A Dirichlet process assigns exponential mass to a finite-partition simplex neighborhood

If $\alpha (U_0)=0$ and $N=1$, [F1]–[F2] give $F(U_0)=0$ and $F(U_1)=w_1=1$ almost surely, so the event has probability one. In all remaining cases there are at least two positive-parameter cells. By [F1]–[F2], if $\alpha (U_0)>0$ the vector $(X_0,X_1,\dots ,X_N):=(F(U_0),F(U_1),\dots ,F(U_N))$ has the displayed Dirichlet density; if $\alpha (U_0)=0$, set $X_0=0$ and use the $N$-coordinate Dirichlet density. Choose an index $i_{\ast }$ with $w_{i_{\ast }}={\max }_j{w}_j\ge 1\mathrm{/}N$. For every $j\mathrm{\ne }{i}_{\ast }$ restrict $x_j$ to $[\max (w_j-{\eta }^2,{\eta }^2\mathrm{/}2),\min (w_j+{\eta }^2,1)]$, an interval of length at least ${\eta }^2\mathrm{/}2$. If $\alpha (U_0)>0$, also restrict $x_0$ to $[{\eta }^2\mathrm{/}2,{\eta }^2]$ and define $x_{i_{\ast }}=1-x_0-{\sum }_{j\mathrm{\ne }{i}_{\ast }}{x}_j$; if $\alpha (U_0)=0$, define $x_{i_{\ast }}=1-{\sum }_{j\mathrm{\ne }{i}_{\ast }}{x}_j$. The deviations of the coordinates other than $i_{\ast }$ sum to at most $N{\eta }^2$, so $\mathrm{\mid }{x}_{i_{\ast }}-w_{i_{\ast }}\mathrm{\mid }\le N{\eta }^2$. Consequently the total event deviation is at most $2N{\eta }^2+{\eta }^2\le 3\eta$; and $x_{i_{\ast }}\ge 1\mathrm{/}N-N{\eta }^2\ge {\eta }^2\mathrm{/}2$ because $N\eta \le 1\mathrm{/}4$. These choices lie in the simplex, satisfy the event, and their integration region has coordinate volume at least $({\eta }^2\mathrm{/}2{)}^N$ when $\alpha (U_0)>0$ and $({\eta }^2\mathrm{/}2{)}^{N-1}$ when $\alpha (U_0)=0$.

For $0<a_j\le 1$, split the Euler integral for $\mathrm{\Gamma }(1+a_j)$ at $1$. On $(0,1]$, $t^{a_j}\le 1$, and on $[1,\mathrm{\infty })$, $t^{a_j}\le t$, so $\mathrm{\Gamma }(1+a_j)\le 1+{\int }_1^{\mathrm{\infty }}t{e}^{-t}dt=1+2\mathrm{/}e$. The functional equation in [F3] gives $\mathrm{\Gamma }(a_j)\le (1+2\mathrm{/}e)\mathrm{/}{a}_j$. Put $c_{\mathrm{\Gamma }}=(1+2\mathrm{/}e{)}^{-1}$. When $\alpha (U_0)>0$, the normalizing coefficient in [F2] is at least $[\mathrm{\Gamma }(m)\mathrm{/}\mathrm{\Gamma }(\alpha (U_0))]c_{\mathrm{\Gamma }}^N{\prod }_{j=1}^N{a}_j$; when $\alpha (U_0)=0$ and $N\ge 2$, it is at least $\mathrm{\Gamma }(m)c_{\mathrm{\Gamma }}^N{\prod }_{j=1}^N{a}_j$. These Gamma factors are positive by [F3]. On the region from step 1.1, each interval coordinate satisfies $x_j^{a_j-1}\ge 1$. If $\alpha (U_0)>1$, its factor is at least $({\eta }^2\mathrm{/}2{)}^m$; if $0<\alpha (U_0)\le 1$, it is at least one. For $\alpha (U_0)>0$, integrating the density over that region is therefore at least $[\mathrm{\Gamma }(m)\mathrm{/}\mathrm{\Gamma }(\alpha (U_0))](c_{\mathrm{\Gamma }}a\mathrm{/}2{)}^N{\eta }^{(b+2)N}$, multiplied by $({\eta }^2\mathrm{/}2{)}^m$ when $\alpha (U_0)>1$. For $\alpha (U_0)=0$ and $N\ge 2$, the coordinate volume $({\eta }^2\mathrm{/}2{)}^{N-1}\ge ({\eta }^2\mathrm{/}2{)}^N$ gives a lower bound $\mathrm{\Gamma }(m)(c_{\mathrm{\Gamma }}a\mathrm{/}2{)}^N{\eta }^{(b+2)N}$. If $\alpha (U_0)$ ranges in a compact $J\subset (0,m]$, [F4] bounds $\mathrm{\Gamma }(\alpha (U_0))$ above uniformly, so $\mathrm{\Gamma }(m)\mathrm{/}\mathrm{\Gamma }(\alpha (U_0))$ is uniformly bounded below; all other Gamma factors here are fixed and positive. Since $a,b$ are fixed, $(c_{\mathrm{\Gamma }}a\mathrm{/}2{)}^N{\eta }^{(b+2)N}\ge \exp \{-c_{1}N\log (e\mathrm{/}\eta )\}$ for some $c_1>0$. When present, the residual factor $({\eta }^2\mathrm{/}2{)}^m$ is at least $\exp \{-c_{2}N\log (e\mathrm{/}\eta )\}$ because $m$ is fixed and $N\ge 1$. Combining these bounds gives $C\exp \{-cN\log (e\mathrm{/}\eta )\}$ with constants depending only on $a,b,m,\alpha (U_0)$, uniformly when $\alpha (U_0)\in J$. The probability-one case in step 1.1 also satisfies the claimed bound after taking $C\le 1$. The resulting probability is the one in the Statement by [F1].