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

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.

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

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

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.