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The signal of a stochastic gravitational wave (GW) background has been long sought after for several decades and has just come within the observational reach due to the pulsar timing array. While several sources of GW background have been analysed in the literature, the phenomenon of scalar-induced GW turns out to be a simple and inevitable candidate. An important property of scalar-induced GW is that it is highly sensitive to higher-order correlations, such as three-point and four-point correlations of the primordial scalar perturbations, which are known to be difficult to access with probes such as cosmic microwave background (CMB). In this talk, I would like to discuss the unique imprints of scalar bispectrum and trispectrum on the scalar-induced GW. Using representative inflationary models, I shall illustrate that the bispectrum significantly alters the spectral density of GW. Further, a parity-odd component, if present in the scalar trispectrum, shall induce preferential helicity in GW. Thus, the detection or absence of chirality in GW shall directly constrain the strength of parity violation in the trispectrum and hence the underlying model. These inferences from GW will complement our knowledge of primordial non-Gaussianity and parity-violation that are gleaned from CMB and galaxy surveys. Time permitting, I shall also discuss potential tests of statistical homogeneity and isotropy of inflationary perturbations, by examining the cross-correlation between unequal wavenumbers and different polarizations of GW. Lastly, I shall close with the outlook of these results in light of ongoing and upcoming observational missions.