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Scientific seminar of the Faculty of Physics and Mathematics of Smolensk State University
June 24, 2024 15:30–16:30, Smolensk, SmolSU, Physics&Mathematics Faculty, 2 corp., 230
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On the proximity of distributions of successive sums on convex sets and in
the Prokhorov metric
A. Yu. Zaitsev St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
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Abstract:
Let $X_1$, $X_2$, ... be independent identically distributed random
vectors in a $d$-dimensional Euclidean space with distribution $F$. Then
$S_n=X_1+...+X_n$ has distribution $F^n$ (degrees of measures are
understood in the sense of convolution). Let $R(F,G)=\sup|F(A)-G(A)|$,
where the supremum is taken over all convex subsets of $d$-dimensional
Euclidean space. Then for any nontrivial distributions $F$ there is $c(F)$
depending only on $F$ and such that $R(F^n,F^{n+1})$ does not exceed
$c(F)$ divided by the square root of $n$, for any natural $n$. A
distribution $F$ is considered trivial if it is concentrated on an affine
hyperplane that does not contain the origin. It is clear that for such $F$
we have $R(F^n,F^{n+1})=1$.
A similar result is also obtained for the Prokhorov distance between the
distributions of vectors $S_n$ and $S_{n+1}$ normalized by the square root
of $n$. Moreover, the statement remains true for arbitrary distributions,
including trivial ones.
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