PRECISE RATE IN THE LAW OF ITERATED LOGARITHM FOR ρ-MIXING SEQUENCE

Let {X,X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set S n=n k=1X k,M n=max k≤n｜S k｜,n≥1. Suppose lim n→∞ES2 n/n=∶σ2>0 and ∞n=1ρ 2/d(2n)<∞, where d=2,if -1<b<0 and d>2(b+1),if b≥0. It is proved that,for any b>-1, limε0ε 2(b+1)∞n=1(loglogn)bnlognP{M n≥εσ2nloglogn}= 2(b+1)πГ(b+3/2)∞k=0(-1)k(2k+1) 2b+2,where Г(·) is a Gamma function.

Study on a soil erosion sampling survey in the Pan-Third Pole region based on higher-resolution images

Soil erosion is one of the most severe global environmental problems,and soil erosion surveys are the scientific basis for planning soil conservation and ecological development.To improve soil erosion sampling survey methods and accurately and rapidly estimate the actual rates of soil erosion,a Pan-Third Pole region was taken as an example to study a methodology of soil erosion sampling survey based on high-spatial-resolution remote sensing images.The sampling units were designed using a stratified variable probability systematic sampling method.The spatiotemporal characteristics of soil erosion and conservation were taken into account,and finer-resolution freely available and accessible images in Google Earth were used.Through the visual interpretation of the free high-resolution remote sensing images,detailed information on land use and soil conservation measures was obtained.Then,combined with the regional soil erosion factor data products,such as rainfall-runoff erosivity factor(R),soil erodibility factor(K),and slope length and steepness factor(LS),the soil loss rates of some sampling units were calculated.The results show that,based on these high-resolution remote sensing images,the land use and soil conservation measures of the sampling units can be quickly and accurately extracted.The interpretation accuracy in 4 typical cross sections was more than 80%,and sampling accuracy,described by histogram similarity in 11 large sampling sites,show that the landuse of sampling uints can represent the structural characteristics of regional land use.Based on the interpretation of data from the sample survey and the regional soil erosion factor data products,the calculation of the soil erosion rate can be completed quickly.The calculation results can reflect the actual conditions of soil erosion better than the potential soil erosion rates calculated by using the coarse-resolution remote sensing method.