
ASAO ARAI
Keyword(s):  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Scaling Limit ◽  
Relativistic Limit ◽  
Maxwell Operator ◽  
Adjoint Extension ◽  
General Limit ◽  
Adjoint Operators ◽  
Abstract Framework ◽  
General Limit Theorem

The non-relativistic (scaling) limit of a particle-field Hamiltonian H, called a Dirac–Maxwell operator, in relativistic quantum electrodynamics is considered. It is proven that the non-relativistic limit of H yields a self-adjoint extension of the Pauli–Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics. This is done by establishing in an abstract framework a general limit theorem on a family of self-adjoint operators partially formed out of strongly anticommuting self-adjoint operators and then by applying it to H.

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