Weighted Random Averages and Recursive Interpolation in Fibonacci Sequences

We investigate the multifractal geometry of irregular sets arising from weighted averages of random variables, where the weights (wn) form a positive sequence with exponential growth. Our analysis applies in particular to sequences generated by linear recurrence relations of Fibonacci type, including higher-order generalizations such as the Tetranacci sequence (Tn). Using a Cantor-type construction built from alternating free and forced blocks, we show that the associated exceptional sets may attain full Hausdorff and packing dimension, independently of the precise form of the recurrence. We further develop a probabilistic interpretation of (Tn) through an appropriate Markov representation that encodes its combinatorial evolution and yields sharp asymptotic behavior. Finally, given n+1 consecutive terms of a Fibonacci-type sequence, one may construct a polynomial Pn(x) of degree at most n via Lagrange interpolation; we show that this polynomial admits an implicit recursive representation consistent with the underlying recurrence.