Let (ak)k∈N be an increasing sequence of positive integers satisfying the Hadamard gap condition a_{k+1}/a_k > q > 1 for all k ∈ N, and let
S_n(ω) = \sum_{k=1}^n cos(2πa_kω), n ∈ N, ω ∈ [0, 1].
Then S_n is called a lacunary trigonometric sum, and can be viewed as a random variable
defined on the probability space Ω = [0, 1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for (S_n)_{n∈N} has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition ak+1/ak → ∞, we prove that the sequence (Sn/n)n∈N does indeed satisfy a large deviation principle with speed n and the same rate function I as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when ak = qk for some q ∈ {2, 3, . . .}, (S_n/n)_{n∈N} satisfies a
large deviation principle (with speed n) and a rate function I_q that is different from I, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of Iq . In addition, we also prove that Iq converges pointwise to I as q → ∞. Furthermore, we construct a random perturbation (a_k)_{k∈N} of the sequence (2^k)_{k∈N} for which a_{k+1}/a_k → 2 as k → ∞, but for which at the same time (S_n/n)n∈N satisfies a large deviation principle with the same rate function I as in the independent case, which is surprisingly different from the rate function I_2 one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence (a_k)_{k∈N}. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem or in the law of the iterated logarithm for lacunary trigonometric sums. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.
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Heavytailed distributions, correlations, kurtosis and Taylor’s Law of fluctuation scaling
Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44 , 2089–2097; p. 2091) speculated that ‘the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·’. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative halfline. We show that the asymptotic scaling of Taylor’s Law (a powerlaw variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavytailed data similarly. We show that the leastsquares estimator of the slope in a linear model with heavytailed predictor and noise unexpectedly converges much faster than when they have finite variances.
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 Award ID(s):
 2015379
 NSFPAR ID:
 10233354
 Date Published:
 Journal Name:
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
 Volume:
 476
 Issue:
 2244
 ISSN:
 13645021
 Page Range / eLocation ID:
 20200610
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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First row indicates header; second row indicates units 1. Spreadsheet: annual precip_drainage Description: Precipitation measured from nearby Kellogg Biological Station (KBS) Long Term Ecological Research (LTER) Weather station, over 20092016 study period. Data shown in Figure 1; original data source for precipitation (https://lter.kbs.msu.edu/datatables/7). Drainage estimated from SALUS crop model. Note that drainage is percolation out of the root zone (0125 cm). Annual precipitation and drainage values shown here are calculated for growing and nongrowing crop periods. Variate Description year year of the observation crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” precip_G precipitation during growing period (milliMeter) precip_NG precipitation during nongrowing period (milliMeter) drainage_G drainage during growing period (milliMeter) drainage_NG drainage during nongrowing period (milliMeter) 2. Spreadsheet: biomass_corn, perennial grasses Description: Maximum aboveground biomass measurements from corn, switchgrass, miscanthus, native grass and restored prairie plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20092015. Data shown in Figure 2. Variate Description year year of the observation date day of the observation (mm/dd/yyyy) crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” replicate each crop has four replicated plots, R1, R2, R3 and R4 station stations (S1, S2 and S3) of samplings within the plot. For more details, refer to link (https://data.sustainability.glbrc.org/protocols/156) species plant species that are rooted within the quadrat during the time of maximum biomass harvest. See protocol for more information, refer to link (http://lter.kbs.msu.edu/datatables/36) For maize biomass, grain and whole biomass reported in the paper (weed biomass or surface litter are excluded). Surface litter biomass not included in any crops; weed biomass not included in switchgrass and miscanthus, but included in grass mixture and prairie. fraction Fraction of biomass biomass_plot biomass per plot on dryweight basis (Grams_Per_SquareMeter) biomass_ha biomass (megaGrams_Per_Hectare) by multiplying column biomass per plot with 0.01 3. Spreadsheet: