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A renewal equation in a multidimensional space. (English. Russian original) Zbl 1099.60060
Theory Probab. Appl. 49, No. 4, 737-744 (2005); translation from Teor. Veroyatn. Primen. 49, No. 4, 779-785 (2004).
Consider the integral equation \[ f(x)=g(x)+\int K(x-t)f(t)\,dt\tag{*} \] in the unknown function \(f\), where \(x\in \mathbb{R}^n\), \(t\in \mathbb{R}^n\), \(n>1\), \(g\in L_1\) and \(K\) is a probability density on \(\mathbb{R}^n\). The paper studies unicity and existence of the solution \(f\). This problem is solved in the Banach spaces \(L_{\infty 1}\) and \(ML\). Here \(L_{\infty 1}\) is the set of those \(h(x_1,\dots,x_n)\) where \(k(x_1)= \int|h(x_1,\dots,x_n)|dx_2\dots dx_n\) is essentially bounded and \(ML\) is the set of those \(h\) where \(k (x_1)\) is the sum of an \(L_\infty(\mathbb{R})\) and an \(L_1(\mathbb{R})\) function. It is shown by Fourier methods that the homogeneous equation only has the trivial constant solution in \(L_{\infty 1}\) and by extension also in \(ML\). By iteration it is shown, when \(\int\|x\|K(x)dx<\infty\) and \(\int x_1K(x)dx> 0\), that there is a solution of (*) in \(ML\). Rotation of the coordinate system extends this result to general nonzero moment vectors. This solution leads to the construction of the renewal density for \(g=K\). The author applies results from his preceding paper [Stochastic Processes Appl. 85, No. 2, 237–247 (2000; Zbl 0997.60096)] on the essentially different case \(n=1\).
60K05 Renewal theory
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