Monotonicity of min-entropy Let $Z^t = (Y_1,\ldots,Y_t)$ be a sequence of random variables each taking values in $Y$. The random variables are not necessarily i.i.d but we know the joint distributions. i.e for every $z = (z_1,...,z_t)$ we know $P^{Z^t}(z)$

The min-entropy of a random variable $X$ is defined as $-\log_2(\max P(x) )$ for x in the value set of $X$.

Finally, we define a sequence of values $H^\infty(t) =  -\log_2(\max P^{Z^t}(z))$ for $Z^t$ defined as above.

How can we show that $H^\infty(t)$  is monotonically increasing in t? i.e for $t' \geq t$ it is the case that $H^\infty(t') \geq H^\infty(t)$.

What I am trying without success is to show that $\max P^{Z^t}(z^t) \geq \max P^{Z^{t'}}(z^{t'})$ where $t \leq t'$.
Let's make it slightly simpler by proving the case of two variables with no tuple indexing to clutter it up.  Fix two random variables $X$ and $Y$.  Is $H_\infty[X] \leq H_\infty[(X, Y)]$?

For each $x$, we have $$\Pr[X = x] = \sum_y \Pr[X = x, Y = y].$$  Note that since probability masses are always positive, $\max \Pr[\cdots] \leq \sum \Pr[\cdots]$; then the sense will get reversed because $p \mapsto -\log p$ is a decreasing function:

\begin{align}
  H_\infty[X]
  &amp;= -\log \max_x \Pr[X = x] \\
  &amp;= -\log \max_x \sum_y \Pr[X = x, Y = y] \\
  &amp;\leq -\log \max_x \max_y \Pr[X = x, Y = y] \\
  &amp;= -\log \max_{x,y} \Pr[X = x, Y = y] \\
  &amp;= H_\infty[(X, Y)].
\end{align}

Then to prove that $t \mapsto H_\infty[(Z_1, \dots, Z_t)]$ is increasing, take $X = (Z_1, \dots, Z_t)$ and $Y = Z_{t+1}$.