biomass_poplar Description: Maximum aboveground biomass measurements from poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20092015. Data shown in Figure 2. Note that poplar biomass was estimated from crop growth curves until the poplar was harvested in the winter of 201314. Variate Description year year of the observation method methods of poplar biomass sampling date day of the observation (mm/dd/yyyy) replicate each crop has four replicated plots, R1, R2, R3 and R4 diameter_at_ground poplar diameter (milliMeter) at the ground diameter_at_15cm poplar diameter (milliMeter) at 15 cm height biomass_tree biomass per plot (Grams_Per_Tree) biomass_ha biomass (megaGrams_Per_Hectare) by multiplying biomass per tree with 0.01 4. Spreadsheet: annual N leaching_volwtd conc Description: Annual leaching rate (kiloGrams_N_Per_Hectare) and volumeweighted mean N concentrations (milliGrams_N_Per_Liter) of nitrate (no3) and dissolved organic nitrogen (don) in the leachate samples collected from corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20092016. Data for nitrogen leached and volumewtd mean N concentration shown in Figure 3a and Figure 3b, respectively. Note that ammonium (nh4) concentration were much lower and often undetectable (<0.07 milliGrams_N_Per_Liter). Also note that in 2009 and 2010 cropyears, data from some replicates are missing. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” cropyear year of the observation replicate each crop has four replicated plots, R1, R2, R3 and R4 no3 leached annual leaching rates of nitrate (kiloGrams_N_Per_Hectare) don leached annual leaching rates of don (kiloGrams_N_Per_Hectare) volwtd no3 conc. Volumeweighted mean no3 concentration (milliGrams_N_Per_Liter) volwtd don conc. Volumeweighted mean don concentration (milliGrams_N_Per_Liter) 5. Spreadsheet: summary_N leached Description: Summary of total amount and forms of N leached (kiloGrams_N_Per_Hectare) and the percent of applied N lost to leaching over the seven years for corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20092016. Data for nitrogen amount leached shown in Figure 4a and percent of applied N lost shown in Figure 4b. Note the fraction of unleached N includes in harvest, accumulation in root biomass, soil organic matter or gaseous N emissions were not measured in the study. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” no3 leached annual leaching rates of nitrate (kiloGrams_N_Per_Hectare) don leached annual leaching rates of don (kiloGrams_N_Per_Hectare) N unleached N unleached (kiloGrams_N_Per_Hectare) in other sources are not studied % of N applied N lost to leaching % of N applied N lost to leaching 6. Spreadsheet: annual DOC leachin_volwtd conc Description: Annual leaching rate (kiloGrams_Per_Hectare) and volumeweighted mean N concentrations (milliGrams_Per_Liter) of dissolved organic carbon (DOC) in the leachate samples collected from corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20092016. Data for DOC leached and volumewtd mean DOC concentration shown in Figure 5a and Figure 5b, respectively. Note that in 2009 and 2010 cropyears, water samples were not available for DOC measurements. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” cropyear year of the observation replicate each crop has four replicated plots, R1, R2, R3 and R4 doc leached annual leaching rates of nitrate (kiloGrams_Per_Hectare) volwtd doc conc. volumeweighted mean doc concentration (milliGrams_Per_Liter) 7. Spreadsheet: growing season length Description: Growing season length (days) of corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in the Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20092015. Date shown in Figure S2. Note that growing season is from the date of planting or emergence to the date of harvest (or leaf senescence in case of poplar). Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” year year of the observation growing season length growing season length (days) 8. Spreadsheet: correlation_nh4 VS no3 Description: Correlation of ammonium (nh4+) and nitrate (no3) concentrations (milliGrams_N_Per_Liter) in the leachate samples from corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 20132015. Data shown in Figure S3. Note that nh4+ concentration in the leachates was very low compared to no3 and don concentration and often undetectable in three cropyears (20132015) when measurements are available. 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Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” year year of the observation don don concentration (milliGrams_N_Per_Liter) no3 no3 concentration (milliGrams_N_Per_Liter) doc doc concentration (milliGrams_Per_Liter)more » « less

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