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Improvements of Plachky–Steinebach theorem
H. Comman Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile
Abstract:
We show that the conclusion of the Plachky–Steinebach theorem holds true for
intervals of the form
$]\overline{L}_r'(\lambda),y[$, where
$\overline{L}_r'(\lambda)$ is the right derivative (but not necessarily
a derivative) of the generalized
$\mathrm{log}$-moment generating function
$\overline{L}$ with some
$\lambda> 0$ and
$y\in\,]\overline{L}_r'(\lambda),+\infty]$, under only the following two
conditions: (a)
$\overline{L}'_r(\lambda)$ is a limit point of the set
$\{\overline{L}'_r(t)\colon t>\lambda\}$, and (b)
$\overline{L}(t_i)$ has
limit with
$t_i$ belonging to some decreasing sequence converging to
$\sup\{t>\lambda\colon\overline{L}_{|]\lambda,t]}\ \text{is affine}\}$. By
replacing
$\overline{L}_r'(\lambda)$ with
$\overline{L}_r'(\lambda^+)$, the
above result extends verbatim to the case
$\lambda=0$ (replacing (a) by the
right continuity of
$\overline{L}$ at zero when
$\overline{L}_r'(0^+)=-\infty$). No hypothesis is made on
$\overline{L}_{]-\infty,\lambda[}$ (for example,
$\overline{L}_{]-\infty,\lambda[}$ may be the constant
$+\infty$ when
$\lambda=0$);
$\lambda\ge 0$ may be a nondifferentiability point
of
$\overline{L}$ and, moreover, a limit point of nondifferentiability
points of
$\overline{L}$;
$\lambda=0$ may be a left and right discontinuity
point of
$\overline{L}$. The map
$\overline{L}_{|]\lambda,\lambda+\varepsilon[}$ may fail to be strictly
convex for all
$\varepsilon>0$. If we drop the assumption (b), then the same
conclusion holds with upper limits in place of limits. Furthermore, the
foregoing is valid for general nets
$(\mu_\alpha,c_\alpha)$ of Borel
probability measures and powers (in place of the sequence
$(\mu_n,n^{-1})$)
and replacing the intervals
$]\overline{L}_r'(\lambda^+),y[$ by
$]x_\alpha,y_\alpha[$ or
$[x_\alpha,y_\alpha]$, where
$(x_\alpha,y_\alpha)$
is any net such that
$(x_\alpha)$ converges to
$\overline{L}_r'(\lambda^+)$
and $\liminf_\alpha y_\alpha>\overline{L}_r'(\lambda^+)$.
Keywords:
large deviation, $\mathrm{log}$-moment generating function, convexity, differentiability. Received: 30.07.2015
Revised: 11.05.2016
Accepted: 30.06.2016
Language: English
DOI:
10.4213/tvp5156
Fulltext:
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