Domains of Normal Attraction for $\mathcal{G}$-Stable Measures on $R^d$

Context: The quality of running mechanics is often characterized by limb pattern symmetry and used to support clinical decisions throughout the rehabilitation of lower-extremity injuries. It is valuable to ensure that gait analyses provide stable measures while not asking an individual to complete an excessive number of running strides. The present study aimed to determine the minimum number of strides required to establish a stable mean symmetry index (SMSI) of discrete-level measures of spatiotemporal parameters, joint kinematics, and joint kinetics. Further, the study aimed to determine if differences occurred between random and consecutive strides for directional and absolute symmetry indices. Design: Descriptive laboratory study. Methods: A sequential average was used to determine how many strides were required to achieve a SMSI within a 60-second trial. Multiple 2-factor repeated-measure analysis of variances were used to determine if differences between bins of strides and symmetry calculations were significantly different. Results: A median SMSI was achieved in 15 strides for all biomechanical variables. There were no significant differences (P > .05) found between consecutive and random bins of 15 strides within a 60-second trial. Although there were significant differences between symmetry calculation values for most variables (P < .05), there appeared to be no systematic difference between the numbers of strides required for stable symmetry for either index. Conclusions: As 15 strides were sufficient to achieve a SMSI during running, a continued emphasis should be placed on the number of strides collected when examining interlimb symmetry.

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A Note on Normal Attraction to a Stable Law

Canadian Mathematical Bulletin ◽

10.4153/cmb-1971-081-5 ◽

1971 ◽

Vol 14 (3) ◽

pp. 451-452

Author(s):

M. V. Menon ◽

V. Seshadri

Keyword(s):

Distribution Function ◽

Sufficient Conditions ◽

Characteristic Exponent ◽

Random Variables ◽

Stable Law ◽

Common Distribution ◽

Normal Attraction ◽

Common Distribution Function ◽

The Common ◽

Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

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