## 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 [6] 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

#### References

• M. Hino, On equivalence of product measures by random translation, J. Math. Kyoto Univ. 34 (1994), 755-765.
• S. Kakutani, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224.
• K. Kitada and H. Sato, On the absolute continuity of infinite product measure and its convolution, Probab. Theory Related Fields, 81 (1989), 609-627.
• Y. Okazaki, On equivalence of product measure by symmetric random $\ell_4$-translation, J. Funct. Anal., 115 (1993), 100-103.
• Y. Okazaki and H. Sato, Distinguishing a random sequence from a random translate of itself, Ann. Probab., 22 (1994), 1092-1096.
• H. Sato and M. Tamashiro, Absolute continuity of one-sided random translations, Stochastic Process. Appl., 58 (1995), 187-204.
• H. Sato and C. Watari, Some integral inequalities and absolute continuity of a symmetric random translation, J. Funct. Anal., 114 (1993), 257-266.
• L.A. Shepp, Distinguishing a sequence of random variables from a translate of itself, Ann. Math. Statist.36 (1965), 1107-1112.