Professor Ken-ichi Yoshihara died in Yokohama, Japan, on October 29, 2016. He was born on September 20, 1932 in Zushi, a small town near Yokohama. He graduated with a BA from Yokohama National University in 1954; and received a master’s degree (1956) and PhD (1965) from the Tokyo University of Education (later Tsukuba University). He held faculty appointments at Yokohama National University in Yokohama (1963–97) and at Soka University in Hachioji (1997–2007). He was awarded the Japanese government’s Medal of Honor with Purple Ribbon in 2011, for his outstanding contributions in mathematics and in education.
Professor Yoshihara was a pioneer of probability theory and statistics in the field of analysis of weakly dependent random variables. He established a breakthrough method to approximate a sequence of dependent random variables satisfying some mixing conditions by a sequence of independent random variables constructed carefully according to the joint distributions of the original sequence. He estimated the error terms very accurately and obtained the almost best possible error evaluation in the approximation (see ). He especially studied dependent random variables with the absolutely regular mixing condition. The absolutely regular mixing condition satisfies the ϕ-mixing condition and the strong mixing condition satisfies the absolute regular mixing condition (see , ,  and ).
In the field of time series analysis, we investigate its property according to the equation of its modeling. For example, in the autoregressive (AR) model, the random variable at time t is defined by sums of time series with some weights defined before t and some noises. Since each random variable at time t can be written by sums of infinitely many random noises, we need very complicated calculations for such research. On the other hand, from the viewpoint of mixing properties, a large family of linear models of time series, like AR models, satisfies some mixing conditions. Therefore Yoshihara’s approximation method for random variables with mixing conditions is very useful in time series analysis. (See .)
Using his approximation method, he extended some limit theorems such as the central limit theorem and law of large numbers for independent random variables to weakly dependent random variables. In particular, he paid attention to symmetric statistics like U-statistics and V-statistics, and showed the asymptotic normality of such statistics for dependent random variables satisfying some mixing condition. (See ,  and .)
He also developed the theory of extreme value statistics for weakly dependent random variables. Recently, the rise in the risk of natural disasters due to climate change has been causing concern. Since extreme value statistics is deeply involved with such risk analysis, it is increasingly important. Originally, extreme value statistics had been investigated for independent random variables. As mentioned previously, lots of time series described by some linear models satisfy mixing conditions. Therefore extreme value statistics can be applied to time series by Yoshihara’s approximation method, and has improved its availability. (See .)
In , Yoshihara collected recent developments of analysis for stochastic sequences of weakly dependent random variables in probability theory and statistics into a significant and substantial 15 volumes.
Finally, I mention Professor Yoshihara’s interest in education, not only for university students but also for high school students. He wrote some textbooks of mathematics for high school students, which were approved by Japan’s Ministry of Education.
Shuya Kanagawa, Tokyo City University
 Yoshihara, K. (1976) Limiting behavior of U-statistics for stationary absolutely regular processes. Z. Wahrsch. Verw. Gebiete 35: 237–252.
 Yoshihara, K. (1978) Limiting behavior of one-sample rank order statistics for absolutely regular processes. Z. Wahrsch. Verw. Gebiete 43: 101–127.
 Yoshihara, K. (1978) Probability inequalities for sums of stationary absolutely regular processes and their applications. Z. Wahrsch. Verw. Gebiete 43: 319–329.
 Kanagawa. S. and Yoshihara, K. (1994) The almost sure invariance principles of degenerate U-statistics of degree two for stationary random variables. Stoch. Proc.Appl. 43: 347–356.
 Yoshihara, K., Weakly Dependent Stoch-astic Sequences and their Applications. Vol.1–15 (1992–2005) Sanseido, Tokyo.
Comments on “Obituary: Ken-ichi Yoshihara, 1932–2016”