{
  "id": "1502.01426",
  "version": "v1",
  "published": "2015-02-05T03:42:16.000Z",
  "updated": "2015-02-05T03:42:16.000Z",
  "title": "Strong law of large numbers for supercritical superprocesses under second moment condition",
  "authors": [
    "Zhen-Qing Chen",
    "Yan-Xia Ren",
    "Renming Song",
    "Rui Zhang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Suppose that $X=\\{X_t, t\\ge 0\\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$ \\psi(x,\\lambda)=-a(x)\\lambda+b(x)\\lambda^2+\\int_{(0,+\\infty)}(e^{-\\lambda y}-1+\\lambda y)n(x,dy), \\quad x\\in E, \\quad\\lambda> 0, $$ where $a\\in \\mathcal{B}_b(E)$, $b\\in \\mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\\infty)$ satisfying $$ \\sup_{x\\in E}\\int_0^\\infty y^2 n(x,dy)<\\infty. $$ Put $T_tf(x)=\\P_{\\delta_x}\\langle f,X_t\\rangle$. Let $\\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\\phi_0$ and $\\wh{\\phi}_0$ be the eigenfunctions of $L$ and $\\widehat{L}$ (the dural of $L$) respectively associated with $\\lambda_0$. Under some conditions on the spatial motion and the $\\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$ \\lim_{t\\rightarrow\\infty}e^{-\\lambda_0 t}\\langle f, X_t\\rangle = W_\\infty\\int_E\\wh{\\phi}_0(y)f(y)m(dy),\\quad \\P_{\\mu}\\mbox{-a.s.}, $$ for any finite initial measure $\\mu$ on $E$ with compact support, where $W_\\infty$ is the martingale limit defined by $W_\\infty:=\\lim_{t\\to\\infty}e^{-\\lambda_0t}\\langle \\phi_0, X_t\\rangle$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\\mu$ and the function $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-05T03:42:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "second moment condition",
      "strong law",
      "large numbers",
      "supercritical superprocesses",
      "spatial motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150201426C"
    }
  }
}