## Taiwanese Journal of Mathematics

### EQUIVALENCE OF NON-NEGATIVE RANDOM TRANSLATES OF AN IID RANDOM SEQUENCE

#### Abstract

Let $\bf X=\{X_k\}$ be an IID random sequence and $\bf Y=\{Y_k\}$ be an independent random sequence also independent of $\bf X$. Denote by $\mu_{\bf X}$ and $\mu_{\bf X+\bf Y}$ the probability measures on the sequence space induced by $\bf X$ and $\bf X+\bf Y =\{X_k+Y_k\}$, respectively. The problem is to characterize $\mu_{\bf X+\bf Y}\sim\mu_{\bf X}$ in terms of $\mu_{\bf Y}$ in the case where $\bf X$ is non-negative. Sato and Tamashiro  first discussed this problem assuming the existence of ${f_{\bf X}(x)=\frac{d\mu_{X_1}}{dx}(x)}$. They gave several necessary or sufficient conditions for $\mu_{\bf X+\bf Y}\sim\mu_{\bf X}$ under some additional assumptions on $f_{\bf X}$ or on $\bf Y$. The authors precisely improve these results. First they rationalize the assumption of the existence of $f_{\bf X}$. Then they prove that the condition (C.6) is necessary for wider classes of $f_\bf X$ with local regularities. They also prove if the $p$-integral $I_p^0(\bf X)\lt \infty$ and $\bf Y\in \ell_p^+$ a.s., then (C.6) is necessary and sufficient. Furthermore, in the typical case where $\bf X$ is exponentially distributed, they prove an explicit necessary and sufficient condition for $\mu_{\bf X+\bf Y}\sim \mu_\bf X$.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 1 (2009), 269-286.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405283

Digital Object Identifier
doi:10.11650/twjm/1500405283

Mathematical Reviews number (MathSciNet)
MR2489318

Zentralblatt MATH identifier
1178.60029

#### Citation

Honda, Aoi; Okazaki, Yoshiaki; Sato, Hiroshi. EQUIVALENCE OF NON-NEGATIVE RANDOM TRANSLATES OF AN IID RANDOM SEQUENCE. Taiwanese J. Math. 13 (2009), no. 1, 269--286. doi:10.11650/twjm/1500405283. https://projecteuclid.org/euclid.twjm/1500405283